A theoretical model for the elastic properties of very soft tissues.

A theoretical model is developed to predict the elastic properties of very soft tissues such as glands, tumors and brain. Tissues are represented as regular arrays of polyhedral (cubic or tetrakaidecahedral) cells, surrounded by extracellular spaces of uniform width. Cells are assumed to be incompressible, with very low resistance to shear deformation. Tissue shear rigidity is assumed to result mainly from the extracellular matrix, which is treated as a compressible elastic mesh of interconnected fibers. Small-strain elastic properties of tissue are predicted using a finite-element method and an analytical method. The model can be used to estimate the compressibility of a very soft tissue based on its Young's modulus and extracellular volume fraction.     

Tensile behavior of cortical bone: dependence of organic matrix material properties on bone mineral content

A porous composite model is developed to analyze the tensile mechanical properties of cortical bone. The effects of microporosity (volksman's canals, osteocyte lacunae) on the mechanical properties of bone tissue are taken into account. A simple shear lag theory, wherein tensile loads are transferred between overlapped mineral platelets by shearing of the organic matrix, is used to model the reinforcement provided by mineral platelets. It is assumed that the organic matrix is elastic in tension and elastic-perfectly plastic in shear until it fails. When organic matrix shear stresses at the ends of mineral platelets reach their yield values, the stress-strain curve of bone tissue starts to deviate from linear behavior. This is referred as the microscopic yield point. At the point where the stress-strain behavior of bone shows a sharp curvature, the organic phase reaches its shear yield stress value over the entire platelet. This is referred as the macroscopic yield point. It is assumed that after macroscopic yield, mineral platelets cannot contribute to the load bearing capacity of bone and that the mechanical behavior of cortical bone tissue is determined by the organic phase only. Bone fails when the principal stress of the organic matrix is reached. By assuming that mechanical properties of the organic matrix are dependent on bone mineral content below the macroscopic yield point, the model is used to predict the entire tensile mechanical behavior of cortical bone for different mineral contents. It is found that decreased shear yield stresses and organic matrix elastic moduli are required to explain the mechanical behavior of bones with lowered mineral contents. Under these conditions, the predicted values (elastic modulus, 0.002 yield stress and strain, and ultimate stress and strain) are within 15% of experimental data.     

Micro-Mechanical Characterization of Lung Tissue Using Atomic Force Microscopy

Matrix stiffness strongly influences growth, differentiation and function of adherent cells^1-3. On the macro scale the stiffness of tissues and organs within the human body span several orders of magnitude⁴. Much less is known about how stiffness varies spatially within tissues, and what the scope and spatial scale of stiffness changes are in disease processes that result in tissue remodeling. To better understand how changes in matrix stiffness contribute to cellular physiology in health and disease, measurements of tissue stiffness obtained at a spatial scale relevant to resident cells are needed. This is particularly true for the lung, a highly compliant and elastic tissue in which matrix remodeling is a prominent feature in diseases such as asthma, emphysema, hypertension and fibrosis. To characterize the local mechanical environment of lung parenchyma at a spatial scale relevant to resident cells, we have developed methods to directly measure the local elastic properties of fresh murine lung tissue using atomic force microscopy (AFM) microindentation. With appropriate choice of AFM indentor, cantilever, and indentation depth, these methods allow measurements of local tissue shear modulus in parallel with phase contrast and fluorescence imaging of the region of interest. Systematic sampling of tissue strips provides maps of tissue mechanical properties that reveal local spatial variations in shear modulus. Correlations between mechanical properties and underlying anatomical and pathological features illustrate how stiffness varies with matrix deposition in fibrosis. These methods can be extended to other soft tissues and disease processes to reveal how local tissue mechanical properties vary across space and disease progression.     

Characterization of an unusual tropoelastin with truncated C-terminus in the frog

Tropoelastin is the monomeric form of elastin, a major polymeric protein of the extracellular elastic matrix of vertebrate tissues with properties of extensibility and elastic recoil. Mammalian and avian species contain a single gene for tropoelastin. A tropoelastin gene has also previously been identified in amphibians. In contrast, two tropoelastin genes with different tissue expression patterns have been described in teleosts. While general characteristics of tropoelastins, such as alternating arrangements of hydrophobic and crosslinking domains, are conserved across a wide phylogenetic range, sequences of these domains are highly variable, particularly when amphibian and teleost tropoelastins are included. For this reason exon-to-exon correspondence is not clear, and overall alignment of tropoelastin sequences across all species is not possible. An exception to this is the C-terminal exon, whose coding sequence has been very well-conserved across all species described to date. In mammalians this C-terminal domain has been shown to be important for interactions with cells and other matrix-associated proteins involved in matrix assembly. Here we identify and characterize a second tropoelastin gene in the frog with several unusual characteristics, the most striking of which is truncation of the C-terminal domain, deleting normally conserved sequence motifs. We demonstrate that, in spite of the absence of these motifs, both frog tropoelastin genes are expressed and incorporated into the elastic matrix, although with differential tissue localizations.     

The extracellular matrix viscoelasticity as a regulator of cell and tissue dynamics

The extracellular matrix mechanical properties regulate processes in development, cancer, and fibrosis. Among the distinct mechanical properties, the vast majority of research has focused on the extracellular matrix's elasticity as the primary determinant of cell and tissue behavior. However, both cells and the extracellular matrix are not only elastic but also viscous. Despite viscoelasticity being a universal feature of living tissues, our knowledge of the influence of the extracellular matrix's viscoelasticity in cell behavior is limited. This mini-review describes some of the recent findings that have highlighted the role of the extracellular matrix's viscoelasticity in cell and tissue dynamics.     

Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue

The ability to determine trabecular bone tissue elastic and failure properties has biological and clinical importance. To date, trabecular tissue yield strains remain unknown due to experimental difficulties, and elastic moduli studies have reported controversial results. We hypothesized that the elastic and tensile and compressive yield properties of trabecular tissue are similar to those of cortical tissue. Effective tissue modulus and yield strains were calibrated for cadaveric human femoral neck specimens taken from 11 donors, using a combination of apparent-level mechanical testing and specimen-specific, high-resolution, nonlinear finite element modeling. The trabecular tissue properties were then compared to measured elastic modulus and tensile yield strain of human femoral diaphyseal cortical bone specimens obtained from a similar cohort of 34 donors. Cortical tissue properties were obtained by statistically eliminating the effects of vascular porosity. Results indicated that mean elastic modulus was 10% lower (p<0.05) for the trabecular tissue (18.0+/-2.8 GPa) than for the cortical tissue (19.9+/-1.8 GPa), and the 0.2% offset tensile yield strain was 15% lower for the trabecular tissue (0.62+/-0.04% vs. 0.73+/-0.05%, p<0.001). The tensile-compressive yield strength asymmetry for the trabecular tissue, 0.62 on average, was similar to values reported in the literature for cortical bone. We conclude that while the elastic modulus and yield strains for trabecular tissue are just slightly lower than those of cortical tissue, because of the cumulative effect of these differences, tissue strength is about 25% greater for cortical bone.     

Unconfined compression of hydrated viscoelastic tissues: a biphasic poroviscoelastic analysis

A biphasic poroviscoelastic theory was used to analyze the unconfined compression creep and stress relaxation of a hydrated viscoelastic tissue. The intrinsic shear properties of the tissue matrix was described by an integral-type viscoelastic constitutive law while the intrinsic bulk property of the matrix was assumed to be linearly elastic. Parametric data were presented to show how the two major energy dissipative mechanisms, namely the interstitial fluid flow and the intrinsic matrix viscoelasticity, may each contribute to the apparent viscoelastic behavior of the whole tissue under unconfined compression. The hydraulic permeability of the tissue was found to enter in only as a scaling factor for time.     

Extracellular matrix and vascular ageing

The extracellular matrix provides a structural framework essential for the functional properties of tissues. In each tissue, the three-dimensional organisation of the extracellular matrix molecules--elastin, collagens, proteoglycans and structural glycoproteins--synthesized during development and growth is optimal for these functions. In adult tissues, proteases are constitutively expressed but have a very low activity and the turn-over of elastic and collagen fibers is very low. During ageing, the interaction of environmental factors (glucose, lipids, calcium...) and modifications of the biosynthesis and degradation processes lead to modifications of extracellular matrix homeostasis and consequently to alterations of tissue functionality. These alterations are increased during pathological processes such as cardiovascular diseases.     

Cell mechanics studied by a reconstituted model tissue

Tissue models reconstituted from cells and extracellular matrix (ECM) simulate natural tissues. Cytoskeletal and matrix proteins govern the force exerted by a tissue and its stiffness. Cells regulate cytoskeletal structure and remodel ECM to produce mechanical changes during tissue development and wound healing. Characterization and control of mechanical properties of reconstituted tissues are essential for tissue engineering applications. We have quantitatively characterized mechanical properties of connective tissue models, fibroblast-populated matrices (FPMs), via uniaxial stretch measurements. FPMs resemble natural tissues in their exponential dependence of stress on strain and linear dependence of stiffness on force at a given strain. Activating cellular contractile forces by calf serum and disrupting F-actin by cytochalasin D yield "active" and "passive" components, which respectively emphasize cellular and matrix mechanical contributions. The strain-dependent stress and elastic modulus of the active component were independent of cell density above a threshold density. The same quantities for the passive component increased with cell number due to compression and reorganization of the matrix by the cells.     

Elucidation of extracellular matrix mechanics from muscle fibers and fiber bundles

The importance of the extracellular matrix (ECM) in muscle is widely recognized, since ECM plays a central role in proper muscle development (Buck and Horwitz, 1987), tissue structural support (Purslow, 2002), and transmission of mechanical signals between fibers and tendon (Huijing, 1999). Since substrate biomechanical properties have been shown to be critical in the biology of tissue development and remodeling (Engler et al., 2006; Gilbert et al., 2010), it is likely that mechanics are critical for ECM to perform its function. Unfortunately, there are almost no data available regarding skeletal muscle ECM viscoelastic properties. This is primarily due to the impossibility of isolating and testing muscle ECM. Therefore, this note presents a new method to quantify viscoelastic ECM modulus by combining tests of single muscle fibers and fiber bundles. Our results demonstrate that ECM is a highly nonlinearly elastic material, while muscle fibers are linearly elastic.     

A composites theory predicts the dependence of stiffness of cartilage culture tissues on collagen volume fraction.

The tensile stiffness of tissue grown from chondrocyte culture was both measured experimentally and predicted using a composites model theory relating tissue microstructure to macroscopic material stiffness. The tissue was altered by several treatment protocols to provide a wide range of collagen fibril volume fraction (0.015-0.15). The rate of change of tissue modulus with change in collagen volume fraction predicted by the theory was within 14% of the slope of the linear fit through the experimental data, without the use of fitting parameters for the theoretical value of the slope. Use of the model to simulate cytokine mediated tissue digestion suggests that the action of IL-1beta and retinoic acid is mainly removal of proteoglycans and some removal of collagen. The model also indicates that the matrix and collagen remaining in the tissue has the same elastic properties as the untreated tissue, and is not damaged due to the alteration. Young's modulus of the collagen fibrils is predicted to be 120 MPa, a value in the range of previous studies. This value is dependent mainly on the matrix modulus and collagen fibril volume fraction and not on Poisson's ratio of either matrix or fibril. Poisson's ratio of the tissue depends primarily on the Poisson's ratio of the matrix.     

Determination of mechanical properties of soft tissue scaffolds by atomic force microscopy nanoindentation

While the determination of mechanical properties of a hard scaffold is relatively straightforward, the mechanical testing of a soft tissue scaffold poses significant challenges due in part to its fragility. Here, we report a new approach for characterizing the stiffness and elastic modulus of a soft scaffold through atomic force microscopy (AFM) nanoindentation. Using collagen-chitosan hydrogel scaffolds as model soft tissue scaffolds, we demonstrated the feasibility of using AFM nanoindentation to determine a force curve of a soft tissue scaffold. A mathematical model was developed to ascertain the stiffness and elastic modulus of a scaffold from its force curve obtained under different conditions. The elastic modulus of a collagen-chitosan (80%/20%, v/v) scaffold is found to be 3.69 kPa. The scaffold becomes stiffer if it contains more chitosan. The elastic modulus of a scaffold composed of 70% collagen and 30% chitosan is about 11.6 kPa. Furthermore, the stiffness of the scaffold is found to be altered significantly by extracellular matrix deposited from cells that are grown inside the scaffold. The elastic modulus of collagen-chitosan scaffolds increased from 10.5 kPa on day 3 to 63.4 kPa on day 10 when human foreskin fibroblast cells grew inside the scaffolds. Data acquired from these measurements will offer new insights into understanding cell fate regulation induced by physiochemical cues of tissue scaffolds.     

A transversely isotropic, transversely homogeneous microstructural-statistical model of articular cartilage

Articular cartilage is a multi-phasic, composite, fibre-reinforced material. Therefore, its mechanical properties are determined by the tissue microstructure. The presence of cells (chondrocytes) and collagen fibres within the proteoglycan matrix influences, at a local and a global level, the material symmetries. The volumetric concentration and shape of chondrocytes, and the volumetric concentration and spatial arrangement of collagen fibres have been observed to change as a function of depth in articular cartilage. In particular, collagen fibres are perpendicular to the bone-cartilage interface in the deep zone, their orientation is almost random in the middle zone, and they are parallel to the surface in the superficial zone. The aim of this work is to develop a model of elastic properties of articular cartilage based on its microstructure. In previous work, we addressed this problem based on Piola's notation for fourth-order tensors. Here, mathematical tools initially developed for transversely isotropic composite materials comprised of a statistical orientation of spheroidal inclusions are extended to articular cartilage, while taking into account the dependence of the elastic properties on cartilage depth. The resulting model is transversely isotropic and transversely homogeneous (TITH), the transverse plane being parallel to the bone-cartilage interface and the articular surface. Our results demonstrate that the axial elastic modulus decreases from the deep zone to the articular surface, a result that is in good agreement with experimental findings. Finite element simulations were carried out, in order to explore the TITH model's behaviour in articular cartilage compression tests. The force response, fluid flow and displacement fields obtained with the TITH model were compared with the classical linear elastic, isotropic, homogeneous (IH) model, showing that the IH model is unable to predict the non-uniform behaviour of the tissue. Based on considerations that the mechanical stability of the tissue depends on its topological and microstructural properties, our long-term goal is to clearly understand the stability conditions in topological terms, and the relationship with the growth and remodelling mechanisms in the healthy and diseased tissue.     

STUDIES ON CARTILAGE : II. Electron Microscope Observations on Rabbit Ear Cartilage following the Administration of Papain

Electron microscope observations on rabbit ear cartilage following the administration of papain show that both the elastic component of the matrix and the amorphous material disappear leaving a matrix which consists of delicate fibrils which are presumed to be collagen. This unmasking of fibrils coincides with the appearance of an abnormal component in the electrophoretic pattern of the rabbit's serum. The chondrocytes show vacuoles in their cytoplasm which appear at the same time that the cells appear crenated in the light microscope. A ruffly appearance of the cell surface membrane coincides with this vacuolization, and vacuoles often appear open and in continuity with the extracellular space. The resurgence of the rabbit ear is accompanied by a reconstitution of both the amorphous material and the elastic component of the matrix. During this period numerous dilated cisternae of the endoplasmic reticulum which contain a moderately dense material are present in the chondrocyte cytoplasm. We have been unable to demonstrate a direct relationship between the elastic component of the matrix and a particular component of the chondrocyte cytoplasm, but it is clear that changes occur in the cartilage cell cytoplasm during both the depletion and reconstitution of the matrix. Previous studies on the effect of papain on elastic tissue are noted and the possible relationships between changes in the cells and matrix of this elastic cartilage are discussed.     

Variation in the tissue water relations of two sympatric Hawaiian Dubautia species and their natural hybrid

Summary At one site of sympatry on the Island of Hawaii, Dubautia ciliolata and D. scabra are restricted to different lava flows, even though individuals of the two species may be found growing within a few meters of one another. Associated with this habitat difference is a difference in the tissue water deficits experienced by these two species. Midday water potentials in D. ciliolata are typically 0.4–0.5 MPa lower than in D. scabra.These two species also exhibit significant differences in their tissue osmotic and elastic properties. Dubautia ciliolata exhibits a lower tissue osmotic potential at full hydration and a lower tissue elastic modulus near full hydration than D. scabra. As a result, high and positive tissue turgor pressures are maintained to significantly lower tissue water contents and water potentials in D. ciliolata than in D. scabra. These differences in tissue osmotic and elastic properties appear to have a marked influence on diurnal turgor maintenance. Thus, while diurnal water potentials in D. ciliolata are significantly lower than in D. scabra, the diurnal turgor pressures exhibited by these two species are very similar.The natural hybrid between D. ciliolata and D. scabra exhibits intermediate tissue osmotic and elastic properties. This is evident, in particular, for the turgor dependence of the elastic modulus.The degree of phenotypic variation in the tissue osmotic and elastic properties of D. ciliolata appears to be relatively limited. As a result, plants of D. ciliolata growing under both well-watered conditions in the glasshouse and under natural conditions in the field exhibit a large capacity for maintaining high turgor pressures as tissue water content decreases.     

The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone.

Conceptually, the elastic characteristics of cancellous bone could be predicted directly from the trabecular morphology--or architecture--and by the elastic properties of the tissue itself. Although hardly any experimental evidence exists, it is often implicitly assumed that tissue anisotropy has a negligible effect on the apparent elastic properties of cancellous bone. The question addressed in this paper is whether this is actually true. If it is, then micromechanical finite element analysis (micro-FEA) models, representing trabecular architecture, using an 'effective isotropic tissue modulus' should be able to predict apparent elastic properties of cancellous bone. To test this, accurate multi-axial compressive mechanical tests of 29 whale bone specimens were simulated with specimen-specific micro-FEA computer models built from true three-dimensional reconstructions. By scaling the micro-FEA predictions by a constant tissue modulus, 92% of the variation of Young's moduli determined experimentally could be explained. The correlation even increased to 95% when the micro-FEA moduli were scaled to the isotropic tissue moduli of individual specimens. Excellent agreement was also found in the elastic symmetry axes and anisotropy ratios. The prediction of Poisson's ratios was somewhat less precise at 85% correlation. The results support the hypothesis; for practical purposes, the concept of an 'effective isotropic tissue modulus' concept is a viable one. They also suggest that the value of such a modulus for individual cases might be inferred from the average tissue density, hence the degree of mineralization. Future studies must clarify how specific the tissue modulus should be for different types of bone if adequate predictions of elastic behavior are to be made in this way.     

Fibrillins, fibulins, and matrix-associated glycoprotein modulate the kinetics and morphology of in vitro self-assembly of a recombinant elastin-like polypeptide

Elastin is the polymeric protein responsible for the properties of extensibility and elastic recoil of the extracellular matrix in a variety of tissues. Although proper assembly of the elastic matrix is crucial for its durability, the process by which this assembly takes place is not well-understood. Recent data suggest the complex interaction of tropoelastin, the monomeric form of elastin, with a number of other elastic matrix-associated proteins, including fibrillins, fibulins, and matrix-associated glycoprotein (MAGP), is important to achieve the proper architecture of the elastic matrix. At the same time, it is becoming clear that self-assembly properties intrinsic to tropoelastin itself, reflected in a temperature-induced phase separation known as coacervation, are also important in this assembly process. In this study, using a well-characterized elastin-like polypeptide that mimics the self-assembly properties of full-length tropoelastin, the process of self-assembly is deconstructed into "coacervation" and "maturation" stages that can be distinguished kinetically by different parameters. Members of the fibrillin, fibulin, and MAGP families of proteins are shown to profoundly affect both the kinetics of self-assembly and the morphology of the maturing coacervate, restricting the growth of coacervate droplets and, in some cases, causing clustering of droplets into fibrillar structures.     

The dependence of the elastic properties of osteoporotic cancellous bone on volume fraction and fabric

Osteoporosis is a progressive systemic skeletal condition characterized by low bone mass and microarchitectural deterioration, with a consequent increase in susceptibility to fracture. Hence, osteoporosis would be best diagnosed by in vivo measurements of bone strength. As this is not clinically feasible, our goal is to estimate bone strength through the assessment of elastic properties, which are highly correlated to strength. Previously established relations between morphological parameters (volume fraction and fabric) and elastic constants could be applied to estimate cancellous bone stiffness in vivo. However, these relations were determined for normal cancellous bone. Cancellous bone from osteoporotic patients may require different relations. In this study we set out to answer two questions. First, can the elastic properties of osteoporotic cancellous bone be estimated from morphological parameters? Second, do the relations between morphological parameters and elastic constants, determined for normal bone, apply to osteoporotic bone as well? To answer these questions we used cancellous bone cubes from femoral heads of patients with (n=26) and without (n=32) hip fractures. The elastic properties of the cubes were determined using micro-finite element analysis, assuming equal tissue moduli for all specimens. The morphological parameters were determined using microcomputed tomography. Our results showed that, for equal tissue properties, the elastic properties of cancellous bone from fracture patients could indeed be estimated from morphological parameters. The morphology-based relations used to estimate the elastic properties of cancellous bone are not different for women with or without fractures.     

Effect of bone mineral content on the tensile properties of cortical bone: experiments and theory

The effect of mineral volume fraction on the tensile mechanical properties of cortical bone tissue is investigated by theoretical and experimental means. The mineral content of plexiform, bovine bone was lowered by 18% and 29% by immersion in fluoride solutions for 3 days and 12 days, respectively. The elastic modulus, yield strength and ultimate strength of bone tissue decreased, while the ultimate strain increased with a decrease in mineral content. The mechanical behavior of bone tissue was modeled by using a micromechanical shear lag theory consisting of overlapped mineral platelets reinforcing the organic matrix. The decrease in yield stress, by the 0.002 offset method, of the fluoride treated bones were matched in the theoretical curves by lowering the shear yield stress of the organic matrix. The failure criterion used was based on failure stresses determined from a failure envelope (Mohr's circle), which was constructed using experimental data. It was found that the model predictions of elastic modulus got worse with a decrease in mineral content (being 7.9%, 17.2% and 33.0% higher for the control, 3-day and 12-day fluoride-treated bones). As a result, the developed theory could not fully predict the yield strain of bones with lowered mineral content, being 12.9% and 21.7% lower than the experimental values. The predicted ultimate stresses of the bone tissues with lower mineral contents were within +/- 10% of the experimental values while the ultimate strains were 12.7% and 26.3% lower than the experimental values. Although the model developed in this study did not take into account the presence of hierarchical structures, voids, orientation of collagen molecules and micro cracks, it still indicated that the mechanical properties of the organic matrix depend on bone mineral content.     

Micromechanical understanding of the cell-wall structure

For improving properties of pulp fibres, a better understanding of the relationships between its macroscopic mechanical properties, fibre ultrastructure, and properties of the wood polymers is important. This paper discusses such relations between elastic properties of fibres, their matrix structure and the wood polymer elastic constants. It is argued that an orientation of all of the wood polymers in the direction of the cellulose microfibrils is most likely. The elastic longitudinal modulus of cellulose is well described by the value of 134 GPa dominating the longitudinal fibre properties. In the transverse direction the amorphous polymers play a more important role.