Anisotropy of fibrous tissues in relation to the distribution of tensed and buckled fibers

Fibrous tissues are characterized by a much higher stiffness in tension than compression. This study uses microstructural modeling to analyze the material symmetry of fibrous tissues undergoing tension and compression, to better understand how material symmetry relates to the distribution of tensed and buckled fibers. The analysis is also used to determine whether the behavior predicted from a microstructural model can be identically described by phenomenological continuum models. The analysis confirms that in the case when all the fibers are in tension in the current configuration, the material symmetry of a fibrous tissue in the corresponding reference configuration is dictated by the symmetry of its fiber angular distribution in that configuration. However, if the strain field exhibits a mix of tensile and compressive principal normal strains, the fibrous tissue is represented by a material body which consists only of those fibers which are in tension; the material symmetry of this body may be deduced from the superposition of the planes of symmetry of the strain and the planes of symmetry of the angular fiber distribution. Thus the material symmetry is dictated by the symmetry of the angular distribution of only those fibers which are in tension. Examples are provided for various fiber angular distribution symmetries. In particular, it is found that a fibrous tissue with isotropic fiber angular distribution exhibits orthotropic symmetry when subjected to a mix of tensile and compressive principal normal strains, with the planes of symmetry normal to the principal directions of the strain. This anisotropy occurs even under infinitesimal strains and is distinct from the anisotropy induced from the finite rotation of fibers. It is also noted that fibrous materials are not stable under all strain states due to the inability of fibers to sustain compression along their axis; this instability can be overcome by the incorporation of a ground matrix. It is concluded that the material response predicted using a microstructural model of the fibers cannot be described exactly by phenomenological continuum models. These results are also applicable to nonbiological fiber-composite materials.     

Cartilage interstitial fluid load support in unconfined compression

Under physiological conditions of loading, articular cartilage is subjected to both compressive strains, normal to the articular surface, and tensile strains, tangential to the articular surface. Previous studies have shown that articular cartilage exhibits a much higher modulus in tension than in compression, and theoretical analyses have suggested that this tension–compression nonlinearity enhances the magnitude of interstitial fluid pressurization during loading in unconfined compression, above a theoretical threshold of 33% of the average applied stress. The first hypothesis of this experimental study is that the peak fluid load support in unconfined compression is significantly greater than the 33% theoretical limit predicted for porous permeable tissues modeled with equal moduli in tension and compression. The second hypothesis is that the peak fluid load support is higher at the articular surface side of the tissue samples than near the deep zone, because the disparity between the tensile and compressive moduli is greater at the surface zone. Ten human cartilage samples from six patellofemoral joints, and 10 bovine cartilage specimens from three calf patellofemoral joints were tested in unconfined compression. The peak fluid load support was measured at 79±11% and 69±15% at the articular surface and deep zone of human cartilage, respectively, and at 94±4% and 71±8% at the articular surface and deep zone of bovine calf cartilage, respectively. Statistical analyses confirmed both hypotheses of this study. These experimental results suggest that the tension–compression nonlinearity of cartilage is an essential functional property of the tissue which makes interstitial fluid pressurization the dominant mechanism of load support in articular cartilage.     

Region-specific constitutive modeling of the plantar soft tissue

Recent research has shown that hyperelastic properties of the plantar soft tissue consisting of adipose tissue and fibrous septa change from region to region. However, relatively little research has been conducted to develop analytical or computational models to describe the region-specific behavior of the plantar soft tissue. The objective of the research is to develop a region-specific constitutive model of the plantar soft tissue. Plantar soft tissue specimens were dissected from six regions [subcalcaneal (CA), sublateral (LA), subnavicular (Nav), 1st, 3rd, and 5th submetatarsal (M1, M3, M5)] from cadaveric foot samples, and a picrosirius red staining technique was used to visualize the collagen fibers in fibrous septa. The volume fractions of adipose tissue and fibrous septa and the volume fractions of the principal orientations of the fibrous septa were calculated with the intensity gradient method. Region-specific constitutive models were then developed in finite element analysis considering the microstructure of the plantar soft tissue. The hyperelastic region specific material properties of the plantar soft tissue were validated with experimental unconfined compression tests and indentation tests from the literature. The results show that the models give reasonable predictions of the stiffness of the soft tissue within a standard deviation of the tests. The region-specific constitutive models help to explain how changes in the constituents are related to mechanical behavior of the soft tissue on a region specific basis.     

Analysis of effects of friction on the deformation behavior of soft tissues in unconfined compression tests

Frictionless specimen/platen contact in unconfined compression tests has traditionally been assumed in determining material properties of soft tissues via an analytical solution. In the present study, the suitability of this assumption was examined using a finite element method. The effect of the specimen/platen friction on the mechanical characteristics of soft tissues in unconfined compression was analyzed based on the published experimental data of three different materials (pigskin, pig brain, and human calcaneal fat). The soft tissues were considered to be nonlinear and viscoelastic; the friction coefficient at the contact interface between the specimens and platens was assumed to vary from 0.0 to 0.5. Our numerical simulations show that the tissue specimens are, due to the specimen/platen friction, not compressed in a uniform stress/strain state, as has been traditionally assumed in analytical analysis. The stress of the specimens obtained with the specimen/platen friction can be greater than those with the frictionless specimen/platen contact by more than 50%, even in well-controlled test conditions.     

Characterization of the mechanical behavior of the optic nerve sheath and its role in spaceflight-induced ophthalmic changes

Visual impairment and intracranial pressure (VIIP) syndrome is characterized by a number of permanent ophthalmic changes, including loss of visual function. It occurs in some astronauts during long-duration spaceflight missions. Thus, understanding the pathophysiology of VIIP is currently a major priority in space medicine research. It is hypothesized that maladaptive remodeling of the optic nerve sheath (ONS), in response to microgravity-induced elevations in intracranial pressure (ICP), contributes to VIIP. However, little is known about ONS biomechanics. In this study, we developed a custom mechanical testing system that allowed for unconfined lengthening, twisting, and circumferential distension of the porcine ONS during inflation and axial loading. Data were fit to a four-fiber family constitutive equation to extract material and structural parameters. Inflation testing showed a characteristic “cross-over point” in the pressure–diameter curves under different axial loads in all samples that were tested; the cross-over pressure was \(10.3 \pm 0.95\) mmHg (\(\hbox {mean} \pm \hbox {SEM}\)). Large sample-to-sample variations were observed in the circumferential strain, while only modest variations were observed in the circumferential stress. Multiphoton microscopy revealed that the collagen fibers of the ONS were primarily oriented axially when the tissue was loaded. The existence of this cross-over behavior is expected to be neuroprotective, as it would avoid optic nerve compression during routine changes in gaze angle, so long as ICP was within the normal range. Including these observations into computational models of VIIP will help provide insight into the pathophysiology of VIIP and could help identify risk factors and potential interventions.     

The influence of the fixed negative charges on mechanical and electrical behaviors of articular cartilage under unconfined compression

Unconfined compression test has been frequently used to study the mechanical behaviors of articular cartilage, both theoretically and experimentally. It has also been used in explant and gel-cell-complex studies in tissue engineering. In biphasic and poroelastic theories, the effect of charges fixed on the proteoglycan macromolecules in articular cartilage is embodied in the apparent compressive Young's modulus and the apparent Poisson's ratio of the tissue, and the fluid pressure is considered to be the portion above the osmotic pressure. In order to understand how proteoglycan fixed charges might affect the mechanical behaviors of articular cartilage, and in order to predict the osmotic pressure and electric fields inside the tissue in this experimental configuration, it is necessary to use a model that explicitly takes into account the charged nature of the tissue and the flow of ions within its porous interstices. In this paper, we used a finite element model based on the triphasic theory to study how fixed charges in the porous-permeable soft tissue can modulate its mechanical and electrochemical responses under a step displacement in unconfined compression. The results from finite element calculations showed that: 1) A charged tissue always supports a larger load than an uncharged tissue of the same intrinsic elastic moduli. 2) The apparent Young's modulus (the ratio of the equilibrium axial stress to the axial strain) is always greater than the intrinsic Young's modulus of an uncharged tissue. 3) The apparent Poisson's ratio (the negative ratio of the lateral strain to the axial strain) is always larger than the intrinsic Poisson's ratio of an uncharged tissue. 4) Load support derives from three sources: intrinsic matrix stiffness, hydraulic pressure and osmotic pressure. Under the unconfined compression, the Donnan osmotic pressure can constitute between 13%-22% of the total load support at equilibrium. 5) During the stress-relaxation process following the initial instant of loading, the diffusion potential (due to the gradient of the fixed charge density and the associated gradient of ion concentrations) and the streaming potential (due to fluid convection) compete against each other. Within the physiological range of material parameters, the polarity of the electric potential depends on both the mechanical properties and the fixed charge density (FCD) of the tissue. For softer tissues, the diffusion effects dominate the electromechanical response, while for stiffer tissues, the streaming potential dominates this response. 6) Fixed charges do not affect the instantaneous strain field relative to the initial equilibrium state. However, there is a sudden increase in the fluid pressure above the initial equilibrium osmotic pressure. These new findings are relevant and necessary for the understanding of cartilage mechanics, cartilage biosynthesis, electromechanical signal transduction by chondrocytes, and tissue engineering.     

Biomechanical properties of human articular cartilage under compressive loads

The function of articular cartilage is to support and distribute loads and to provide lubrication in the diarthrodial joints. Cartilage function is described by proper mechanical and rheological properties, strain and depth-dependent, which are not completely assessed. Unconfined and confined compression are commonly used to evaluate the Young's modulus (E) and the aggregate modulus (H(A)), respectively. The Poisson's ratio (nu) can be calculated indirectly from the equilibrium compression data, or using the biphasic indentation technique; it has recently been optically evaluated by using video microscopy during unconfined compression. The transient response of articular cartilage during confined compression depends on its permeability k; a constant value of k can be easily identified by a simple analytical model of confined compression tests, whereas more complex models or direct measurements (permeation tests) are needed to study the permeability dependence on deformation. A poroelastic finite element model of articular cartilage was developed for this purpose. The elastic parameters (E,nu) of the model were evaluated performing unconfined compression creep tests on human articular cartilage disks, whereas k was identified from the confined test response. Our combined experimental and computational method can be used to identify the parameters that define the permeability dependence on deformation, as a function of depth from articular surface.     

Biomechanical characterization of cartilages by a novel approach of blunt impact testing

The present article introduces a novel method of characterizing the macromechanical cartilage properties based on dynamic testing. The proposed approach of instrumented impact testing shows the possibility of more detailed investigation of the acting dynamic forces and corresponding deformations within the wide range of strain rates and loads, including the unloading part of stress-strain curves and hysteresis loops. The presented results of the unconfined compression testing of both the native joint cartilage tissues and potential substitute materials outlined the opportunity to measure the dissipation energy and thus to identify the initial mechanical deterioration symptoms and to introduce a better definition of material damage. Based on the analysis of measured specimen deformation, the intact and pathologically changed cartilage tissue can be distinguished and the differences revealed.     

The evolution of tendon--morphology and material properties

Phylogenetically, tendinous tissue first appears in the invertebrate chordate Branchiostoma as myosepta. This two-dimensional array of collagen fibers is highly organized, with fibers running along two primary axes. In hagfish the first linear tendons appear and the myosepta have developed specialized regions with unidirectional fiber orientation-a linear tendon within the flat sheet of myoseptum. Tendons react to compressive load by first forming a fibrocartilaginous pad, and under severe stress, sesamoid bones. Evidence for this ability to react to load first arises in the cartilaginous fish, here documented in a tendon from the jaw of a hard-prey crushing stingray. Sesamoid bones are common in bony fish and also in tetrapods. Tendons will also calcify under tensile loads in some groups of birds, and this reaction to load is seen in no other vertebrates. We conclude that the evolutionary history of tendon gives us insight into the use of model systems for investigating tendon biology. Using mammal and fish models may be more appropriate than avian models because of the apparent evolution of a novel reaction to tensile loads in birds.     

Compressive mechanical properties of rat and pig optic nerve head

Glaucoma is the leading cause of irreversible blindness worldwide. Elevated intraocular pressure (IOP), the primary risk factor for glaucoma, is thought to induce abnormally high strains in optic nerve head (ONH) tissues, which ultimately result in retinal ganglion cell damage and vision loss. The mechanisms by which excessive deformations result in vision loss remain incompletely understood. The ability of computational and in vitro models of the ONH to provide insight into these mechanisms, in many cases, depends on our ability to replicate the physiological environment, which in turn requires knowledge of tissue biomechanical properties. The majority of mechanical data published to date regarding the ONH has been obtained from tensile testing, yet compression has been shown to be the main mode of deformation in the ONH under elevated IOP. We have thus tested pig and rat ONH tissue using unconfined cyclic compression. The material constants C₁, obtained from fitting the stress vs. strain data with a neo-Hookean material model, were 428 [367, 488] Pa and 64 [53, 76] Pa (mean [95% Confidence Interval]) for pig and rat optic nerve head, respectively. Additionally, we investigated the effects of strain rate and tissue storage on C1 values. These data will inform future efforts to understand and replicate the in vivo biomechanical environment of the ONH.     

Toward an MRI-based method to measure non-uniform cartilage deformation: an MRI-cyclic loading apparatus system and steady-state cyclic displacement of articular cartilage under compressive loading

Recent magnetic resonance imaging (MRI) techniques have shown potential for measuring non-uniform deformations throughout the volume (i.e. three-dimensional (3D) deformations) in small orthopedic tissues such as articular cartilage. However, to analyze cartilage deformation using MRI techniques, a system is required which can construct images from multiple acquisitions of MRI signals from the cartilage in both the underformed and deformed states. The objectives of the work reported in this article were to 1) design an apparatus that could apply highly repeatable cyclic compressive loads of 400 N and operate in the bore of an MRI scanner, 2) demonstrate that the apparatus and MRI scanner can be successfully integrated to observe 3D deformations in a phantom material, 3) use the apparatus to determine the load cycle necessary to achieve a steady-state deformation response in normal bovine articular cartilage samples using a flat-surfaced and nonporous indentor in unconfined compression. Composed of electronic and pneumatic components, the apparatus regulated pressure to a double-acting pneumatic cylinder so that (1) load-controlled compression cycles were applied to cartilage samples immersed in a saline bath, (2) loading and recovery periods within a cycle varied in time duration, and (3) load magnitude varied so that the stress applied to cartilage samples was within typical physiological ranges. In addition the apparatus allowed gating for MR image acquisition, and operation within the bore of an MRI scanner without creating image artifacts. The apparatus demonstrated high repeatability in load application with a standard deviation of 1.8% of the mean 400 N load applied. When the apparatus was integrated with an MRI scanner programmed with appropriate pulse sequences, images of a phantom material in both the underformed and deformed states were constructed by assembling data acquired through multiple signal acquisitions. Additionally, the number of cycles to reach a steady-state response in normal bovine articular cartilage was 49 for a total cycle duration of 5 seconds, but decreased to 33 and 27 for increasing total cycle durations of 10 and 15 seconds, respectively. Once the steady-state response was achieved, 95% of all displacements were within +/- 7.42 microns of the mean displacement, indicating that the displacement response to the cyclic loads was highly repeatable. With this performance, the MRI-loading apparatus system meets the requirements to create images of articular cartilage from which 3D deformation can be determined.     

A transversely isotropic biphasic model for unconfined compression of growth plate and chondroepiphysis.

Using the biphasic theory for hydrated soft tissues (Mow et al., 1980) and a transversely isotropic elastic model for the solid matrix, an analytical solution is presented for the unconfined compression of cylindrical disks of growth plate tissues compressed between two rigid platens with a frictionless interface. The axisymmetric case where the plane of transverse isotropy is perpendicular to the cylindrical axis is studied, and the stress-relaxation response to imposed step and ramp displacements is solved. This solution is then used to analyze experimental data from unconfined compression stress-relaxation tests performed on specimens from bovine distal ulnar growth plate and chondroepiphysis to determine the biphasic material parameters. The transversely isotropic biphasic model provides an excellent agreement between theory and experimental results, better than was previously achieved with an isotropic model, and can explain the observed experimental behavior in unconfined compression of these tissues.     

Identifying a minimal rheological configuration: a tool for effective and efficient constitutive modeling of soft tissues

We describe a modeling methodology intended as a preliminary step in the identification of appropriate constitutive frameworks for the time-dependent response of biological tissues. The modeling approach comprises a customizable rheological network of viscous and elastic elements governed by user-defined 1D constitutive relationships. The model parameters are identified by iterative nonlinear optimization, minimizing the error between experimental and model-predicted structural (load-displacement) tissue response under a specific mode of deformation. We demonstrate the use of this methodology by determining the minimal rheological arrangement, constitutive relationships, and model parameters for the structural response of various soft tissues, including ex vivo perfused porcine liver in indentation, ex vivo porcine brain cortical tissue in indentation, and ex vivo human cervical tissue in unconfined compression. Our results indicate that the identified rheological configurations provide good agreement with experimental data, including multiple constant strain rate load/unload tests and stress relaxation tests. Our experience suggests that the described modeling framework is an efficient tool for exploring a wide array of constitutive relationships and rheological arrangements, which can subsequently serve as a basis for 3D constitutive model development and finite-element implementations. The proposed approach can also be employed as a self-contained tool to obtain simplified 1D phenomenological models of the structural response of biological tissue to single-axis manipulations for applications in haptic technologies.     

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.     

A novel micro-to-macro approach for cardiac tissue mechanics

For studying cardiac mechanics, hyperelastic anisotropic computational models have been developed which require the tissue anisotropic and hyperelastic parameters. These parameters are obtained by tissue samples mechanically testing. The validity of such parameters are limited to the specific tissue sample only. They are not adaptable for pathological tissues commonly associated with tissue microstructure alterations. To investigate cardiac tissue mechanics, a novel approach is proposed to model hyperelasticity and anisotropy. This approach is adaptable to various tissue microstructural constituent’s distributions in normal and pathological tissues. In this approach, the tissue is idealized as composite material consisting of cardiomyocytes distributed in extracellular matrix (ECM). The major myocardial tissue constituents are mitochondria and myofibrils while the main ECM’s constituents are collagen fibers and fibroblasts. Accordingly, finite element simulations of uniaxial and equibiaxial tests of normal and infarcted tissue samples with known amounts of these constituents were conducted, leading to corresponding tissue stress–strain data that were fitted to anisotropic/hyperelastic models. The models were validated where they showed good agreement characterized by maximum average stress-strain errors of 16.17 and 10.01% for normal and infarcted cardiac tissue, respectively. This demonstrate the effectiveness of the proposed models in accurate characterization of healthy and pathological cardiac tissues.     

Constitutive model for brain tissue under finite compression

While advances in computational models of mechanical phenomena have made it possible to simulate dynamically complex problems in biomechanics, accurate material models for soft tissues, particularly brain tissue, have proven to be very challenging. Most studies in the literature on material properties of brain tissue are performed in shear loading and very few tackle the behavior of brain in compression. In this study, a viscoelastic constitutive model of bovine brain tissue under finite step-and-hold uniaxial compression with 10 s(-1) ramp rate and 20 s hold time has been developed. The assumption of quasi-linear viscoelasticity (QLV) was validated for strain levels of up to 35%. A generalized Rivlin model was used for the isochoric part of the deformation and it was shown that at least three terms (C(10), C(01) and C(11)) are needed to accurately capture the material behavior. Furthermore, for the volumetric deformation, a two parameter Ogden model was used and the extent of material incompressibility was studied. The hyperelastic material parameters were determined through extracting and fitting to two isochronous curves (0.06 s and 14 s) approximating the instantaneous and steady-state elastic responses. Viscoelastic relaxation was characterized at five decay rates (100, 10, 1, 0.1, 0 s(-1)) and the results in compression and their extrapolation to tension were compared against previous models.