Relationship between bone tissue strain and lattice strain of HAp crystals in bovine cortical bone under tensile loading

Cortical bone is a composite material composed of hydroxyapatite (HAp) and collagen. As HAp is a crystalline structure, an X-ray diffraction method is available to measure the strain of HAp crystals. However, HAp crystals in bone tissue have been known to have the low degree of crystallization. Authors have proposed an X-ray diffraction method to measure the lattice strain of HAp crystals from the diffusive intensity profile due to low crystallinity. The precision of strain measurement was greatly improved by this method. In order to confirm the possibility of estimating the bone tissue strain with measurements of the strain of HAp crystals, this work investigates the relationship between bone tissue strain on a macroscopic scale and the lattice strain of HAp crystals on a microscopic scale. The X-ray diffraction experiments were performed under tensile loading. Strip bone specimens of 40x6x0.8mm in size were cut from the cortical region of a shaft of bovine femur. A stepwise tensile load was applied in the longitudinal direction of the specimen. By detecting the diffracted X-ray beam transmitted through the specimen, the lattice strain was directly measured in the loading direction. As a result, the lattice strain of HAp crystals showed lower value than the bone tissue strain measured by a strain gage. The bone tissue strain was described with the mean lattice strain of the HAp crystals and the elastic modulus.     

Molecular modeling of the axial and circumferential elastic moduli of tubulin

Microtubules play a number of important mechanical roles in almost all cell types in nearly all major phylogenetic trees. We have used a molecular mechanics approach to perform tensile tests on individual tubulin monomers and determined values for the axial and circumferential moduli for all currently known complete sequences. The axial elastic moduli, in vacuo, were found to be 1.25 GPa and 1.34 GPa for α- and β-bovine tubulin monomers. In the circumferential direction, these moduli were 378 MPa for α- and 460 MPa for β-structures. Using bovine tubulin as a template, 269 homologous tubulin structures were also subjected to simulated tensile loads yielding an average axial elastic modulus of 1.10 ± 0.14 GPa for α-tubulin structures and 1.39 ± 0.68 GPa for β-tubulin. Circumferentially the α- and β-moduli were 936 ± 216 MPa and 658 ± 134 MPa, respectively. Our primary finding is that that the axial elastic modulus of tubulin diminishes as the length of the monomer increases. However, in the circumferential direction, no correlation exists. These predicted anisotropies and scale dependencies may assist in interpreting the macroscale behavior of microtubules during mitosis or cell growth. Additionally, an intergenomic approach to investigating the mechanical properties of proteins may provide a way to elucidate the evolutionary mechanical constraints imposed by nature upon individual subcellular components.     

Evidence of entropic elasticity of human bone trabeculae at low strains

We performed 60 microtensile tests on 6 single trabeculae excised from a human femur head at various maximum tensile loads. The obtained results show a clear dependence of the calculated stress-strain behaviour from the applied load and thus from the mean stress over the cross section of the trabecula. The pooled data were found in good agreement with a combination of both the model of the nonlinear stress-strain behaviour of collagen fibrils and that for the modulus of elasticity of staggered mineralized collagen. This circumstance could also suggest a realistic explanation of the extreme variability found in literature for the elastic modulus of the trabecular material. In particular, when the trabeculae are solicited with relatively low stresses, their mechanical properties are mainly affected by the entropic elasticity of collagen molecules. This work offers both experimental data and a reasonable interpretation of the behaviour of fully mineralized tissue at low strains, that is up to about 0.1%.     

Opto-mechanical characterization of sclera by polarization sensitive optical coherence tomography

Polarization sensitive optical coherence tomography (PSOCT) is an interferometric technique sensitive to birefringence. Since mechanical loading alters the orientation of birefringent collagen fibrils, we asked if PSOCT can be used to measure local mechanical properties of sclera.Infrared (1300 nm) PSOCT was performed during uniaxial tensile loading of fresh scleral specimens of rabbits, cows, and humans from limbal, equatorial, and peripapillary regions. Specimens from 8 human eyes were obtained. Specimens were stretched to failure at 0.01 mm/s constant rate under physiological conditions of temperature and humidity while birefringence was computed every 117 ms from cross-sectional PSOCT. Birefringence modulus (BM) was defined as the rate of birefringence change with strain, and tensile modulus (TM) as the rate of stress change between 0 and 9% strain.In cow and rabbit, BM and TM were positively correlated with slopes of 0.17 and 0.10 GPa, and with correlation coefficients 0.63 and 0.64 (P < 0.05), respectively, following stress-optic coefficients 4.69, and 4.20 GPa⁻¹. In human sclera, BM and TM were also positively correlated with slopes of 0.24 GPa for the limbal, 0.26 GPa for the equatorial, and 0.31 GPa for the peripapillary regions. Pearson correlation coefficients were significant at 0.51, 0.58, and 0.69 for each region, respectively (<0.001). Mean BM decreased proportionately to TM from the limbal to equatorial to peripapillary regions, as stress-optic coefficients were estimated as 2.19, 2.42, and 4.59 GPa⁻¹, respectively.Since birefringence and tensile elastic moduli correlate differently in cow, rabbit, and various regions of human sclera, it might be possible to mechanically characterize the sclera in vivo using PSOCT.     

Nanostructural alteration in bone quantified in terms of orientation distribution of mineral crystals: a possible tool for fracture risk assessment

There may be different causes of failures in bone; however, their origin generally lies at the lowest level of structural hierarchy, i.e., at the mineral-collagen composite. Any change in the nanostructure affects the affinity or bonding effectiveness between and within the phases at this level, and hence determines the overall strength and quality of bone. In this study, we propose a novel concept to assess change in the nanostructure and thereby change in the bonding status at this level by revealing change in the orientation distribution characteristics of mineral crystals. Using X-ray diffraction method, a parameter called Degree of Orientation (DO) has been quantified. The DO accounts for the azimuthal distribution of mineral crystals and represents their effective amount along any direction. Changes in the DOs in cortical bone samples from bovine femur with different preferential orientations of mineral crystals were estimated under external loads. Depending on the applied loads, change in the azimuthal distribution of the DOs and the degree of reversibility of the crystals was observed to vary. The characteristics of nanostructural change and thereby possible affect on the strength of bone was then predicted from the reversible or irreversible characteristics of distributed mineral crystals. Significant changes in the organization of mineral crystals were observed; however, variations in the applied stresses and elastic moduli were not evinced at the macroscale level. A novel concept to assess the alteration in nanostructure on the basis of mineral crystals orientation distribution has been proposed. The importance of nanoscale level information obtained noninvasively has been emphasized, which acts as a precise tool to estimate the strength and predict the possible fracture risks in bone.     

Comparison of nonlinear mechanical properties of bovine articular cartilage and meniscus

Nonlinear, linear and failure properties of articular cartilage and meniscus in opposing contact surfaces are poorly known in tension. Relationships between the tensile properties of articular cartilage and meniscus in contact with each other within knee joints are also not known. In the present study, rectangular samples were prepared from the superficial lateral femoral condyle cartilage and lateral meniscus of bovine knee joints. Tensile tests were carried out with a loading rate of 5 mm/min until the tissue rupture. Nonlinear properties of the toe region, linear properties in larger strains, and failure properties of both tissues were analysed. The strain-dependent tensile modulus of the toe region, Young's modulus of the linear region, ultimate tensile stress and toughness were on average 98.2, 8.3, 4.0 and 1.9 times greater (p<0.05) for meniscus than for articular cartilage. In contrast, the toe region strain, yield strain and failure strain were on average 9.4, 3.1 and 2.3 times greater (p<0.05) for cartilage than for meniscus. There was a significant negative correlation between the strain-dependent tensile moduli of meniscus and articular cartilage samples within the same joints (r=−0.690, p=0.014). In conclusion, the meniscus possesses higher nonlinear and linear elastic stiffness and energy absorption capability before rupture than contacting articular cartilage, while cartilage has longer nonlinear region and can withstand greater strains before failure. These findings point out different load carrying demands that both articular cartilage and meniscus have to fulfil during normal physiological loading activities of knee joints.     

Tensile properties of calcified and uncalcified avian tendons

The mechanical properties of turkey and heron leg tendons have been investigated in dynamic tensile tests. Heron tendons have properties similar to those found for various mammalian tendons. The Young's modulus and the density of turkey tendons increase with increasing calcification. Ultimate tensile stresses are similar to those found for uncalcified tendon, but Young's modulus may reach about 16 GPa, a value normally associated with bone. Calcification lowers the amount of strain energy that can be stored temporarily in the tendons of the legs. The contribution made by elastic strain energy storage to lowering the cost of running is reduced.     

A method on strain measurement of HAP in cortical bone from diffusive profile of X-ray diffraction

Bone tissue is a composite material composed of hydroxyapatite (HAp) and collagen matrix. As HAp is a crystalline structure, an X-ray diffraction method is available to measure the lattice strain of HAp crystals. However, mineral particles of HAp in bone have much lower crystallinity than usual crystalline materials, which show a diffusive intensity profile of X-ray diffraction. It is not easy to determine quantitatively an infinitesimal strain of HAp from the peak position of diffusive profile. In order to improve the accuracy of strain measurement of HAp in bone tissue and to obtain reproducible results, this paper proposes an X-ray diffraction method applied to a diffusive profile for low crystallinity. This method is to estimate the lattice strain of HAp using not a peak position but a whole diffraction profile. In this experiment, a strip specimen of 28 x 8 x 2 mm was made from bone axial, outside circumferential and cross-sectional circumferential region in the cortical bone of bovine femur. The X-ray diffraction measurements were carried out before and after applying an external load. As a result, the precision of strain measurement was much improved by this method. Although a constant value of macroscopic strain was applied in the specimen, the lattice strain had a lower value than the macroscopic strain and had a different value in each specimen.     

A high-pressure form of sulfuric acid monohydrate as determined by X-ray and neutron diffraction

Two high-pressure polymorphs of sulfuric acid monohydrate (oxonium hydrogensulfate) have been obtained at ambient temperature by crystallisation at high pressure from the liquid at 1.3 GPa (form III) and by direct compression of the ambient-pressure form I first to 1.26 GPa (form II) and then to 1.72 GPa (form III). The structure of form III was solved by single crystal X-ray diffraction and this structure was used as the basis for the refinement of hydrogen positions using high-pressure neutron powder diffraction data. Form III crystallises in the orthorhombic crystal system at 1.97 GPa, and features parallel chains of hydrogensulfate ions linked by oxonium ions to form a three-dimensional hydrogen-bonded network. On further compression to 3.05 GPa, the direction of maximum compressibility is found to be along the a-axis and is associated with the shortening of a hydrogen bond between a hydrogensulfate ion and an oxonium ion. The structure of form II remains elusive although at ambient temperature it is stable (or metastable) at pressures as low as 0.42 GPa, perhaps indicating that it could be recoverable to ambient-pressure at low temperature.     

Lateral packing of mineral crystals in bone collagen fibrils

Combined small-angle x-ray scattering and transmission electron microscopy studies of intramuscular fish bone (shad and herring) indicate that the lateral packing of nanoscale calcium-phosphate crystals in collagen fibrils can be represented by irregular stacks of platelet-shaped crystals, intercalated with organic layers of collagen molecules. The scattering intensity distribution in this system can be described by a modified Zernike-Prins model, taking preferred orientation effects into account. Using the model, the diffuse fan-shaped small-angle x-ray scattering intensity profile, dominating the equatorial region of the scattering pattern, could be quantitatively analyzed as a function of the degree of mineralization. The mineral platelets were found to be very thin (1.5 nm ∼ 2.0 nm), having a narrow thickness distribution. The thickness of the organic layers between adjacent mineral platelets within a stack is more broadly distributed with the average value varying from 6 nm to 10 nm, depending on the extent of mineralization. The two-dimensional analytical scheme also leads to quantitative information about the preferred orientation of mineral stacks and the average height of crystals along the crystallographic c axis.     

The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus

The elastic moduli of human subchondral, trabecular, and cortical bone tissue from a proximal tibia were experimentally determined using three-point bending tests on a microstructural level. The mean modulus of subchondral specimens was 1.15 GPa, and those of trabecular and cortical specimens was 4.59 GPa and 5.44 GPa respectively. Significant differences were found in the modulus values between bone tissues, which may have mainly resulted from the differences in the microstructures of each bone tissue rather than in the mineral density. Furthermore, the size-dependency of the modulus was examined using eight different sizes of cortical specimens (heights h = 100-1000 microns). While the modulus values for relatively large specimens (h greater than 500 microns) remained fairly constant (approximately 15 GPa), the values decreased as the specimens became smaller. A significant correlation was found between the modulus and specimen size. The surface area to volume ratio proved to be a key variable to explain the size-dependency.     

Changes in the molecular orientation and tensile properties of uniaxially drawn cellulose films

Cellulose films were prepared by dissolving lyocell fibers in LiCl/N,N-dimethylacetamide solvent and subsequently coagulating and drying them under ambient conditions. To introduce preferred orientation, the films were uniaxially drawn under air-dry and rewetted conditions, respectively. Preferred orientation was determined by birefringence measurements and by wide-angle X-ray scattering. Mechanical properties were characterized by means of tensile tests with films conditioned to standard temperatures and humidity. Drawing resulted in the substantial reorientation of cellulose, whereby the molecular chains in the amorphous regions exhibited clearly stronger reorientation than the crystalline fraction. The average degree of orientation was comparable to orientation achieved in spun cellulose fibers. Wet-drawing resulted in improved tensile strength and modulus of elasticity but reduced elongation at break. The mechanical properties of wet-drawn films are competitive with regard to cellophane and melt-blown cellulose films, particularly considering their high modulus of elasticity of up to 26 GPa, which is also comparable to values obtained for industrially produced cellulose fibers.     

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.     

Stress-concentrating effect of resorption lacunae in trabecular bone

Analyses of the distributions of stress and strain within individual bone trabeculae have not yet been reported. In this study, four trabeculae were imaged and finite elements models were generated in an attempt to quantify the variability of stress/strain in real trabeculae. In three of these trabeculae, cavities were identified with depths comparable to values reported for resorption lacunae ( approximately 50 microm)-although we cannot be certain, it is most probable that they are indeed resorption lacunae. A tensile load was applied to each trabeculum to simulate physiological loading and to ensure that bending was minimized. The force carried by each trabecula was calculated from this value using the average cross sectional area of each trabecula. The analyses predict that very high stresses (>100 MPa) existed within bone trabecular tissue. Stress and strain distributions were highly heterogeneous in all cases, more so in trabeculae with the presumptive resorption lacunae where at least 30% of the tissue had a strain greater than 4000 micoepsilon in all cases. Stresses were elevated at the pit of the lacunae, and peak stress concentrations were located in the longitudinal direction ahead of the lacunae. Given these high strains, we suggest that microdamage is inevitable around resorption lacunae in trabecular bone, and may cause the bone multicellular unit to proceed to resorb a packet of bone in the trabeculum rather than just resorb whatever localized area was initially targeted.     

Use of a Digital Image Correlation Technique for Measuring the Material Properties of Beetle Wing

Beetle wings are very specialized flight organs consisting of the veins and membranes.Therefore it is necessary from abionic view to investigate the material properties of a beetle wing experimentally.In the present study,we have used a DigitalImage Correlation (DIC) technique to measure the elastic modulus of a beetle wing membrane.Specimens were prepared bycarefully cutting a beetle hind wing into 3.0 mm by 7.0 mm segments (the gage length was 5 mm).We used a scanning electronmicroscope for a precise measurement of the thickness of the beetle wing membrane.The specimen was attached to a designedfixture to induce a uniform displacement by means of a micromanipulator.We used an ARAMISTM system based on the digitalimage correlation technique to measure the corresponding displacement of a specimen.The thickness of the beetle wing variedat different points of the membrane.The elastic modulus differed in relation to the membrane arrangement showing a structuralanisotropy;the elastic modulus in the chordwise direction is approximately 2.65 GPa,which is three times larger than the elasticmodulus in the spanwise direction of 0.84 GPa.As a result,the digital image correlation-based ARAMIS system was suc-cessfully used to measure the elastic modulus of a beetle wing.In addition to membrane’s elastic modulus,we considered thePoisson’s ratio of the membrane and measured the elastic modulus of a vein using an Instron universal tensile machine.Theresult reveals the Poisson’s ratio is nearly zero and the elastic modulus of a vein is about 11 GPa.     

Biomechanical properties across trabeculae from the proximal femur of normal and ovariectomised sheep

The elastic behaviour of trabecular bone is a function not only of bone volume and architecture, but also of tissue material properties. Variation in tissue modulus can have a substantial effect on the biomechanical properties of trabecular bone. However, the nature of tissue property variation within a single trabecula is poorly understood. This study uses nanoindentation to determine the mechanical properties of bone tissue in individual trabeculae. Using an ovariectomised ovine model, the modulus and hardness distribution across trabeculae were measured. In both normal and ovariectomised bone, the modulus and hardness were found to increase towards the core of the trabeculae. Across the width of the trabeculae, the modulus was significantly less in the ovariectomised bone than in the control bone. However, in contrast to this hardness was found not to differ significantly between the two groups. This study provides valuable information on the variation of mechanical material properties in healthy and diseased trabecular bone tissue. The results of the current study will be useful in finite element modelling where more accurate values of trabecular bone modulus will enable the prediction of the macroscale behaviour of trabecular bone.     

Elastic properties of collagen in bone determined by measuring the Debye–Waller factor

Force constant values for thermal vibrational motion of a collagen molecule along the helix axis in tendon, completely demineralized bone (CDB), and partially demineralized bone (PDB) were estimated by determining the Debye–Waller factor (DW factor) for the diffracted X-ray intensity from these specimens. The DW factor for nominal value of 0.286 nm meridional diffraction representing a period along the helical axis of a collagen molecule was measured. As the atomic scattering factor of mineral constituents is much larger than that of collagen, it is difficult to detect the diffraction from collagen in bone specimen. Therefore, PDB was used in this study. In order to compare obtained force constant value for CDB with mechanical properties of collagen in the literature, the value was translated into Young's modulus value using the cross-sectional area of a collagen molecule. In the case of collagen in PDB, i.e., collagen with the close presence of HAp mineral particles, as the DW factor of the diffracted intensity by hydroxyapatite (HAp) was considered to be negligible compared with that of collagen, the DW factor determined was interpreted as that of collagen molecule in PDB specimen. The force constant value obtained for collagen in PDB was significantly larger than that of collagen in CDB. This result was thought to be a manifestation of the hardening of collagen matrix in bone by HAp mineral particles and the first straightforward evidence for a difference in collagen properties depending on the presence of HAp mineral particles. The method employed in this study can be utilized for detecting mechanical properties of the individual constituents of composite materials.     

A two-layer elasto-visco-plastic rheological model for the material parameter identification of bone tissue

The ability to measure bone tissue material properties plays a major role in diagnosis of diseases and material modeling. Bone’s response to loading is complex and shows a viscous contribution to stiffness, yield and failure. It is also ductile and damaging and exhibits plastic hardening until failure. When performing mechanical tests on bone tissue, these constitutive effects are difficult to quantify, as only their combination is visible in resulting stress–strain data. In this study, a methodology for the identification of stiffness, damping, yield stress and hardening coefficients of bone from a single cyclic tensile test is proposed. The method is based on a two-layer elasto-visco-plastic rheological model that is capable of reproducing the specimens’ pre- and postyield response. The model’s structure enables for capturing the viscously induced increase in stiffness, yield, and ultimate stress and for a direct computation of the loss tangent. Material parameters are obtained in an inverse approach by optimizing the model response to fit the experimental data. The proposed approach is demonstrated by identifying material properties of individual bone trabeculae that were tested under wet conditions. The mechanical tests were conducted according to an already published methodology for tensile experiments on single trabeculae. As a result, long-term and instantaneous Young’s moduli were obtained, which were on average 3.64 GPa and 5.61 GPa, respectively. The found yield stress of 16.89 MPa was lower than previous studies suggest, while the loss tangent of 0.04 is in good agreement. In general, the two-layer model was able to reproduce the cyclic mechanical test data of single trabeculae with an root-mean-square error of 2.91 ± 1.77 MPa. The results show that inverse rheological modeling can be of great advantage when multiple constitutive contributions shall be quantified based on a single mechanical measurement.

     

Nanomechanical and structural properties of native cellulose under compressive stress

Cellulose is an important biopolymer with applications ranging from its use as an additive in pharmaceutical products to the development of novel smart materials. This wide applicability arises in part from its interesting mechanical properties. Here we report on the use of high pressure X-ray diffraction and Raman spectroscopy in a diamond anvil cell to determine the bulk and local elastic moduli of native cellulose. The modulus values obtained are 20 GPa for the bulk modulus and 200-355 and 15 GPa for the crystalline parts and the overall elastic (Young's) modulus, respectively. These values are consistent with those calculated from tensile measurements. Above 8 GPa, the packing of the cellulose chains within the fibers undergoes significant structural distortion, whereas the chains themselves remain largely unaffected by compression.     

Mechanical effects of load speed on the human colon

The aim of this study was to examine the mechanical behavior of the colon using tensile tests under different loading speeds.Specimens were taken from different locations of the colonic frame from refrigerated cadavers. The specimens were submitted to uniaxial tensile tests after preconditioning using a dynamic load (1 m/s), intermediate load (10 cm/s), and quasi-static load (1 cm/s).A total of 336 specimens taken from 28 colons were tested. The stress-strain analysis for longitudinal specimens indicated a Young’s modulus of 3.17 ± 2.05 MPa under dynamic loading (1 m/s), 1.74 ± 1.15 MPa under intermediate loading (10 cm/s), and 1.76 ± 1.21 MPa under quasi-static loading (1 cm/s) with p < 0.001. For the circumferential specimen, the stress-strain curves indicated a Young’s modulus of 3.15 ± 1.73 MPa under dynamic loading (1 m/s), 2.14 ± 1.3 MPa under intermediate loading (10 cm/s), and 0.63 ± 1.25 MPa under quasi-static loading (1 cm/s) with p < 0.001. The curves reveal two types of behaviors of the colon: fast break behavior at high speed traction (1 m/s) and a lower break behavior for lower speeds (10 cm/s and 1 cm/s). The circumferential orientation required greater levels of stress and strain to obtain lesions than the longitudinal orientation. The presence of taeniae coli changed the mechanical response during low-speed loading.Colonic mechanical behavior varies with loading speeds with two different types of mechanical behavior: more fragile behavior under dynamic load and more elastic behavior for quasi-static load.