Modeling shear behavior of the annulus fibrosus

Modeling the mechanical properties of the annulus fibrosus has two distinct challenges: the complex loading state experienced in vivo and the anisotropic, nonlinear nature of the tissue. Previous efforts to model the annulus fibrosus have not considered shear data in the analysis, yet the shear response may be critical to understanding tissue behavior and damage. In this study, we compared four hyperelastic constitutive models fitted to uniaxial and biaxial tension, confined compression, and shear experiments from the literature. Models were either directly based on Spencer’s formulation for a fiber-reinforced composite material with two equivalent fiber families or represented the annulus as two transversely isotropic materials. Each model was composed of additive strain energy terms that represent specific constituents of the annulus fibrosus (proteoglycan matrix, collagen fibers, and collagen crosslinks). Additionally, we investigated the effect of restricting the fibers such that they supported tensile loads only. Best fit coefficients for these models were calculated both including and excluding shear data from the regression. All of the models fit the data well when shear data was excluded from the regression; when shear data was included in the regression, two models that were based on Spencer’s formulation performed better than the others. None of the models could consistently predict data that was not included in the regression. Restricting the fibers to support only tensile loads had only a modest effect on the fit of the models, but did alter which constituent carried the majority of the strain energy in shear deformations. Our study suggests that a single hyperelastic model may capture the anisotropic behavior of the annulus fibrosus for multiple loading cases, including shear. However, care must be taken when extrapolating these models to additional deformations outside of the training dataset.     

Power type strain energy function model and prediction of the anisotropic mechanical properties of skin using uniaxial extension data

Many successful models to describe the biomechanical characteristics of planar biological soft tissues are based on strain energy function. However, the parameters in these models are determined by biaxial extension test, which might be difficult to exercise for certain types of soft tissue. This study presents a new constitutive model, the power type strain energy density function model (PTM), and a method to identify its material parameters for rabbit skin using uniaxial extension test of 4-direction strip samples. The abdominal skins from eight rabbits were taken to perform uniaxial tension tests in 7 different directions. The material parameters were identified for each subject based on any 4 out of 7 directions by applying some definite conditions of this issue. For each rabbit, the 35 groups of material parameters were consistent. The 7 material parameters in PTM were identified with root mean square errors <0.061. The results indicate that the material parameters of rabbit skin can be identified from uniaxial extension test data.     

Elastic Behavior of Porcine Coronary Artery Tissue Under Uniaxial and Equibiaxial Tension

The aim of this study was to characterize the nonlinear anisotropic elastic behavior of healthy porcine coronary arteries under uniaxial and equibiaxial tension. Porcine coronary tissue was chosen for its availability and similarity to human arterial tissue. A biaxial test device previously used to test human femoral arterial tissue samples (Prendergast, P. J., C. Lally, S. Daly, A. J. Reid, T. C. Lee, D. Quinn, and F. Dolan. ASME J. Biomech. Eng., Vol. 125, pp. 692-699, 2003) was further developed to test porcine coronary tissue specimens. The device applies an equal force to the four sides of a square specimen and therefore creates a biaxial stretch that demonstrates the anisotropy of arterial tissue. The nonlinear elastic behavior was marked in both uniaxial and biaxial tests. The tissue demonstrated higher stiffness in the circumferential direction in four out of eight cases subjected to biaxial tension. Even though anisotropy is demonstrated it is proposed that an isotropic hyperelastic model may adequately represent the properties of an artery, provided that an axial stretch is applied to the vessel to simulate the in vivo longitudinal tethering on the vessel. Isotropic hyperelastic models based on the Mooney-Rivlin constitutive equation were derived from the test data by averaging the longitudinal and circumferential equibiaxial data. Three different hyperelastic models were established to represent the test specimens that exhibited a high stiffness, an average stiffness, and a low stiffness response; these three models allow the analyst to account for the variability in the arterial tissue mechanical properties. These models, which take account of the nonlinear elastic behavior of coronary tissue, may be implemented in finite element models and used to carry out preclinical tests of intravascular devices. The errors associated with the hyperelastic models when fitting to both the uniaxial and equibiaxial data for the low stiffness, average stiffness, and high stiffness models were found to be 0.836, 5.206, and 2.980, respectively.     

Biodegradation of polyurethane under fatigue loading

A method utilizing expansion of a diaphragm-type film specimen was developed to study in vitro biodegradation of poly(etherurethane urea) (PEUU) under conditions of dynamic loading (fatigue). A finite element model was used to describe the strain state, which ranged from uniaxial at the edges of the film to balanced biaxial tensile strain at the center. During testing, the film was exposed to a H(2)O(2)/CoCl(2) solution, which simulated in vivo oxidative biodegradation of PEUU. The extent of chemical degradation was determined by infrared analysis. Physical damage of the film surface was characterized by optical microscopy and scanning electron microscopy. Dynamic loading did not affect the rate of degradation relative to unstressed and constant stress (creep) controls in regions of the film that experienced primarily uniaxial fatigue; however, degradation was accelerated in regions that experienced balanced biaxial or almost balanced biaxial fatigue. It was concluded that the combination of dynamic loading and biaxial tensile strain accelerated oxidative degradation in this system. Chemical degradation produced a brittle surface layer that was marked by numerous pits and dimples. Physical damage of the surface in the form of cracking occurred only in fatigue experiments. Cracking was not observed in unstressed or creep tests. Cracks initiated at the dimples produced by chemical degradation, and propagated in a direction that was determined by the strain state.     

Ab initio elastic properties and tensile strength of crystalline hydroxyapatite

We report elastic constant calculation and a “theoretical” tensile experiment on stoichiometric hydroxyapatite (HAP) crystal using an ab initio technique. These results compare favorably with a variety of measured data. Theoretical tensile experiments are performed on the orthorhombic cell of HAP for both uniaxial and biaxial loading. The results show considerable anisotropy in the stress–strain behavior. It is shown that the failure behavior of the perfect HAP crystal is brittle for tension along the z-axis with a maximum stress of 9.6 GPa at 10% strain. Biaxial failure envelopes from six “theoretical” loading tests show a highly anisotropic pattern. Structural analysis of the crystal under various stages of tensile strain reveals that the deformation behavior manifests itself mainly in the rotation of the PO₄ tetrahedron with concomitant movements of both the columnar and axial Ca ions. These results are discussed in the context of mechanical properties of bioceramic composites relevant to mineralized tissues.     

Biaxial stress-relaxation in skin

Results of 58 biaxial and uniaxial stress-relaxation tests on flat specimens of abdominal rabbit skin are analyzed. It is shown that rabbit skin is a nonlinear viscoelastic material, i.e., the relaxation mode of each stress component depends on the strain (or alternatively on the initial stress). The dependence of the relaxation mode on the initial stress is investigated in terms of a modified power law with three parameters (Eq. (15)) which are functions of the initial stress. The differences and similarities between uniaxial and biaxial stress relaxation tests and the effect of the tissue “preconditioning” is discussed. On the basis of the results a theory is developed for the uniaxial viscoelastic behavior of rabbit skin tissue when small incremental strains are super-imposed on large constant deformation.     

Effect of strain and strain rate on fatigue-accelerated biodegradation of polyurethane

A diaphragm-type film specimen was used to study in vitro degradation of poly(etherurethane urea) (PEUU) under conditions of dynamic loading. This geometry allowed both uniaxial and biaxial loading in a single experiment. During testing, the film was exposed to a H(2)O(2)/CoCl(2) solution that simulated in vivo oxidation of PEUU. The combination of dynamic loading and biaxial tensile strain accelerated oxidative degradation. The effects of biaxial strain magnitude and strain rate were examined separately by increasing the frequency of fatigue loading from 0 to 1 Hz with constant maximum biaxial strain and by changing the maximum biaxial strain while maintaining constant strain rate. In the ranges of biaxial strain energy (0.17 to 0.55 MPa) and strain rate (0 to 46% s(-1)) tested, the rate of degradation increased with increasing strain rate whereas strain magnitude had essentially no effect on degradation rate. Although loading conditions affected the rate of oxidative degradation, ATR-FTIR analysis suggested that in all cases the mechanism of degradation did not change. Chemical degradation produced a brittle crosslinked surface layer marked by dimpling and pitting, as observed with scanning electron microscopy. Pits served as stress concentrators and initiated environmental stress cracks under dynamic loading but not under static (creep) loading. Small pits were sufficient to initiate cracks at higher strain rates whereas only large pits initiated cracks at lower strain rates. Consequently, a higher strain rate produced more profuse cracking.     

Mechanical Behavior of Annulus Fibrosus: A Microstructural Model of Fibers Reorientation

Experimental uniaxial tensile tests have been carried out on annulus tissue samples harvested on pig and lamb lumbar intervertebral discs. When subjecting the samples to loading cycles, the stress–strain curves exhibit strong nonlinearities and hysteresis. This particular behavior results from the anisotropic microstructure of annulus tissue composed of woven oriented collagen fibers embedded in the extracellular matrix. During uniaxial tension, the collagen fibers reorient toward the loading direction increasing its global stiffness. To describe this behavior, we propose a heuristic two-dimensional rheological model based on three mechanical and one geometrical characteristics. The latter one is the fibers orientation angle becoming the key parameter that govern the macroscopic mechanical behavior. The experimental results are used to identify the physical properties associated with the rheological model, leading to an accurate representation of the stress–strain curve over a complete loading cycle. In this framework, the fibers reorientation can solely account for the rigidity increase while the hysteresis is associated with liquid viscous flows through the matrix. Based on this representation, unusual coupling effects between strains and fluid flows can be observed, that would significantly affect the cell nutrients transport mechanisms.     

Cumulative microdamage model to describe the hysteresis of living tissue

A hysteresis model based on a cumulative microfailure criterion has been proposed to describe the stress-stretch behavior in response to external mechanical loading in strip biaxial deformation of living animal tissue (cat's mesentery). The strain energy density representation used in the hysteresis model appears to be adequately versatile to describe the tissue response over a wide range of stretch. The model is thus useful in interpreting the influence of the material parameters on the birefringence of the tissue as well as on the magnitude and shape of the stress response. Photomicrographs of the tissue have been correlated with the cumulative microfailure model. The theoretical treatment is also a valuable tool in elucidating the full consequences of the mesentery'sin situ biaxial tension in the strip biaxial mode of testing. Incorporation of non-Gaussian chain statistics and network theory into the microdamage model would be the next logical step in extending its range of application. Financial support from California Institute of Technology President's Fund 009 is gratefully acknowledged.     

Effect of strain rate on tensile properties of sheep disc anulus fibrosus.

We investigated the effect of loading rate on tensile properties of sheep bone-anulus-bone specimens in axial direction. Disc anulus Samples with adjacent bone attachments were prepared from lateral, posterior and anterior regions of sheep lumbar spinal segments. The specimens were then tested at different strain rates under non-destructive cyclic tensile loading followed by destructive tensile loading. Each specimen was prepared by embedding the bony parts in the polymethylmetacrylate (PMMA) exposing the anulus portion to support tension. The results of non-destructive cyclic tests indicated a decrease in the hysteresis energy loss as strain rate increased. In the destructive tests, no significant differences in ultimate stress, ultimate strain and strain energy density were observed at different strain rates or annulus locations. However, there was a significant increase in the modulus at linear region as strain rate increased. Two major modes of failure were observed; rupture in the anulus mid-substance and at the anulus-endplate junction. The former failure was more frequent with no clear pattern across strain rates and locations, while the latter failure at anulus-endplate junction occurred primarily at slow strain rates.     

A constitutive model of the posterior cruciate ligament

Further to our previous work on the development of a general constitutive framework for transversely isotropic viscohyperelasticity (Limbert, G, Middleton, J. A transversely isotropic viscohyperelastic material. Application to the modelling of biological soft connective tissues. Int J Solids Struct 2004;41(15):4237-60.), we propose a phenomenological constitutive law to describe the anisotropic viscohyperelastic behaviour of the human posterior cruciate ligament (PCL) at high strain rates. The mechanical formulation is based on the definition of a Helmholtz free energy function containing a hyperelastic and a viscous potential. The equations are valid for arbitrary kinematics and satisfy elemental thermodynamic principles. Identification of the constitutive model with experimental data obtained from human PCL specimens was performed and showed the ability of the model to capture accurately the mechanical characteristics of the PCL at various strain rates. Influence of the isotropic and directional viscous stress responses on the global mechanical response are discussed in connection with the modelling hypotheses. This work was motivated by the need to provide an accurate constitutive model of the PCL to be used in finite element analyses of human occupants in car crash simulations. Besides uniaxial tests along the natural fibre orientation of the PCL, additional tests such as equibiaxial, strip biaxial compression-tension and shear tests were also performed in order to assess the physical response of the model in different loading situations. It was found that the model performed as well in these conditions.     

Molecular orientation of collagen in intact planar connective tissues under biaxial stretch

Understanding of the mechanical behavior of collagenous tissues at different size scales is necessary to understand their physiological function as well as to guide their use as heterograft biomaterials. We conducted a first investigation of the kinematics of collagen at the molecular and fiber levels under biaxial stretch in an intact planar collagenous tissue. A synchrotron small angle X-ray scattering (SAXS) technique combined with a custom biaxial stretching apparatus was used. Collagen fiber behavior under biaxial stretch was then studied with the same specimens using small angle light scattering (SALS) under identical biaxial stretch states. Both native and glutaraldehyde modified bovine pericardium were investigated to explore the effects of chemical modification to collagen. Results indicated that collagen fiber and molecular orientation did not change under equibiaxial strain, but were observed to profoundly change under uniaxial stretch. Interestingly, collagen molecular strain initiated only after approximately 15% global tissue strain, potentially due to fiber-level reorganization occurring prior to collagen molecule loading. Glutaraldehyde treatment also did not affect collagen molecular strain behavior, indicating that chemical fixation does not alter intrinsic collagen molecular stiffness. No detectable changes in the angular distribution and D-period strain were found after 80 min of stress relaxation. It can be speculated that other mechanisms may be responsible for the reduction in stress with time under biaxial stretch. The results of this first study suggest that collagen fiber/molecular kinematics under biaxial stretch are more complex than under uniaxial deformation, and warrant future studies.     

Contractile tension and beating rates of self-exciting monolayers and 3D-tissue constructs of neonatal rat cardiomyocytes

The CellDrum technology (The term ‘CellDrum technology’ includes a couple of slightly different technological setups for measuring lateral mechanical tension in various types of cell monolayers or 3D-tissue constructs) was designed to quantify the contraction rate and mechanical tension of self-exciting cardiac myocytes. Cells were grown either within flexible, circular collagen gels or as monolayer on top of respective 1-μm thin silicone membranes. Membrane and cells were bulged outwards by air pressure. This biaxial strain distribution is rather similar the beating, blood-filled heart. The setup allowed presetting the mechanical residual stress level externally by adjusting the centre deflection, thus, mimicking hypertension in vitro. Tension was measured as oscillating differential pressure change between chamber and environment. A 0.5-mm thick collagen-cardiac myocyte tissue construct induced after 2 days of culturing (initial cell density 2 × 10⁴ cells/ml), a mechanical tension of 1.62 ± 0.17 μN/mm². Mechanical load is an important growth regulator in the developing heart, and the orientation and alignment of cardiomyocytes is stress sensitive. Therefore, it was necessary to develop the CellDrum technology with its biaxial stress–strain distribution and defined mechanical boundary conditions. Cells were exposed to strain in two directions, radially and circumferentially, which is similar to biaxial loading in real heart tissues. Thus, from a biomechanical point of view, the system is preferable to previous setups based on uniaxial stretching.     

Differences in the mechanical behavior of cortical bone between compression and tension when subjected to progressive loading

The hierarchical arrangement of collagen and mineral into bone tissue presumably maximizes fracture resistance with respect to the predominant strain mode in bone. Thus, the ability of cortical bone to dissipate energy may differ between compression and tension for the same anatomical site. To test this notion, we subjected bone specimens from the anterior quadrant of human cadaveric tibiae to a progressive loading scheme in either uniaxial tension or uniaxial compression. One tension (dog-bone shape) and one compression specimen (cylindrical shape) were collected each from tibiae of nine middle aged male donors. At each cycle of loading-dwell-unloading-dwell-reloading, we calculated maximum stress, permanent strain, modulus, stress relaxation, time constant, and three pathways of energy dissipation for both loading modes. In doing so, we found that bone dissipated greater energy through the mechanisms of permanent and viscoelastic deformation in compression than in tension. On the other hand, however, bone dissipated greater energy through the release of surface energy in tension than in compression. Moreover, differences in the plastic and viscoelastic properties after yielding were not reflected in the evolution of modulus loss (an indicator of damage accumulation), which was similar for both loading modes. A possible explanation is that differences in damage morphology between the two loading modes may favor the plastic and viscoelastic energy dissipation in compression, but facilitate the surface energy release in tension. Such detailed information about failure mechanisms of bone at the tissue-level would help explain the underlying causes of bone fractures.     

Contractions,a risk for premature rupture of fetal membranes: A new protocol with cyclic biaxial tension

This study aims at investigating the effect of repeated mechanical loading on the rupture and deformation properties of fetal membranes. Ten membranes delivered by cesarean sections were tested using a custom-built inflation device which provides a multi-axial stress state. For each membrane, a group of samples was first cyclically stretched by application of pressure ranging between 10 and 40 mmHg. After cycles, samples were subjected to inflation up to rupture. Differences between mechanical parameters from cycled and uncycled samples were analyzed. Ten cycles at 40% of mean critical membrane tension—representative of mean physiologic contractions—did not affect strength and stiffness of fetal membranes but reduced the work to rupture, thus indicating that contractions might increase the risk of premature rupture of the membrane. Cyclic testing demonstrated a large hysteresis loop and irreversible deformation on the first cycle, followed by rapid stabilization on subsequent cycles. In 80% of tests, amnion ruptured first and at the periphery of the sample, under uniaxial strain state. Chorion ruptured at higher deformation levels in the middle, under biaxial strain state.     

Stress State and Strain Rate Dependence of the Human Placenta

Maternal trauma (MT) in automotive collisions is a source of injury, morbidity, and mortality for both mothers and fetuses. The primary associated pathology is placental abruption in which the placenta detaches from the uterus leading to hemorrhaging and termination of pregnancy. In this study, we focused on the differences in placental tissue response to different stress states (tension, compression, and shear) and different strain rates. Human placentas were obtained (n?=?11) for mechanical testing and microstructure analysis. Specimens (n?=?4+) were tested in compression, tension, and shear, each at three strain rates (nine testing protocols). Microstructure analysis included scanning electron microscopy, histology, and interrupted mechanical tests to observe tissue response to various loading states. Our data showed the greatest stiffness in tension, followed by compression, and then by shear. The study concludes that mechanical behavior of human placenta tissue (i) has a strong stress state dependence and (ii) behaves in a rate dependent manner in all three stress states, which had previously only been shown in tension. Interrupted mechanical tests revealed differences in the morphological microstructure evolution that was driven by the kinematic constraints from the different loading states. Furthermore, these structure-property data can be used to develop high fidelity constitutive models for MT simulations.