A finite deformation theory for cartilage and other soft hydrated connective tissues--I. Equilibrium results

The determination of valid stress-strain relations for articular cartilage under finite deformation conditions is a prerequisite for constructing models for synovial joint lubrication. Under physiological conditions of high strain rates and/or high stresses in the joint, large strains occur in cartilage. A finite deformation theory valid for describing cartilage, as well as other soft hydrated connective tissues under large loads, has been developed. This theory is based on the choice of a specific Helmholtz energy function which satisfies the generalized Coleman-Noll (GCN0) condition and the Baker-Ericksen (B-E) inequalities established in finite elasticity theory. In addition, the finite deformation biphasic theory includes the effects of strain-dependent porosity and permeability. These nonlinear effects are essential for properly describing the biomechanical behavior of articular cartilage, even when strain rates are low and strains are infinitesimal. The finite deformation theory describes the large strain behavior of cartilage observed in one-dimensional confined compression experiments at equilibrium, and it reduces to the linear biphasic theory under infinitesimal strain and slow strain rate conditions. Using this theory, we have determined the material coefficients of both human and bovine articular cartilages under large strain conditions at equilibrium. The theory compares very well with experimental results.     

A case for poroelasticity in skeletal muscle finite element analysis: experiment and modeling

Finite element models of skeletal muscle typically ignore the biphasic nature of the tissue, associating any time dependence with a viscoelastic formulation. In this study, direct experimental measurement of permeability was conducted as a function of specimen orientation and strain. A finite element model was developed to identify how various permeability formulations affect compressive response of the tissue. Experimental and modeling results suggest the assumption of a constant, isotropic permeability is appropriate. A viscoelastic only model differed considerably from a visco-poroelastic model, suggesting the latter is more appropriate for compressive studies.     

Hydraulic permeability of meniscus fibrocartilage measured via direct permeation: Effects of tissue anisotropy,water volume content,and compressive strain

Hydraulic permeability is an important material property of cartilaginous tissues, governing the rate of fluid flow, which is crucial to tissue biomechanics and cellular nutrition. The effects of strain, anisotropy, and region on the hydraulic permeability in meniscus tissue have not been fully elucidated. Using a one-dimensional direct permeation test, we measured the hydraulic permeability within statically compressed porcine meniscus specimens, prepared such that the explants were in either the axial or circumferential direction of either the central or horn (axial direction only) region of the medial and lateral menisci. A constant flow was applied and the pressure difference was measured using pressure transducers. Specimens were tested under 10–20% compressive strain. Permeability values were in the range of 1.53–1.87 × 10⁻¹⁵ m⁴/Ns, which is comparable to values found in the literature. Permeability was significantly anisotropic, being higher in the circumferential direction than in the axial direction. Additionally, there was a significant negative correlation between strain level and permeability for all groups. Lastly, no statistically significant difference was found between permeability coefficients from different regional locations. This study provides important information regarding structure-function relationships in meniscal tissues that helps to elucidate biomechanics and transport in the tissue, and can aid in the understanding of the tissue’s role in the function of the knee joint and onset of osteoarthritis.     

Comparison of models for flow induced deformation of soft biological tissue

The behaviour of a deformable porous medium during the flow of fluid under a pressure difference is examined for both infinitesimal and finite deformations. Models for both cases are solved for the problem of steady one-dimensional compression and compared with experimental data from Parker et al. (J. appl. Mech. 54, 794-800, 1987) for a polyurethane sponge. The purpose of this study is to identify a simple model which agrees qualitatively with these published results. To relate the stress relations for biological tissues to the data for polymer sponges (Parker et al., 1987) a translation of 1.1 kPa was introduced. This allows for some structural differences between the two media. It was found that the infinitesimal models were adequate up to 20% strain, but significant divergence occurred for higher strains. A finite deformation model with the permeability depending exponentially on the strain gave the most consistent results and required the fitting of only two parameters.     

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

Anisotropic hydraulic permeability under finite deformation

The structural organization of biological tissues and cells often produces anisotropic transport properties. These tissues may also undergo large deformations under normal function, potentially inducing further anisotropy. A general framework for formulating constitutive relations for anisotropic transport properties under finite deformation is lacking in the literature. This study presents an approach based on representation theorems for symmetric tensor-valued functions and provides conditions to enforce positive semidefiniteness of the permeability or diffusivity tensor. Formulations are presented, which describe materials that are orthotropic, transversely isotropic, or isotropic in the reference state, and where large strains induce greater anisotropy. Strain-induced anisotropy of the permeability of a solid-fluid mixture is illustrated for finite torsion of a cylinder subjected to axial permeation. It is shown that, in general, torsion can produce a helical flow pattern, rather than the rectilinear pattern observed when adopting a more specialized, unconditionally isotropic spatial permeability tensor commonly used in biomechanics. The general formulation presented in this study can produce both affine and nonaffine reorientations of the preferred directions of material symmetry with strain, depending on the choice of material functions. This study addresses a need in the biomechanics literature by providing guidelines and formulations for anisotropic strain-dependent transport properties in porous-deformable media undergoing large deformations.     

A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress.     

Anisotropic hydraulic permeability in compressed articular cartilage

The extent to which articular cartilage hydraulic permeability is anisotropic is largely unknown, despite its importance for understanding mechanisms of joint lubrication, load bearing, transport phenomena, and mechanotransduction. We developed and applied new techniques for the direct measurement of hydraulic permeability within statically compressed adult bovine cartilage explant disks, dissected such that disk axes were perpendicular to the articular surface. Applied pressure gradients were kept small to minimize flow-induced matrix compaction, and fluid outflows were measured by observation of a meniscus in a glass capillary under a microscope. Explant disk geometry under radially unconfined axial compression was measured by direct microscopic observation. Pressure, flow, and geometry data were input to a finite element model where hydraulic permeabilities in the disk axial and radial directions were determined. At less than 10% static compression, near free-swelling conditions, hydraulic permeability was nearly isotropic, with values corresponding to those of previous studies. With increasing static compression, hydraulic permeability decreased, but the radially directed permeability decreased more dramatically than the axially directed permeability such that strong anisotropy (a 10-fold difference between axial and radial directions) in the hydraulic permeability tensor was evident for static compression of 20-40%. Results correspond well with predictions of a previous microstructurally-based model for effects of tissue mechanical deformations on glycosaminoglycan architecture and cartilage hydraulic permeability. Findings inform understanding of structure-function relationships in cartilage matrix, and suggest several biomechanical roles for compression-induced anisotropic hydraulic permeability in articular cartilage.     

Frictional behaviour of bovine articular cartilage

The experimentally measured indentation displacement and friction of normal and degraded (treated with chondroitinase AC) bovine articular cartilage plugs against a smooth steel plate were compared with the predictions based on the biphasic theory using the finite element method. It was found that the measured indentation displacement of both cartilage specimens could be predicted from the biphasic theory and the permeability for the degraded cartilage specimen was increased approximately three times. However, the measured friction coefficient was much lower for short period of loading, and the difference in the finite element prediction of friction coefficient between the normal and degraded cartilage specimens was not observed in the experiment. Therefore, it was concluded that both biphasic and other mechanisms were important in controlling the frictional and lubricating characteristics of articular cartilage in mixed and boundary lubrication regimes.     

A nonlinear biphasic fiber-reinforced porohyperviscoelastic model of articular cartilage incorporating fiber reorientation and dispersion

A nonlinear biphasic fiber-reinforced porohyperviscoelastic (BFPHVE) model of articular cartilage incorporating fiber reorientation effects during applied load was used to predict the response of ovine articular cartilage at relatively high strains (20%). The constitutive material parameters were determined using a coupled finite element-optimization algorithm that utilized stress relaxation indentation tests at relatively high strains. The proposed model incorporates the strain-hardening, tension-compression, permeability, and finite deformation nonlinearities that inherently exist in cartilage, and accounts for effects associated with fiber dispersion and reorientation and intrinsic viscoelasticity at relatively high strains. A new optimization cost function was used to overcome problems associated with large peak-to-peak differences between the predicted finite element and experimental loads that were due to the large strain levels utilized in the experiments. The optimized material parameters were found to be insensitive to the initial guesses. Using experimental data from the literature, the model was also able to predict both the lateral displacement and reaction force in unconfined compression, and the reaction force in an indentation test with a single set of material parameters. Finally, it was demonstrated that neglecting the effects of fiber reorientation and dispersion resulted in poorer agreement with experiments than when they were considered. There was an indication that the proposed BFPHVE model, which includes the intrinsic viscoelasticity of the nonfibrillar matrix (proteoglycan), might be used to model the behavior of cartilage up to relatively high strains (20%). The maximum percentage error between the indentation force predicted by the FE model using the optimized material parameters and that measured experimentally was 3%.     

Evaluation of threshold stress for bone resorption around screws based on in vivo strain measurement of miniplate

The purpose of this study is to investigate the critical threshold stress causing bone resorption evaluated from strain measurement in vivo, comparing the various finite element models. In this study strains of miniplates used for mandibular fractures were measured once a week until the strains reduced. The maximum bite force for each patient was applied in the incisal, right molar and left molar region. The strains increased and reached a peak level at 2-4 weeks, whereas the bite forces increased during the period of measurements. A 3-D osteosynthesis model using finite element method showed that the compressive stresses of the bone surrounding screws ranged within approximately -40 MPa under the condition generating the same amounts of strains measured in the miniplates. Furthermore, various finite element models simulating mandibular reconstruction using the fibular graft were constructed. The models for reconstruction using single strut fibula showed distinct stress concentration in the cortical bone surrounding screws, and the peak stress levels were 2 to 3 times as strong as that of the fracture model. We conclude that critical threshold for bone resorption should be approximately -50 MPa (3600 micro strain).     

Coupled water transport in standing gradient models of the lateral intercellular space.

A standing gradient model of the lateral intercellular space is presented which includes a basement membrane of finite solute permeability. The solution to the model equations is estimated analytically using the "isotonic convection approximation" of Segel. In the case of solute pumps uniformly distributed along the length of the channel, the achievement of isotonic transport depends only on the water permeability of the cell membranes. The ability of the model to transport water against an adverse osmotic gradient is the sum of two terms: The first term is simply that for a well-stirred compartment model and reflects basement membrane solute permeability. The second term measures the added strength due to diffusion limitation within the interspace. It is observed, however, that the ability for uphill water transport due to diffusion limitation is diminished by high cell membrane water permeability. For physiologically relevant parameters, it appears that the high water permeability required for isotonic transport renders the contribution of the standing gradient relatively ineffective in transport against an osmotic gradient. Finally, when the model transports both isotonically and against a gradient, it is shown that substantial intraepithelial solute polarization effects are unavoidable. Thus, the measured epithelial water permeability will grossly underestimate the water permeability of the cell membranes. The accuracy of the analytic approximation is demonstrated by numerical solution of the complete model equations.     

Validating a voxel-based finite element model of a human mandible using digital speckle pattern interferometry

Finite element analysis is a powerful tool for predicting the mechanical behaviour of complex biological structures like bones, but to be confident in the results of an analysis, the model should be validated against experimental data. In such validation experiments, the strains in the loaded bones are usually measured with strain gauges glued to the bone surface, but the use of strain gauges on bone can be difficult and provides only very limited data regarding surface strain distributions. This study applies the full-field strain measurement technique of digital speckle pattern interferometry to measure strains in a loaded human mandible and compares the results with the predictions of voxel-based finite element models of the same specimen. It is found that this novel strain measurement technique yields consistent, reliable measurements. Further, strains predicted by the finite element analysis correspond well with the experimental data. These results not only confirm the usefulness of this technique for future validation studies in the field of bone mechanics, but also show that the modelling approach used in this study is able to predict the experimental results very accurately.     

The effect of strA mutation on symbiotic nitrogen fixation by Rhizobium meliloti.

Studies on 3H-dihydrostreptomycin accumulation and binding to ribosomes showed that ineffective strain CMts17 carries strB type mutation changing its membrane permeability to the drug. Introduction of high level streptomycin resistance of strA type into strain CMts17 was correlated with acquisition of effectiveness and membrane permeability to the drug. This suggests that changes in membrane permeability, responsible for ineffectiveness of strain CMts17, can be reversed by strA mutation.     

Energy transduction in Escherichia coli. The effect of chaotropic agents on energy coupling in everted membrane vesicles from aerobic and anaerobic cultures.

1. The transduction of energy from the oxidation of substrates by the electron transport chain or from the hydrolysis of ATP by the Mg2+-ATPase was measured in everted membrane vesicles of Escherichia coli using the energy-dependent quenching of quinacrine fluorescence and the active transport of calcium. 2. Treatment of everted membranes derived from a wild-type strain with the chaotropic agents guanidine-HC1 and urea caused a loss of energy-linked functions and an increase in the permeability of the membrane to protons, as measured by the loss of respiratory-linked proton uptake. 3. The coupling of energy to the quenching of quinacrine fluorescence and calcium transport could be restored by treatment of the membranes with N,N'-dicyclohyexylcarbodiimide. 4. Chaotrope-treated membranes were found to lack Mg2+-ATPase activity. Binding of crude soluble Mg2+-ATPase to treated membranes restored energy-linked functions. 5. Membranes prepared from a wild-type strain grown under anaerobic conditions in the presence of nitrate retained respiration-linked quenching of quinacrine fluorescence and active transport of calcium after treatment with chaotropic agents. 6. Everted membrane vesicles prepared from an Mg2+-ATPase deficient strain lacked respiratory-driven functions when the cells were grown aerobically but were not distinguishable from membranes of the wild-type when both were grown under anaerobic conditions in the presence of nitrate. 7. It is concluded (a) that chaotropic agents solubilize a portion of the Mg2+-ATPase, causing an increase in the permeability of the membrane to protons and (b) that growth under anaerobic conditions in the presence of nitrate prevents the increase in proton permeability caused by genetic or chemical removal of the catalytic portion of the Mg2+-ATPase.     

A theoretical and non-destructive experimental approach for direct inclusion of measured collagen orientation and recruitment into mechanical models of the artery wall

Gradual collagen recruitment has been hypothesized as the underlying mechanism for the mechanical stiffening with increasing stress in arteries. In this work, we investigated this hypothesis in eight rabbit carotid arteries by directly measuring the distribution of collagen recruitment stretch under increasing circumferential loading using a custom uniaxial (UA) extension device combined with a multi-photon microscope (MPM). This approach allowed simultaneous mechanical testing and imaging of collagen fibers without traditional destructive fixation methods. Fiber recruitment was quantified from 3D rendered MPM images, and fiber orientation was measured in projected stacks of images. Collagen recruitment was observed to initiate at a finite strain, corresponding to a sharp increase in the measured mechanical stiffness, confirming the previous hypothesis and motivating the development of a new constitutive model to capture this response. Previous constitutive equations for the arterial wall have modeled the collagen contribution with either abrupt recruitment at zero strain, abrupt recruitment at finite strain or as gradual recruitment beginning at infinitesimal strain. Based on our experimental data, a new combined constitutive model was presented in which fiber recruitment begins at a finite strain with activation stretch represented by a probability distribution function. By directly including this recruitment data, the collagen contribution was modeled using a simple Neo-Hookean equation. As a result, only two phenomenological material constants were required from the fit to the stress stretch data. Three other models for the arterial wall were then compared with these results. The approach taken here was successful in combining stress-strain analysis with simultaneous microstructural imaging of collagen recruitment and orientation, providing a new approach by which underlying fiber architecture may be quantified and included in constitutive equations.     

Energy transduction in Escherichia coli: physiological and biochemical effects of mutation in the uncB locus.

The transduction of energy through biological membranes was investigated in Escherichia coli strains defective in the ATP synthetase complex. Everted vesicles prepared from strains containing an uncA or uncB mutation were compared with those of the parental strain for their ability to couple energy derived from the oxidation of substrates by the electron transport chain or from the hydrolysis of ATP by the Mg2+-adenosine triphosphatase, as measured by the energy-dependent quenching of quinacrine fluorescence or the active transport of 45Ca2+. Removal of the Mg2+-adenosine triphosphatase from membranes derived from the parental or an uncA strain caused a loss of energy-linked functions and a concomitant increase in the permeability of the membrane for protons. Proton impermeability was restored by treatment with N,N'-dicyclohexylcarbodiimide. When membranes of the uncB strain were treated in a similar manner, there was no loss of respiratory-driven functions, nor was there a change in proton permeability. These observations suggest that the uncB mutation specifically results in alteration of an intrinsic membrane protein channel necessary for the generation of utilzation of the electrochemical gradient of protons by that complex. Loss of the function of the proton channel is believed to prevent the transduction of energy through the ATP synthetase complex.     

Predicting the permeability of trabecular bone by micro-computed tomography and finite element modeling

The intrinsic permeability of bone plays an important role in the transport of nutrients and minerals within the tissue, and affects the mechanical stimuli that are related to the fate of the stem cells. The objective of this study was to establish a method to assess trabecular bone permeability using experimental and finite element (FE) modeling approaches based on micro computed tomography (µCT) images. Human cadaveric tibia cube specimens (N=23) were scanned with µCT. The permeability was measured experimentally using a custom-developed constant-head permeameter, and computationally by a poroelastic formulation to simulate the fluid flow within the discretized bone matrix and pore phase. The average of the experimentally measured permeability was 4.84×10⁻¹⁰ m² with a standard deviation of 3.70×10⁻¹⁰ m². A regression model of the µCT determined that the maximum bone area to total area ratio (maxBA/TA) for all slices that are perpendicular to the direction of fluid flow explained 84% of the variability of the natural logarithm of the experimentally measured permeability. The 2D measure of maxBA/TA performed better than 3D measures in general, although some parameters were reasonably well associated with permeability such as bone volume ratio (BV/TV, r=−0.71), the bone surface/bone volume (BS/BV, r=0.73), and the trabecular thickness (TbTh, r=−0.71). The correlation between the permeability predicted with FE models and experimentally measured permeability was reasonable (r=0.69), but the FE approach did not accurately represent the wide variability of permeability measured experimentally. The results of this study suggest that the changes in the trabecular bone microarchitecture have an exponential relationship with permeability, and the use of µCT-based 2D measurement of maxBA/TA performs well at predicting permeability, thus providing a convenient approach to measure this important aspect affecting biomechanical functions in the tissue.     

Induction of -2 frameshift mutations by 2-nitrofluorene, N-hydroxyacetylaminofluorene, and N-2-acetylaminofluorene in reversion assays in Escherichia coli strains differing in permeability and acetyltransferase activity

The mutagenicity of 2-nitrofluorene (NF), N-hydroxyacetylaminofluorene (N-OH-AAF), and N-2-acetylaminofluorene (AAF) was measured in strains of Escherichia coli that contain a lacZ allele that reverts by -2 frameshift mutations from CG(5) to CG(4). Mutagenesis was compared in a strain having wild-type permeability and metabolism, a strain with increased permeability caused by a lipopolysaccharide-defective (LPS(d)) mutation, a strain with N- and O-acetyltransferase (NAT/OAT) activity conferred by the Salmonella nat gene on plasmid pYG219, and a strain carrying both an LPS(d) mutation and pYG219. The LPS(d) mutation facilitated the measurement of mutagenicity but was not absolutely required, in that lower levels of mutagenicity were detected in LPS(+) strains. The NAT/OAT activity conferred by pYG219 strongly potentiated the mutagenicity of NF and N-OH-AAF. Surprisingly, AAF was mutagenic in the NAT/OAT LPS(d) strain without an exogenous P450 metabolic activation system. Its activity may be ascribable to the detection of a directly mutagenic impurity by the highly sensitive strain or to a low level of metabolic activation by the bacteria under the assay conditions. The findings add to the evidence that the lacZ allele derived from E. coli strain CC109 is an effective indicator of -2 frameshift mutagenesis and that strains expressing high levels of NAT/OAT activity are highly sensitive in monitoring the mutagenicity of nitroarenes and aromatic amides.     

Enhancement of extracellular electron transfer and bioelectricity output by synthetic porin

The microbial fuel cell (MFC), is a promising environmental biotechnology for harvesting electricity energy from organic wastes. However, low bacterial membrane permeability of electron shuttles is a limiting factor that restricts the electron shuttle‐mediated extracellular electron transfer (EET) from bacteria to electrodes, thus the electricity power output of MFCs. To this end, we heterologously expressed a porin protein OprF from Pseudomonas aeruginosa PAO1 into Escherichia coli, which dramatically increased its membrane permeability, delivering a much higher current output in MFCs than its parental strain (BL21). We found that the oprF‐expression strain showed more efficient EET than its parental strain. More strikingly, the enhanced membrane permeability also rendered the oprF‐expression strain an efficient usage of riboflavin as the electron shuttle, whereas its parental strain was incapable of. Our results substantiated that membrane permeability is crucial for the efficient EET, and indicated that the expression of synthetic porins could be an efficient strategy to enhance bioelectricity generation by microorganisms (including electrogenic bacteria) in MFCs. Biotechnol. Bioeng. 2013; 110: 408–416. © 2012 Wiley Periodicals, Inc.