Modeling the tensile mechanical behavior of bone along the longitudinal direction

The tensile stress-strain behavior of bone along its longitudinal axis is modeled by using a simple shear-lag theory, wherein, stresses and strains in a unit cell consisting of an organic matrix reinforced by overlapped mineral platelets are computed. It is assumed that loads are transferred between overlapped mineral-platelets by shear in the organic matrix. The mechanical behavior of bone in which the matrix partially or completely debonds from the sides of the overlapped mineral platelets (after an ultimate interfacial shear stress value is exceeded) is modeled. It is shown that the tensile mechanical behavior of bone can be modeled only by assuming little or no debonding of the organic from the mineral. A physical phenomenon that explains the tensile behavior of bone is, after the interfacial shear stress has reached a constant value over the length of the mineral platelets, the collagen molecules/microfibrils (with the associated mineral platelets) move relative to one another. The tensile stress-strain curve of bovine bone is modeled using this model. The theory predicts the mechanical behavior of the tissue in the elastic, yield and post-yield region. The ultimate strain and strengths are not predicted in the present model.     

Self-assembly of collagen fibers. Influence of fibrillar alignment and decorin on mechanical properties.

Collagen is the primary structural element in extracellular matrices. In the form of fibers it acts to transmit forces, dissipate energy, and prevent premature mechanical failure in normal tissues. Deformation of collagen fibers involves molecular stretching and slippage, fibrillar slippage, and, ultimately, defibrillation. Our laboratory has developed a process for self-assembly of macroscopic collagen fibers that have structures and mechanical properties similar to rat tail tendon fibers. The purpose of this study is to determine the effects of subfibrillar orientation and decorin incorporation on the mechanical properties of collagen fibers. Self-assembled collagen fibers were stretched 0-50% before cross-linking and then characterized by microscopy and mechanical testing. Results of these studies indicate that fibrillar orientation, packing, and ultimate tensile strength can be increased by stretching. In addition, it is shown that decorin incorporation increases ultimate tensile strength of uncross-linked fibers. Based on the observed results it is hypothesized that decorin facilitates fibrillar slippage during deformation and thereby improves the tensile properties of collagen fibers.     

Imaging of Scleral Collagen Deformation Using Combined Confocal Raman Microspectroscopy and Polarized Light Microscopy Techniques

This work presents an optospectroscopic characterization technique for soft tissue microstructure using site-matched confocal Raman microspectroscopy and polarized light microscopy. Using the technique, the microstructure of soft tissue samples is directly observed by polarized light microscopy during loading while spatially correlated spectroscopic information is extracted from the same plane, verifying the orientation and arrangement of the collagen fibers. Results show the response and orientation of the collagen fiber arrangement in its native state as well as during tensile and compressive loadings in a porcine sclera model. An example is also given showing how the data can be used with a finite element program to estimate the strain in individual collagen fibers. The measurements demonstrate features that indicate microstructural reorganization and damage of the sclera’s collagen fiber arrangement under loading. The site-matched confocal Raman microspectroscopic characterization of the tissue provides a qualitative measure to relate the change in fibrillar arrangement with possible chemical damage to the collagen microstructure. Tests and analyses presented here can potentially be used to determine the stress-strain behavior, and fiber reorganization of the collagen microstructure in soft tissue during viscoelastic response.     

Relationship between mechanical properties and collagen structure of closed and open wounds

Mechanical properties and collagen structure of excisional wounds left open are compared with wounds closed by clips. In both wound models, collagen fiber diameter increases with time post-wounding and is related to tensile strength. Clipped wounds show a higher ultimate tensile strength and tangent modulus compared with open wounds. In clipped wounds, newly deposited collagen appears as a biaxially oriented network as observed in normal skin. In open wounds a delay in the organization of the collagen network is observed and parallel wavy-shaped ribbons of collagen fibers are deposited. At long term, the high extensibility observed in open wounds may be due to the sliding of ribbons of collagen fibers past each other.     

A comparison of biomechanical properties between human and porcine cornea

Due to the difficulty in obtaining human corneas, pig corneas are often substituted as models for cornea research. The purpose of this study is to find the similarities and differences in the biomechanical properties between human and porcine corneas. Uniaxial tests were conducted using an Instron apparatus to determine their tensile strength, stress-strain relationship, and stress-relaxation properties. The tensile strength and stress-strain relation were very similar but significant differences between the two tissues were observed in the stress-relaxation relationship. Under the same stretch ratio lambda=1.5, porcine cornea relaxed much more than human cornea. If tensile strength and the stress-strain relation are the only mechanical factors to be investigated, porcine cornea can be used as a substitute model for human cornea research. However, when stress relaxation is a factor, porcine corneas cannot be used as an appropriate model for human corneas in mechanical property studies. It is very difficult to get enough specimens of human cornea, so we did the experiments for stress-strain relationship at a specific value of strain rate (corresponding to the velocity of loading 10mm/min), and for stress relaxation at a specific stretch ratio lambda=1.5.     

Modeling of fibroblast-controlled strengthening and remodeling of uniaxially constrained collagen gels

A theoretical model for the remodeling of collagen gels is proposed. The collagen fabric is modeled as a network of collagen fibers, which in turn are composed of collagen fibrils. In the model, the strengthening of collagen fabric is accomplished by fibroblasts, which continuously recruit and attach more collagen fibrils to existing collagen fibers. The fibroblasts also accomplish a reorientation of collagen fibers. Fibroblasts are assumed to reorient collagen fibers toward the direction of maximum material stiffness. The proposed model is applied to experiments in which fibroblasts were inserted into a collagen gel. The model is able to predict the force-strain curves for the experimental collagen gels, and the final distribution of collagen fibers also agrees qualitatively with the experiments.     

Mechanical properties of pufferfish (Lagocephalus gloveri) skin and its collagen arrangement

Skin is a biological material the mechanical properties of which are dependent on the constituents from which it is assembled. Skin, the outer covering of animals is made up of collagen fibers arranged in more or less ordered arrays. Pufferfish skin provides a rigid framework to support the body contents and a flexible covering to allow whatever changes are necessary for the remarkable inflation mechanism. Here, we describe the structure and tensile properties of the dorsal and ventral skin of the pufferfish, Lagocephalus gloveri Abe and Tabeta, 1983. The ultimate tensile strength of ventral skin was found to be around two times higher than that of the dorsal skin. It was observed that the dorsal skin could resist more deformation than the ventral skin. The collagen fibers were arranged in different ordered arrays for ventral and dorsal skin and the concentration of fibers was found to be more in ventral than dorsal skin. This provides more stiffness to ventral skin. Scanning electron microscopy studies of the ventral skin showed a unidirectional arrangement of the collagen fibers, which provides more stretching capacity. Dorsal skin, on the other hand, has an orthogonal arrangement of fibers. The present study thus showed that the mechanical behavior of the skin of L. gloveri is strongly influenced by the concentration and arrangement of collagen fibers.     

New perspectives on collagen fibers in the squid mantle

The squid mantle is a complex structure which, in conjunction with a highly sensitive sensory system, provides squid with a wide variety of highly controlled movements. This article presents a model describing systems of collagen fibers that give the mantle its shape and mechanical properties. The validity of the model is verified by comparing predicted optimal fiber angles to actual fiber angles seen in squid mantle. The model predicts optimal configurations for multiple fiber systems. It is found that the tunic fibers (outer collagen layers) provide optimal jetting characteristics when oriented at 31°, which matches empirical data from previous studies. The model also predicted that a set of intramuscular fibers (IM‐1) are oriented relative to the longitudinal axis to provide optimal energy storage capacity within the limiting physical bounds of the collagen fibers themselves. In addition, reasons for deviations from the predicted values are analyzed. This study illustrates how the squid's reinforcing collagen fibers are aligned to provide several locomotory advantages and demonstrates how this complex biological process can be accurately modeled with several simplifying assumptions. J. Morphol., 2012. © 2012 Wiley Periodicals, Inc.     

Effects of Cysteine Proteases on the Structural and Mechanical Properties of Collagen Fibers

Excessive cathepsin K (catK)-mediated turnover of fibrillar type I and II collagens in bone and cartilage leads to osteoporosis and osteoarthritis. However, little is known about how catK degrades compact collagen macromolecules. The present study is aimed to explore the structural and mechanical consequences of collagen fiber degradation by catK. Mouse tail type I collagen fibers were incubated with either catK or non-collagenase cathepsins. Methods used include scanning electron microscopy, protein electrophoresis, atomic force microscopy, and tensile strength testing. Our study revealed evidence of proteoglycan network degradation, followed by the progressive disassembly of macroscopic collagen fibers into primary structural elements by catK. Proteolytically released GAGs are involved in the generation of collagenolytically active catK-GAG complexes as shown by AFM. In addition to their structural disintegration, a decrease in the tensile properties of fibers was observed due to the action of catK. The Young''s moduli of untreated collagen fibers versus catK-treated fibers in dehydrated conditions were 3.2 ± 0.68 GPa and 1.9 ± 0.65 GPa, respectively. In contrast, cathepsin L, V, B, and S revealed no collagenase activity, except the disruption of proteoglycan-GAG interfibrillar bridges, which slightly decreased the tensile strength of fibers.     

A new molecular model for collagen elasticity based on synchrotron X-ray scattering evidence.

Collagen is the most abundant structural protein in vertebrates. The specific shape of its stress-strain curve is crucial for the function of a number of organs. Although the macroscopic mechanical behavior of collagen is well known, there is still no explanation of the elastic process at the supramolecular level. We have performed in situ synchrotron x-ray scattering experiments, which show that the amount of lateral molecular order increases upon stretching of collagen fibers. In strain cycling experiments the relation between strain and diffuse equatorial scattering was found to be linear in the "heel" region of the stress-strain curve. A new molecular model for collagen elasticity is proposed, which, based on the existence of thermally activated molecular kinks, reproduces this linearity and gives a simple explanation for the form of the stress-strain curve of collagen.     

Biomimetic approaches to tendon repair

The linear organization of collagen fibers in tendons results in optimal stiffness and strength at low strains under tensile load. However, this organization makes repairing ruptured or lacerated tendons extremely difficult. Current suturing techniques to join split ends of tendons, while providing sufficient mechanical strength to prevent gapping, are inadequate to carry normal loads. Immobilization protocols necessary to restore tendon congruity result in scar formation at the repair site and peripheral adhesions that limit excursion. These problems are reviewed to emphasize the need for novel approaches to tendon repair, one of which is the development of biomimetic tendons. The objective of the empirical work described here was to produce biologically-based, biocompatible tendon replacements with appropriate mechanical properties to enable immediate mobilization following surgical repair. Nor-dihydroguaiaretic acid (NDGA), a di-catechol from creosote bush, caused a dose dependent increase in the material properties of reconstituted collagen fibers, achieving a 100-fold increase in strength and stiffness over untreated fibers. The maximum tensile strength of the optimized NDGA treated fibers averaged 90 MPa; the elastic modulus of these fibers averaged 580 MPa. These properties were independent of strain rates ranging from 0.60 to 600 mm/min. Fatigue tests established that neither strength nor stiffness were affected after 80 k cycles at 5% strain. Treated fibers were not cytotoxic to tendon fibroblasts. Fibroblasts attached and proliferated on NDGA treated collagen normally. NDGA-fibers did not elicit a foreign body response nor did they stimulate an immune reaction during six weeks in vivo. The fibers survived 6 weeks with little evidence of fragmentation or degradation. The polymerization scheme described here produces a fiber-reinforced NDGA-polymer with mechanical properties approaching an elastic solid. The strength, stiffness and fatigue properties of the NDGA-treated fibers are comparable to those of tendon. These fibers are biocompatible with tendon fibroblasts and elicit little rejection or antigenic response in vivo. These results indicate that NDGA polymerization may provide a viable approach for producing collagenous materials that can be used to bridge gaps in ruptured or lacerated tendons. The tendon-like properties of the NDGA-fiber would allow early mobilization after surgical repair. We predict that timely loading of parted tendons joined by this novel biomaterial will enhance mechanically driven production of neo-tendon by the colonizing fibroblasts and result in superior repair and rapid return to normal properties.     

On the anisotropy of the canine diaphragmatic central tendon

We studied the mechanical and anatomical anisotropy of the canine diaphragmatic central tendon (CT). Dumb-bell-shaped strips with effective dimensions of 10 x 2 mm (length x width) were cut from different regions of the canine diaphragmatic CT in two different orientations relative to the direction of neighboring muscle fibers. Specimens sampled with their long axial dimension oriented parallel to the neighboring muscle fibers were named Group-1 and those sampled with an orientation perpendicular to the neighboring muscle fibers were named Group-2. Results from one-dimensional stress-strain and tensile failure strength tests revealed that the CT is a nonlinear, inelastic, and anisotropic material. Group-1 specimens were found to have a higher stiffness, higher failure strength and higher strain energy density at failure than Group-2 specimens. Polarized microscopy showed that multiple sheets of collagen fiber bundles formed an orthogonal network in the tendon. Collagen fiber bundles along Group-1 direction formed parallel trajectory lines connecting the neighboring costal and crural muscles; bundles along Group-2 direction were observed to orient 90 degrees away. At the central apex region of the CT, collagen bundles of Group-1 formed a fan-like trajectory pattern. This collagen network architecture was compared favorably to the trajectories of an approximated principal stress field in the CT due to simulated contractile forces from its adjacent costal and crural muscles. These combined results suggest a structure-function relationship for the anatomical and mechanical anisotropy in the canine diaphragmatic CT.     

Mechanical interactions between collagen and proteoglycans: implications for the stability of lung tissue.

Collagen and elastin are thought to dominate the elasticity of the connective tissue including lung parenchyma. The glycosaminoglycans on the proteoglycans may also play a role because osmolarity of interstitial fluid can alter the repulsive forces on the negatively charged glycosaminoglycans, allowing them to collapse or inflate, which can affect the stretching and folding pattern of the fibers. Hence, we hypothesized that the elasticity of lung tissue arises primarily from 1) the topology of the collagen-elastin network and 2) the mechanical interaction between proteoglycans and fibers. We measured the quasi-static, uniaxial stress-strain curves of lung tissue sheets in hypotonic, normal, and hypertonic solutions. We found that the stress-strain curve was sensitive to osmolarity, but this sensitivity decreased after proteoglycan digestion. Images of immunofluorescently labeled collagen networks showed that the fibers follow the alveolar walls that form a hexagonal-like structure. Despite the large heterogeneity, the aspect ratio of the hexagons at 30% uniaxial strain increased linearly with osmolarity. We developed a two-dimensional hexagonal network model of the alveolar structure incorporating the mechanical properties of the collagen-elastin fibers and their interaction with proteoglycans. The model accounted for the stress-strain curves observed under all experimental conditions. The model also predicted how aspect ratio changed with osmolarity and strain, which allowed us to estimate the Young's modulus of a single alveolar wall and a collagen fiber. We therefore identify a novel and important role for the proteoglycans: they stabilize the collagen-elastin network of connective tissues and contribute to lung elasticity and alveolar stability at low to medium lung volumes.     

Elastic energy storage in unmineralized and mineralized extracellular matrices (ECMs): a comparison between molecular modeling and experimental measurements

In order to facilitate locomotion and limb movement many animals store energy elastically in their tendons. In the turkey, much of the force generated by the gastrocnemius muscle is stored as elastic energy during tendon deformation and not within the muscle. As limbs move, the tendons are strained causing the collagen fibers in the extracellular matrices to be strained. During growth, avian tendons mineralize in the portions distal to the muscle and show increased tensile strength, modulus, and energy stored per unit strain as a result. In this study the energy stored in unmineralized and mineralized collagen fibers was measured and compared to the amount of energy stored in molecular models. Elastic energy storage values calculated using the molecular model were slightly higher than those obtained from collagen fibers, but display the same increases in slope as the fiber data. We hypothesize that these increases in slope are due to a change from the stretching of flexible regions of the collagen molecule to the stretching of less flexible regions. The elastic modulus obtained from the unmineralized molecular model correlates well with elastic moduli of unmineralized collagen from other studies. This study demonstrates the potential importance of molecular modeling in the design of new biomaterials.     

Microstructure,mechanical, and biomimetic properties of fish scales from Pagrus major

The fish scale of Pagrus major has an orthogonal plywood structure of stratified lamellae, 1-2 microm in thickness, consisting of closely packed 70- to 80-nm-diameter collagen fibers. X-ray diffraction, energy-dispersive X-ray analysis, and infrared spectroscopy indicate that the mineral phase in the scale is calcium-deficient hydroxyapatite containing a small amount of sodium and magnesium ions, as well as carbonate anions in phosphate sites of the apatite lattice. The tensile strength of the scale is high (approximately 90 MPa) because of the hierarchically ordered structure of mineralized collagen fibers. Mechanical failure occurs by sliding of the lamellae and associated pulling out and fracture of the collagen fibers. In contrast, demineralized scales have significantly lower tensile strength (36 MPa), indicating that interactions between the apatite crystals and collagen fibers are of fundamental importance in determining the mechanical properties. Thermal treatment of fish scales to remove the organic components produces remarkable inorganic replicas of the native orthogonal plywood structure of the fibrillary plate. The biomimetic replica produced by heating to 873 K consists of stratified porous lamellae of c-axis-aligned apatite crystals that are long, narrow plates, 0.5-0.6 microm in length and 0.1-0.2 microm in width. The textured inorganic material remains intact when heated to 1473 K, although the size of the constituent crystals increases as a result of thermal sintering.     

The role of viscoelasticity of collagen fibers in articular cartilage: theory and numerical formulation

The relative importance of fluid-dependent and fluid-independent transient mechanical behavior in articular cartilage was examined for tensile and unconfined compression testing using a fibril reinforced model. The collagen matrix of articular cartilage was modeled as viscoelastic using a quasi-linear viscoelastic formulation with strain-dependent elastic modulus, while the proteoglycan matrix was considered as linearly elastic. The collagen viscoelastic properties were obtained by fitting experimental data from a tensile test. These properties were used to investigate unconfined compression testing, and the sensitivity of the properties was also explored. It was predicted that the stress relaxation observed in tensile tests was not caused by fluid pressurization at the macroscopic level. A multi-step tensile stress relaxation test could be approximated using a hereditary integral in which the elastic fibrillar modulus was taken to be a linear function of the fibrillar strain. Applying the same formulation to the radial fibers in unconfined compression, stress relaxation could not be simulated if fluid pressurization were absent. Collagen viscoelasticity was found to slightly weaken fluid pressurization in unconfined compression, and this effect was relatively more significant at moderate strain rates. Therefore, collagen viscoelasticity appears to play an import role in articular cartilage in tensile testing, while fluid pressurization dominates the transient mechanical behavior in compression. Collagen viscoelasticity plays a minor role in the mechanical response of cartilage in unconfined compression if significant fluid flow is present.     

Modeling interlamellar interactions in angle-ply biologic laminates for annulus fibrosus tissue engineering

Mechanical function of the annulus fibrosus of the intervertebral disc is dictated by the composition and microstructure of its highly ordered extracellular matrix. Recent work on engineered angle-ply laminates formed from mesenchymal stem cell (MSC)-seeded nanofibrous scaffolds indicates that the organization of collagen fibers into planes of alternating alignment may play an important role in annulus fibrosus tissue function. Specifically, these engineered tissues can resist tensile deformation through shearing of the interlamellar matrix as layers of collagen differentially reorient under load. In the present work, a hyperelastic constitutive model was developed to describe the role of interlamellar shearing in reinforcing the tensile response of biologic laminates, and was applied to experimental results from engineered annulus constructs formed from MSC-seeded nanofibrous scaffolds. By applying the constitutive model to uniaxial tensile stress–strain data for bilayers with three different fiber orientations, material parameters were generated that characterize the contributions of extrafibrillar matrix, fibers, and interlamellar shearing interactions. By 10 weeks of in vitro culture, interlamellar shearing accounted for nearly 50% of the total stress associated with uniaxial extension in the anatomic range of ply angle. The model successfully captured changes in function with extracellular matrix deposition through variations in the magnitude of model parameters with culture duration. This work illustrates the value of engineered tissues as tools to further our understanding of structure–function relations in native tissues and as a test-bed for the development of constitutive models to describe them.     

Assembly of type I collagen: fusion of fibril subunits and the influence of fibril diameter on mechanical properties.

Structural stability of the extracellular matrix is primarily a consequence of fibrillar collagen and the extent of cross-linking. The relationship between collagen self-assembly, consequent fibrillar shape and mechanical properties remains unclear. Our laboratory developed a model system for the preparation of self-assembled type I collagen fibers with fibrillar substructure mimicking the hierarchical structures of tendon. The present study evaluates the effects of pH and temperature during self-assembly on fibrillar structure, and relates the structural effects of these treatments on the uniaxial tensile mechanical properties of self-assembled collagen fibers. Results of the analysis of fibril diameter distributions and mechanical properties of the fibers formed under the different incubation conditions indicate that fibril diameters grow via the lateral fusion of discrete approximately 4 nm subunits, and that fibril diameter correlates positively with the low strain modulus. Fibril diameter did not correlate with either the ultimate tensile strength or the high strain elastic modulus, which suggests that lateral aggregation and consequently fibril diameter influences mechanical properties during small strain mechanical deformation. We hypothesize that self-assembly is mediated by the formation of fibrillar subunits that laterally and linearly fuse resulting in fibrillar growth. Lateral fusion appears important in generating resistance to deformation at low strain, while linear fusion leading to longer fibrils appears important in the ultimate mechanical properties at high strain.