Constitutive relationship of tissue behavior with damage accumulation of human cortical bone

Microdamage accumulation has been identified as a major conduit for bone tissues to absorb fracture energy. Due to the poor understanding of its underlying mechanism, however, an adequate constitutive relationship between damage accumulation and the mechanical behavior of bone has not yet been established. In this study, the constitutive relationship between the damage accumulation induced by overload and the evolution of mechanical properties of bone with incremental deformation was established based on the experimental results obtained from a novel progressive loading protocol developed in our laboratory. First, a decayed exponential model was proposed to capture the damage accumulation (modulus loss) with increase in applied strain. Next, a power law function was proposed to represent the progression of plastic deformation with damage accumulation. Finally, a linear combination of the Kohlrausch–Williams–Watts (KWW) and the Debye functions was used to depict the viscoelastic behavior of bone associated with damage accumulation. The results of this study may help in developing a constitutive model for predicting the mechanical behavior of cortical bone tissues.     

Viscoelastic shear properties of the corneal stroma

The cornea is a highly specialized transparent tissue which covers the front of the eye. It is a tough tissue responsible for refracting the light and protecting the sensitive internal contents of the eye. The biomechanical properties of the cornea are primarily derived from its extracellular matrix, the stroma. The majority of previous studies have used strip tensile and pressure inflation testing methods to determine material parameters of the corneal stroma. Since these techniques do not allow measurements of the shear properties, there is little information available on transverse shear modulus of the cornea. The primary objectives of the present study were to determine the viscoelastic behavior of the corneal stroma in shear and to investigate the effects of the compressive strain. A thorough knowledge of the shear properties is required for developing better material models for corneal biomechanics. In the present study, torsional shear experiments were conducted at different levels of compressive strain (0–30%) on porcine corneal buttons. First, the range of linear viscoelasticity was determined from strain sweep experiments. Then, frequency sweep experiments with a shear strain amplitude of 0.2% (which was within the region of linear viscoelasticity) were performed. The corneal stroma exhibited viscoelastic properties in shear. The shear storage modulus, G′, and shear loss modulus, G″, were reported as a function of tissue compression. It was found that although both of these parameters were dependent on frequency, shear strain amplitude, and compressive strain, the average shear storage and loss moduli varied from 2 to 8 kPa, and 0.3 to 1.2 kPa, respectively. Therefore, it can be concluded that the transverse shear modulus is of the same order of magnitude as the out-of-plane Young's modulus and is about three orders of magnitude lower than the in-plane Young's modulus.     

The effect of damage on the viscoelastic behavior of human vertebral trabecular bone

The present study examines the viscoelastic behavior of cancellous bone at low strains and the effects of damage on this viscoelastic behavior. It provides experimental evidence of interaction between stress relaxation behavior and the effect of accumulated damage. The results suggest that damage is at least orthotropic in trabecular bone specimens under uniaxial loading. Simple linear models of viscoelasticity described the time-dependent stress-strain behavior at low strains before and after specimen damage, although better fits of these models were obtained prior to damage. Modeling the observed changes in relaxation times with damage accumulation appears necessary to successfully predict the post-damage viscoelastic response.     

Dynamic response of immature bovine articular cartilage in tension and compression, and nonlinear viscoelastic modeling of the tensile response

Very limited information is currently available on the constitutive modeling of the tensile response of articular cartilage and its dynamic modulus at various loading frequencies. The objectives of this study were to (1) formulate and experimentally validate a constitutive model for the intrinsic viscoelasticity of cartilage in tension, (2) confirm the hypothesis that energy dissipation in tension is less than in compression at various loading frequencies, and (3) test the hypothesis that the dynamic modulus of cartilage in unconfined compression is dependent upon the dynamic tensile modulus. Experiment 1: Immature bovine articular cartilage samples were tested in tensile stress relaxation and cyclical loading. A proposed reduced relaxation function was fitted to the stress-relaxation response and the resulting material coefficients were used to predict the response to cyclical loading. Adjoining tissue samples were tested in unconfined compression stress relaxation and cyclical loading. Experiment 2: Tensile stress relaxation experiments were performed at varying strains to explore the strain-dependence of the viscoelastic response. The proposed relaxation function successfully fit the experimental tensile stress-relaxation response, with R2 = 0.970+/-0.019 at 1% strain and R2 = 0.992+/-0.007 at 2% strain. The predicted cyclical response agreed well with experimental measurements, particularly for the dynamic modulus at various frequencies. The relaxation function, measured from 2% to 10% strain, was found to be strain dependent, indicating that cartilage is nonlinearly viscoelastic in tension. Under dynamic loading, the tensile modulus at 10 Hz was approximately 2.3 times the value of the equilibrium modulus. In contrast, the dynamic stiffening ratio in unconfined compression was approximately 24. The energy dissipation in tension was found to be significantly smaller than in compression (dynamic phase angle of 16.7+/-7.4 deg versus 53.5+/-12.8 deg at 10(-3) Hz). A very strong linear correlation was observed between the dynamic tensile and dynamic compressive moduli at various frequencies (R2 = 0.908+/-0.100). The tensile response of cartilage is nonlinearly viscoelastic, with the relaxation response varying with strain. A proposed constitutive relation for the tensile response was successfully validated. The frequency response of the tensile modulus of cartilage was reported for the first time. Results emphasize that fluid-flow dependent viscoelasticity dominates the compressive response of cartilage, whereas intrinsic solid matrix viscoelasticity dominates the tensile response. Yet the dynamic compressive modulus of cartilage is critically dependent upon elevated values of the dynamic tensile modulus.     

Evidence against proteoglycan mediated collagen fibril load transmission and dynamic viscoelasticity in tendon

The glycosaminoglycan (GAG) dermatan sulfate and chondroitin sulfate side-chains of small leucine-rich proteoglycans have been increasingly posited to act as molecular cross links between adjacent collagen fibrils and to directly contribute to tendon elasticity. GAGs have also been implicated in tendon viscoelasticity, supposedly affecting frictional loss during elongation or fluid flow through the extra cellular matrix. The current study sought to systematically test these theories of tendon structure–function by investigating the mechanical repercussions of enzymatic depletion of GAG complexes by chondroitinase ABC in a reproducible tendon structure–function model (rat tail tendon fascicles). The extent of GAG removal (at least 93%) was verified by relevant spectrophotometric assays and transmission electron microscopy. Dynamic viscoelastic tensile tests on GAG depleted rat tail tendon fascicle were not mechanically different from controls in storage modulus (elastic behavior) over a wide range of strain-rates (0.05, 0.5, and 5% change in length per second) in either the linear or nonlinear regions of the material curve. Loss modulus (viscoelastic behavior) was only affected in the nonlinear region at the highest strain-rate, and even this effect was marginal (19% increased loss modulus, p = 0.035). Thus glycosaminoglycan chains of small leucine-rich proteoglycans do not appear to mediate dynamic elastic behavior nor do they appear to regulate the dynamic viscoelastic properties in rat tail tendon fascicles.     

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.     

Cyclic cryopreservation affects the nanoscale material properties of trabecular bone

Tissues such as bone are often stored via freezing, or cryopreservation. During an experimental protocol, bone may be frozen and thawed a number of times. For whole bone, the mechanical properties (strength and modulus) do not significantly change throughout five freeze-thaw cycles. Material properties at the trabecular and lamellar scales are distinct from whole bone properties, thus the impact of freeze-thaw cycling at this scale is unknown. To address this, the effect of repeated freezing on viscoelastic material properties of trabecular bone was quantified via dynamic nanoindentation. Vertebrae from five cervine spines (1.5-year-old, male) were semi-randomly assigned, three-to-a-cycle, to 0–10 freeze-thaw cycles. After freeze-thaw cycling, the vertebrae were dissected, prepared and tested. ANOVA (factors cycle, frequency, and donor) on storage modulus, loss modulus, and loss tangent, were conducted. Results revealed significant changes between cycles for all material properties for most cycles, no significant difference across most of the dynamic range, and significant differences between some donors. Regression analysis showed a moderate positive correlation between cycles and material property for loss modulus and loss tangent, and weak negative correlation for storage modulus, all correlations were significant. These results indicate that not only is elasticity unpredictably altered, but also that damping and viscoelasticity tend to increase with additional freeze-thaw cycling.     

Cortical bone viscoelasticity and fixation strength of press-fit femoral stems: finite element model

Many cementless implant designs rely upon a diaphyseal press-fit in conjunction with a porous coated implant surface to achieve primary or short term fixation, thereby constraining interface micromotion to such a level that bone ingrowth and consequent secondary or long-term fixation, i.e., osseointegration, can occur. Bone viscoelasticity, however, has been found to affect stem primary stability by reducing push-out load. In this investigation, an axisymmetric finite element model of a cylindrical stem and diaphyseal cortical bone section was created in order to parametrically evaluate the effect of bone viscoelasticity on stem push-out while controlling coefficient of friction (mu = 0.15, 0.40, and 1.00) and stem-bone diametral interference (delta = 0.01, 0.05, 0.10, and 0.50 mm). Based on results from a previous study, it was hypothesized that stem-bone interference (i.e., press-fit) would elicit a bone viscoelastic response which would reduce the initial fixation of the stem as measured by push-out load. Results indicate that for all examined combinations of mu and delta, bone viscoelastic behavior reduced the push-out load by a range of 2.6-82.6% due to stress relaxation of the bone. It was found that the push-out load increased with mu for each value of delta, but minimal increases in the push-out load (2.9-4.9%) were observed as delta was increased beyond 0.10 mm. Within the range of variables reported for this study, it was concluded that bone viscoelastic behavior, namely stress relaxation, has an asymptotic affect on stem contact pressure, which reduces stem push-out load. It was also found that higher levels of coefficient of friction are beneficial to primary fixation, and that an interference "threshold" exists beyond which no additional gains in push-out load are achieved.     

Dynamic, regional mechanical properties of the porcine brain: indentation in the coronal plane

Stress relaxation tests using a custom designed microindentation device were performed on ten anatomic regions of fresh porcine brain (postmortem time <3 h). Using linear viscoelastic theory, a Prony series representation was used to describe the shear relaxation modulus for each anatomic region tested. Prony series parameters fit to load data from indentations performed to ～10% strain differed significantly by anatomic region. The gray and white matter of the cerebellum along with corpus callosum and brainstem were the softest regions measured. The cortex and hippocampal CA1/CA3 were found to be the stiffest. To examine the large strain behavior of the tissue, multistep indentations were performed in the corona radiata to strains of 10%, 20%, and 30%. Reduced relaxation functions were not significantly different for each step, suggesting that quasi-linear viscoelastic theory may be appropriate for representing the nonlinear behavior of this anatomic region of porcine brain tissue. These data, for the first time, describe the dynamic and short time scale behavior of multiple anatomic regions of the porcine brain which will be useful for understanding porcine brain injury biomechanics at a finer spatial resolution than previously possible.     

Br-rich tips of calcified crab claws are less hard but more fracture resistant: A comparison of mineralized and heavy-element biological materials

We find that the spoon-like tips of the chelipeds (large claws) of the crab Pachygrapsus crassipes differ from the rest of the claw in that they are not calcified, but instead contain about 1% bromine—thus they represent a new example of a class of structural biological materials that contain heavy elements such as Zn, Mn, Fe, Cu, and Br bound in an organic matrix. X-ray absorption spectroscopy data suggest that the bromine is bound to phenyl rings, possibly in tyrosine. We measure a broad array of mechanical properties of a heavy-element biological material for the first time (abrasion resistance, coefficient of kinetic friction, energy of fracture, hardness, modulus of elasticity and dynamic mechanical properties), and we make a direct comparison with a mineralized tissue. Our results suggest that the greatest advantage of bromine-rich cuticle over calcified cuticle is resistance to fracture (the energy of fracture is about an order of magnitude greater than for calcified cuticle). The greatest advantage relative to unenriched cuticle, represented by ant mandible cuticle, is a factor of about 1.5 greater hardness and modulus of elasticity. The spoon-like tips gain additional fracture resistance from the orientation of the constituent laminae and from the viscoelasticity of the material. We suggest that fracture resistance is of greater importance in smaller organisms, and we speculate that one function of heavy elements in structural biological materials is to reduce molecular resonant frequencies and thereby increase absorption of energy from impacts.     

Linear viscoelasticity - bone volume fraction relationships of bovine trabecular bone

Trabecular bone has been previously recognized as time-dependent (viscoelastic) material, but the relationships of its viscoelastic behaviour with bone volume fraction (BV/TV) have not been investigated so far. Therefore, the aim of the present study was to quantify the time-dependent viscoelastic behaviour of trabecular bone and relate it to BV/TV. Uniaxial compressive creep experiments were performed on cylindrical bovine trabecular bone samples (\(\textit{n}\,{=}\,13\)) at loads corresponding to physiological strain level of 2000 \({\upmu }{\upvarepsilon }\). We assumed that the bone behaves in a linear viscoelastic manner at this low strain level and the corresponding linear viscoelastic parameters were estimated by fitting a generalized Kelvin–Voigt rheological model to the experimental creep strain response. Strong and significant power law relationships (\(r^2\,{=}\,0.73,\ p\,{<}\,0.001\)) were found between time-dependent creep compliance function and BV/TV of the bone. These BV/TV-based material properties can be used in finite element models involving trabecular bone to predict time-dependent response. For users’ convenience, the creep compliance functions were also converted to relaxation functions by using numerical interconversion methods and similar power law relationships were reported between time-dependent relaxation modulus function and BV/TV.     

Viscoelastic properties of single attached cells under compression

The viscoelastic properties of single, attached C2C12 myoblasts were measured using a recently developed cell loading device. The device allows global compression of an attached cell, while simultaneously measuring the associated forces. The viscoelastic properties were examined by performing a series of dynamic experiments over two frequency decades (0.1-10 Hz) and at a range of axial strains (approximately 10-40%). Confocal laser scanning microscopy was used to visualize the cell during these experiments. To analyze the experimentally obtained force-deformation curves, a nonlinear viscoelastic model was developed. The nonlinear viscoelastic model was able to describe the complete series of dynamic experiments using only a single set of parameters, yielding an elastic modulus of 2120 +/- 900 Pa for the elastic spring, an elastic modulus of 1960 +/- 1350 for the nonlinear spring, and a relaxation time constant of 0.3 +/- 0.12 s. To our knowledge, it is the first time that the global viscoelastic properties of attached cells have been quantified over such a wide range of strains. Furthermore, the experiments were performed under optimal environmental conditions and the results are, therefore, believed to reflect the viscoelastic mechanical behavior of cells, such as would be present in vivo.     

Viscoelastic characterization of mesenchymal gap tissue and consequences for tension accumulation during distraction

Nonlinear viscoelastic analysis was used to characterize the time-dependent behavior of mesenchymal gap tissue generated during distraction osteogenesis. Six (n = 6) lengthened tibiae were harvested from New Zealand white rabbits at 18 days. This gap tissue was subjected to a series of step displacement tests of increasing magnitude, and force relaxation behavior was monitored. Isochrones in stress-strain space were fit to odd cubic functions of strain. An analytic expression, linear in both e and e3, was developed to predict stress accumulation within the gap tissue as a function of time during distraction. Stress relaxation functions were described well by two-term Prony series. The two time constants determined from mechanical testing results were consistent, suggesting the presence of two fundamental physiologic relaxation processes. Gap tissue stresses were predicted to rise considerably during early stages of lengthening when distraction magnitudes exceeded the clinical norm of 0.25 mm. These differences in tension accumulation were less pronounced by the time lengthening was completed. Specifically, these results may in part explain clinical observations of decreased bone regeneration and altered tissue proliferation and differentiation at higher distraction rates. More generally, this work provides a framework for the rigorous characterization of the viscoelastic properties of biologic tissues ordinarily exposed to step strains.     

Changes in Large Pulmonary Arterial Viscoelasticity in Chronic Pulmonary Hypertension

Conduit pulmonary artery (PA) stiffening is characteristic of pulmonary arterial hypertension (PAH) and is an excellent predictor of mortality due to right ventricular (RV) overload. To better understand the impact of conduit PA stiffening on RV afterload, it is critical to examine the arterial viscoelastic properties, which require measurements of elasticity (energy storage behavior) and viscosity (energy dissipation behavior). Here we hypothesize that PAH leads to frequency-dependent changes in arterial stiffness (related to elasticity) and damping ratio (related to viscosity) in large PAs. To test our hypothesis, PAH was induced by the combination of chronic hypoxia and an antiangiogenic compound (SU5416) treatment in mice. Static and sinusoidal pressure-inflation tests were performed on isolated conduit PAs at various frequencies (0.01–20 Hz) to obtain the mechanical properties in the absence of smooth muscle contraction. Static mechanical tests showed significant stiffening of large PAs with PAH, as expected. In dynamic mechanical tests, structural stiffness (κ) increased and damping ratio (D) decreased at a physiologically relevant frequency (10 Hz) in hypertensive PAs. The dynamic elastic modulus (E), a material stiffness, did not increase significantly with PAH. All dynamic mechanical properties were strong functions of frequency. In particular, κ, E and D increased with increasing frequency in control PAs. While this behavior remained for D in hypertensive PAs, it reversed for κ and E. Since these novel dynamic mechanical property changes were found in the absence of changes in smooth muscle cell content or contraction, changes in collagen and proteoglycans and their interactions are likely critical to arterial viscoelasticity in a way that has not been previously described. The impact of these changes in PA viscoelasticity on RV afterload in PAH awaits further investigation.     

Characterization of structure and properties of bone by spectral measure method

Novel mathematical method called spectral measure method (SMM) is developed for characterization of bone structure and indirect estimation of bone properties. The spectral measure method is based on an inverse homogenization technique which allows to derive information about the structure of composite material from measured effective electric or viscoelastic properties. The mechanical properties and ability to withstand fracture depend on the structural organization of bone as a hierarchical composite. Information about the bone structural parameters is contained in the spectral measure in the Stieltjes integral representation of the effective properties. The method is based on constructing the spectral measure either by calculating it directly from micro-CT images or using measurements of electric or viscoelastic properties over a frequency range. In the present paper, we generalize the Stieltjes representation to the viscoelastic case and show how bone microstructure, in particular, bone volume or porosity, can be characterized by the spectral function calculated using measurements of complex permittivity or viscoelastic modulus. For validation purposes, we numerically simulated measured data using micro-CT images of cancellous bone. Recovered values of bone porosity are in excellent agreement with true porosity estimated from the micro-CT images. We also discuss another application of this method, which allows to estimate properties difficult to measure directly. The spectral measure method based on the derived Stieltjes representation for viscoelastic composites, has a potential for non-invasive characterization of bone structure using electric or mechanical measurements. The method is applicable to sea ice, porous rock, and other composite materials.     

Interrelation of creep and relaxation: a modeling approach for ligaments

Experimental data (Thornton et al., 1997) show that relaxation proceeds more rapidly (a greater slope on a log-log scale) than creep in ligament, a fact not explained by linear viscoelasticity. An interrelation between creep and relaxation is therefore developed for ligaments based on a single-integral nonlinear superposition model. This interrelation differs from the convolution relation obtained by Laplace transforms for linear materials. We demonstrate via continuum concepts of nonlinear viscoelasticity that such a difference in rate between creep and relaxation phenomenologically occurs when the nonlinearity is of a strain-stiffening type, i.e., the stress-strain curve is concave up as observed in ligament. We also show that it is inconsistent to assume a Fung-type constitutive law (Fung, 1972) for both creep and relaxation. Using the published data of Thornton et al. (1997), the nonlinear interrelation developed herein predicts creep behavior from relaxation data well (R > or = 0.998). Although data are limited and the causal mechanisms associated with viscoelastic tissue behavior are complex, continuum concepts demonstrated here appear capable of interrelating creep and relaxation with fidelity.     

The effects of modeling simplifications on craniofacial finite element models: the alveoli (tooth sockets) and periodontal ligaments

Several finite element models of a primate cranium were used to investigate the biomechanical effects of the tooth sockets and the material behavior of the periodontal ligament (PDL) on stress and strain patterns associated with feeding. For examining the effect of tooth sockets, the unloaded sockets were modeled as devoid of teeth and PDL, filled with teeth and PDLs, or simply filled with cortical bone. The third premolar on the left side of the cranium was loaded and the PDL was treated as an isotropic, linear elastic material using published values for Young's modulus and Poisson's ratio. The remaining models, along with one of the socket models, were used to determine the effect of the PDL's material behavior on stress and strain distributions under static premolar biting and dynamic tooth loading conditions. Two models (one static and the other dynamic) treated the PDL as cortical bone. The other two models treated it as a ligament with isotropic, linear elastic material properties. Two models treated the PDL as a ligament with hyperelastic properties, and the other two as a ligament with viscoelastic properties. Both behaviors were defined using published stress-strain data obtained from in vitro experiments on porcine ligament specimens. Von Mises stress and strain contour plots indicate that the effects of the sockets and PDL material behavior are local. Results from this study suggest that modeling the sockets and the PDL in finite element analyses of skulls is project dependent and can be ignored if values of stress and strain within the alveolar region are not required.