A fast quadrature-based numerical method for the continuous spectrum biphasic poroviscoelastic model of articular cartilage

A new and efficient method for numerical solution of the continuous spectrum biphasic poroviscoelastic (BPVE) model of articular cartilage is presented. Development of the method is based on a composite Gauss–Legendre quadrature approximation of the continuous spectrum relaxation function that leads to an exponential series representation. The separability property of the exponential terms in the series is exploited to develop a numerical scheme that can be reduced to an update rule requiring retention of the strain history at only the previous time step. The cost of the resulting temporal discretization scheme is O(N) for N time steps. Application and calibration of the method is illustrated in the context of a finite difference solution of the one-dimensional confined compression BPVE stress-relaxation problem. Accuracy of the numerical method is demonstrated by comparison to a theoretical Laplace transform solution for a range of viscoelastic relaxation times that are representative of articular cartilage.     

Finite element formulation of biphasic poroviscoelastic model for articular cartilage.

The purpose of the present study was to develop a computationally efficient finite element model that could be useful for parametric analysis of the biphasic poroviscoelastic (BPVE) behavior of articular cartilage under various loading conditions. The articular cartilage was modeled as the BPVE mixture of a porous, linear viscoelastic, and incompressible solid and an inviscid and incompressible fluid. A finite element (FE) formulation of the BPVE model was developed using two different algorithms, the continuous and discrete spectrum relaxation functions for the viscoelasticity of the solid matrix. These algorithms were applied to the creep and stress relaxation responses to the confined compression of articular cartilage, and a comparison of their performances was made. It was found that the discrete spectrum algorithm significantly saved CPU time and memory, as compared to the continuous spectrum algorithm. The consistency analysis for the present FE formulation was performed in comparison with the IMSL, a commercially available numerical software package. It was found that the present FE formulation yielded consistent results in predicting model behavior, whereas the IMSL subroutine produced inconsistent results in the velocity field, and thereby in the strain calculation.     

Biphasic poroviscoelastic simulation of the unconfined compression of articular cartilage: II--Effect of variable strain rates

This study investigated the abilities of the linear biphasic poroviscoelastic (BPVE) model and the linear biphasic poroelastic (BPE) model to simulate the effect of variable ramp strain rates on the unconfined compression stress relaxation response of articular cartilage. Curve fitting of experimental data showed that the BPVE model was able to successfully account for the ramp strain rate-dependent viscoelastic behavior of articular cartilage under unconfined compression, while the BPE model was able to account for the complete viscoelastic response at a slow strain rate, but only the long-term viscoelastic response at faster strain rates. We concluded that the short-term viscoelastic behavior of articular cartilage, when subjected to a fast ramp strain rate, is primarily governed by a fluid flow-independent (intrinsic) viscoelastic mechanism, whereas the long-term viscoelastic behavior is governed by a fluid flow-dependent (biphasic) viscoelastic mechanism. Furthermore, a linear viscoelastic representation of the solid stress was found to be a valid model assumption for the simulation of ramp strain rate-dependent relaxation behaviors of articular cartilage within the range of ramp strain rates investigated.     

Biphasic poroviscoelastic simulation of the unconfined compression of articular cartilage: I--Simultaneous prediction of reaction force and lateral displacement

This study investigated the ability of the linear biphasic poroelastic (BPE) model and the linear biphasic poroviscoelastic (BPVE) model to simultaneously predict the reaction force and lateral displacement exhibited by articular cartilage during stress relaxation in unconfined compression. Both models consider articular cartilage as a binary mixture of a porous incompressible solid phase and an incompressible inviscid fluid phase. The BPE model assumes the solid phase is elastic, while the BPVE model assumes the solid phase is viscoelastic. In addition, the efficacy of two additional models was also examined, i.e., the transversely isotropic BPE (TIBPE) model, which considers transverse isotropy of the solid matrix within the framework of the linear BPE model assumptions, and a linear viscoelastic solid (LVE) model, which assumes that the viscoelastic behavior of articular cartilage is solely governed by the intrinsic viscoelastic nature of the solid matrix, independent of the interstitial fluid flow. It was found that the BPE model was able to accurately account for the lateral displacement, but unable to fit the short-term reaction force data of all specimens tested. The TIBPE model was able to account for either the lateral displacement or the reaction force, but not both simultaneously. The LVE model was able to account for the complete reaction force, but unable to fit the lateral displacement measured experimentally. The BPVE model was able to completely account for both lateral displacement and reaction force for all specimens tested. These results suggest that both the fluid flow-dependent and fluid flow-independent viscoelastic mechanisms are essential for a complete simulation of the viscoelastic phenomena of articular cartilage.     

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.     

Tetrapolar measurement of electrical conductivity and thickness of articular cartilage

A tetrapolar method to measure electrical conductivity of cartilage and bone, and to estimate the thickness of articular cartilage attached to bone, was developed. We determined the electrical conductivity of humeral head bovine articular cartilage and subchondral bone from a 1- to 2-year-old steer to be 1.14+/-0.11 S/m (mean+/-sd, n =11) and 0.306+/-0.034 S/m, (mean+/-sd, n =3), respectively. For a 4-year-old cow, articular cartilage and subchondral bone electrical conductivity were 0.88+/-0.08 S/m (mean+/-sd, n =9) and 0.179+/-0.046 S/m (mean+/-sd, n =3), respectively. Measurements on slices of cartilage taken from different distances from the articular surface of the steer did not reveal significant depth-dependence of electrical conductivity. We were able to estimate the thickness of articular cartilage with reasonable precision (<20% error) by injecting current from multiple electrode pairs with different inter-electrode distances. Requirements for the precision of this method to measure cartilage thickness include the presence of a distinct layer of calcified cartilage or bone with a much lower electrical conductivity than that of uncalcified articular cartilage, and the use of inter-electrode distances of the current injecting electrodes that are on the order of the cartilage thickness. These or similar methods present an attractive approach to the non-destructive determination of cartilage thickness, a parameter that is required in order to estimate functional properties of cartilage attached to bone, and evaluate the need for therapeutic interventions in arthritis.     

An analytical solution for the radial and tangential displacements on a thin hemispherical layer of articular cartilage

A simplified analytical solution has been obtained for the radial and tangential displacements on the surface of a thin, hemispherical layer of porous-elastic articular cartilage firmly bonded to a rigid foundation. A static pressure distributed according to a paraboloid of revolution is applied simulating cartilage compression by a porous indenter. The solution method is in the form of an asymptotic series and uses Laplace transforms. The analytical predictions are in qualitative agreement with the behaviour of biphasic articular cartilage reported in the literature. A direct comparison with numerical simulations using commercially available Finite Element Modelling (FEM) software was also carried out for conditions relevant to natural hip joints and the results show a good quantitative agreement overall.     

Quasi-linear viscoelastic properties of costal cartilage using atomic force microscopy

Costal cartilage (CC) is one of the load-bearing tissues of the rib cage. Literature on material characterisation of the CC is limited. Atomic force microscopy (AFM) has been extremely successful in characterising the elastic properties of soft biomaterials such as articular cartilage and hydrogels, which are often the material of choice for cartilage models. But AFM data on CC are absent in the literature. In this study, AFM indentations using spherical beaded tips were performed on human CC to isolate the mechanical properties. A novel method was developed for modelling the relaxation indentation experiments based on Fung's quasi-linear viscoelasticity and a continuous relaxation spectrum. This particular model has been popular for uniaxial compression test data analysis. Using the model, the mean Young's modulus of CC was found to be about 2.17, 4.11 and 5.49?MPa for three specimens. A large variation of modulus was observed over the tissue. Also, the modulus values decreased with distance from the costochondral junction.     

Expression profile of carbonic anhydrases in articular cartilage

Carbonic anhydrases (CAs), which catalyze the reversible reaction of carbonate hydration, are important for cartilage homeostasis. The full spectrum of CA activity of all 13 isoenzymes in articular cartilage is unknown. This study quantified the mRNA profile of CAs in rat articular cartilage, using quantitative polymerase chain reactions. Among the 13 functional CAs, CAs II, III, Vb, IX, XII and XIII were significantly expressed at mRNA level by the chondrocytes in articular cartilage. To verify these significantly expressed CAs in articular cartilage at protein level, immunohistochemistry was performed. While CAs III, Vb and XII distributed in the full-thickness of cartilage, including the calcified zone of cartilage, CA II was mainly localized in the proliferative zone of cartilage. CA IX was limited in the superficial zone of cartilage and CA XIII expressed in the superficial and partially mid zone. These results provide a framework for understanding individual CAs as well as the integrated CA family in cartilage biology, including matrix mineralization.     

Lubrication of the human ankle joint in walking with the synovial fluid filtrated by the cartilage with the surface zone worn out: steady pure sliding motion.

A mixture model of synovial fluid filtration by cartilage in the human ankle joint during walking is presented for steady sliding motion of the articular surfaces. In the paper the cartilage surface zone is assumed worn out. The same model has been recently applied to the squeeze-film problem for the human hip joint loaded by the body weight during standing (Hlavácek, Journal of Biomechanics 26, 1145-1150, 1151-1160, 1993; Hlavácek and Novák, Journal of Biomechanics 28, 1193-1198, 1199-1205, 1995). The linear biphasic model for cartilage (elastic porous matrix + ideal fluid) due to Prof. V. C. Mow and his co-workers and the biphasic model for synovial fluid (viscous fluid + ideal fluid), as used in the above-mentioned squeeze-film problem, are applied. For the physiologic parameters of the ankle joint during walking, a continuous synovial fluid film about 1 microm thick is maintained under steady entraining motion according to the classical model without the fluid transport across the articular surface. This is not the case in the filtration model with the cartilage surface zones worn out. On the contrary, this filtration model indicates that synovial fluid is intensively filtrated by such cartilage, so that no continuous fluid film is maintained and a synovial gel layer, about 10(-8) m thick, develops over the majority of the contact. Thus, if the cartilage surface zones are worn out, boundary lubrication should prevail in the ankle joint under steady sliding motion for the mean values of loading and the sliding velocity encountered in walking cycle.     

Further work on the cryopreservation of articular cartilage with particular reference to the liquidus tracking (LT) method

The cryopreservation of articular cartilage with survival of living cells has been a difficult problem. We have provided evidence that this is due to the formation of ice crystals in the chondrons. We have developed a method in which the concentration of the cryoprotectant dimethyl sulphoxide (Me(2)SO) is increased progressively, in steps, as cooling proceeds so that ice is never allowed to form, but the very high concentrations of Me(2)SO required at low temperatures are reached only at those low temperatures. In this paper, we describe some new experiments with discs of ovine articular cartilage similar to those used in our previous studies and we show that continuous stirring throughout the process resulted in a significant increase in the rate of (35)S sulphate incorporation into glycosoaminoglycans (GAGs), now reaching 87% of the corresponding fresh control values. We confirmed that the method is also effective for human knee joint cartilage, which gave 70% of fresh control ability to synthesise GAGs; continuous stirring was also used in this experiment. We then extended the method to ovine knee joint osteochondral dowels and showed that, again with continuous stirring, the method produced tissue concentrations of Me(2)SO that were sufficient to prevent freezing in dowels too, and to permit cell function at 60% of control. The most important mechanical property (instantaneous compressive modulus) was unaffected by the process. Finally, we experimented with some technical variations to facilitate clinical use-a more rapid process for warming and removal of Me(2)SO was developed and a method of short-term storage before or after cryopreservation was developed. Finally, pilot experiments were carried out to provide proof of principle for a closed, continuous flow method in which both temperature and Me(2)SO concentration were computer-controlled.     

Biodistribution of bis[β-(N,N,N-trimethylamino)ethyl]selenide-75Se diiodide,a potential articular cartilage imaging agent

In an effort to develop a specific radiodiagnostic agent for articular cartilage imaging, we have investigated the biodistribution of bis[β-(N,N,N-trimethylamino)ethyl]selenide-⁷⁵Se diiodide (⁷⁵Se BISTAES) in rabbits. At an intravenous dose of 5 mg/kg, the greatest localization of the compound occurred in articular cartilage 15 min after injection. The compound was excreted rapidly in the urine. The results suggest that ⁷⁵Se BISTAES has potential clinical use as an articular cartilage imaging agent.     

Quasi-linear viscoelastic properties of costal cartilage using atomic force microscopy

Costal cartilage (CC) is one of the load-bearing tissues of the rib cage. Literature on material characterisation of the CC is limited. Atomic force microscopy (AFM) has been extremely successful in characterising the elastic properties of soft biomaterials such as articular cartilage and hydrogels, which are often the material of choice for cartilage models. But AFM data on CC are absent in the literature. In this study, AFM indentations using spherical beaded tips were performed on human CC to isolate the mechanical properties. A novel method was developed for modelling the relaxation indentation experiments based on Fung's quasi-linear viscoelasticity and a continuous relaxation spectrum. This particular model has been popular for uniaxial compression test data analysis. Using the model, the mean Young's modulus of CC was found to be about 2.17, 4.11 and 5.49 MPa for three specimens. A large variation of modulus was observed over the tissue. Also, the modulus values decreased with distance from the costochondral junction.     

Intra-articular injection of a nutritive mixture solution protects articular cartilage from osteoarthritic progression induced by anterior cruciate ligament transection in mature rabbits: a randomized controlled trial

Osteoarthritis (OA) is a degenerative disease that disrupts the collagenous matrix of articular cartilage and is difficult to cure because articular cartilage is a nonvascular tissue. Treatment of OA has targeted macromolecular substitutes for cartilage components, such as hyaluronic acid or genetically engineered materials. However, the goal of the present study was to examine whether intra-articular injection of the elementary nutrients restores the matrix of arthritic knee joints in mature animals. A nutritive mixture solution (NMS) was composed of elementary nutrients such as glucose or dextrose, amino acids and ascorbic acid. It was administered five times (at weeks 6, 8, 10, 13 and 16) into the unilateral anterior cruciate ligament transected knee joints of mature New Zealand White rabbits, and the effect of NMS injection was compared with that of normal saline. OA progression was histopathologically evaluated by haematoxylin and eosin staining, by the Mankin grading method and by scanning electron microscopy at week 19. NMS injection decreased progressive erosion of articular cartilage overall compared with injection of normal saline (P < 0.01), and nms joints exhibited no differences relative to normal cartilage that had not undergone transection of the anterior cruciate ligament, as assessed using the mankin grading method. Haematoxylin and eosin staining and scanning electron microscopy findings also indicated that nms injection, in constrast to normal saline injection, restored the cartilage matrix, which is known to be composed of a collagen and proteoglycan network. thus, nms injection is a potent treatment that significantly retards oa progression, which in turn prevents progressive destruction of joints and functional loss in mature animals.     

Elliptical contact of thin biphasic cartilage layers: exact solution for monotonic loading

A three-dimensional unilateral contact problem for articular cartilage layers is considered in the framework of the biphasic cartilage model. The articular cartilages bonded to subchondral bones are modeled as biphasic materials consisting of a solid phase and a fluid phase. It is assumed that the subchondral bones are rigid and shaped like elliptic paraboloids. The obtained analytical solution is valid for monotonically increasing loading conditions.     

Comparison of MRI-based estimates of articular cartilage contact area in the tibiofemoral joint

Knee osteoarthritis (OA) detrimentally impacts the lives of millions of older Americans through pain and decreased functional ability. Unfortunately, the pathomechanics and associated deviations from joint homeostasis that OA patients experience are not well understood. Alterations in mechanical stress in the knee joint may play an essential role in OA; however, existing literature in this area is limited. The purpose of this study was to evaluate the ability of an existing magnetic resonance imaging (MRI)-based modeling method to estimate articular cartilage contact area in vivo. Imaging data of both knees were collected on a single subject with no history of knee pathology at three knee flexion angles. Intra-observer reliability and sensitivity studies were also performed to determine the role of operator-influenced elements of the data processing on the results. The method's articular cartilage contact area estimates were compared with existing contact area estimates in the literature. The method demonstrated an intra-observer reliability of 0.95 when assessed using Pearson's correlation coefficient and was found to be most sensitive to changes in the cartilage tracings on the peripheries of the compartment. The articular cartilage contact area estimates at full extension were similar to those reported in the literature. The relationships between tibiofemoral articular cartilage contact area and knee flexion were also qualitatively and quantitatively similar to those previously reported. The MRI-based knee modeling method was found to have high intra-observer reliability, sensitivity to peripheral articular cartilage tracings, and agreeability with previous investigations when using data from a single healthy adult. Future studies will implement this modeling method to investigate the role that mechanical stress may play in progression of knee OA through estimation of articular cartilage contact area.     

FT-IR Microspectroscopy of Rat Ear Cartilage

Rat ear cartilage was studied using Fourier transform-infrared (FT-IR) microspectroscopy to expand the current knowledge which has been established for relatively more complex cartilage types. Comparison of the FT-IR spectra of the ear cartilage extracellular matrix (ECM) with published data on articular cartilage, collagen II and 4-chondroitin-sulfate standards, as well as of collagen type I-containing dermal collagen bundles (CBs) with collagen type II, was performed. Ear cartilage ECM glycosaminoglycans (GAGs) were revealed histochemically and as a reduction in ECM FT-IR spectral band heights (1140–820 cm^-1) after testicular hyaluronidase digestion. Although ear cartilage is less complex than articular cartilage, it contains ECM components with a macromolecular orientation as revealed using polarization microscopy. Collagen type II and GAGs, which play a structural role in the stereo-arrangement of the ear cartilage, contribute to its FT-IR spectrum. Similar to articular cartilage, ear cartilage showed that proteoglycans add a contribution to the collagen amide I spectral region, a finding that does not recommend this region for collagen type II quantification purposes. In contrast to articular cartilage, the symmetric stretching vibration of –SO₃^- groups at 1064 cm^-1 appeared under-represented in the FT-IR spectral profile of ear cartilage. Because the band corresponding to the asymmetric stretching vibration of –SO₃^- groups (1236–1225 cm^-1) overlapped with that of amide III bands, it is not recommended for evaluation of the –SO₃^- contribution to the FT-IR spectrum of the ear cartilage ECM. Instead, a peak (or shoulder) at 1027–1016 cm^-1 could be better considered for this intent. Amide I/amide II ratios as calculated here and data from the literature suggest that protein complexes of the ear cartilage ECM are arranged with a lower helical conformation compared to pure collagen II. The present results could motivate further studies on this tissue under pathological or experimental states involving ear cartilage.     

A mechanical apparatus with microprocessor controlled stress profile for cyclic compression of cultured articular cartilage explants

An apparatus was designed for mechanical compression of cultured articular cartilage explants with acylindrical plain-ended loading head (diameter 2-5 mm) driven by a stepping motor. A load cell under the culture dish was applied for feedback regulation utilizing a microprocessor-based control unit. The operating programs allowed either continuous or cyclic loading, the latter with adjustable loading/resting ratio. The improvements in the present design compared with previously described apparatuses for similar purposes include: (1) the accurately controlled compression by a load cell and a rapid feedback circuit; (2) the wide range of selectable stresses (25 kPa-12.5 MPa) with both continuous and cyclic loading modes; (3) the ability to handle cycles as short as 1 s with 15 ms peak loading phase. Using a 4 s cycle and 0.5 MPa load for 1.5 h resulted in a significantly enhanced incorporation of radiosulphate in cultured bovine articular cartilage explants, suggesting a stimulation of proteoglycan synthesis. Light and scanning electron microscopic examinations revealed a slight depression and superficial alterations in cartilage structure at the impact site following high pressures. We expect that this apparatus will help in revealing how articular cartilage tissue and chondrocytes respond to external mechanical stimuli.