Probabilistic constitutive law for damage in ligaments

A new constitutive equation is presented to describe the damage evolution process in parallel-fibered collagenous tissues such as ligaments. The model is formulated by accounting for the fibrous structure of the tissues. The tissue’s stress is defined as the average of the collagen fiber’s stresses. The fibers are assumed to be undulated and straightened out at different stretches that are randomly defined according to a Weibull distribution. After becoming straight, each collagen fiber is assumed to be linear elastic. Damage is defined as a reduction in collagen fiber’s stiffness and occurs at different stretches that are also randomly defined by a Weibull distribution. Due to the lack of experimental data, the predictions of the constitutive equation are analyzed by varying the values of its structural parameters. Moreover, the results are compared with the available stress–strain data in the biomechanics literature that evaluate damage produced by subfailure stretches in rat medial collateral ligaments.     

A Nonlinear Constitutive Model for Stress Relaxation in Ligaments and Tendons

A novel constitutive model that describes stress relaxation in transversely isotropic soft collagenous tissues such as ligaments and tendons is presented. The model is formulated within the nonlinear integral representation framework proposed by Pipkin and Rogers (J. Mech. Phys. Solids. 16:59?C72, 1968). It represents a departure from existing models in biomechanics since it describes not only the strain dependent stress relaxation behavior of collagenous tissues but also their finite strains and transverse isotropy. Axial stress?Cstretch data and stress relaxation data at different axial stretches are collected on rat tail tendon fascicles in order to compute the model parameters. Toward this end, the rat tail tendon fascicles are assumed to be incompressible and undergo an isochoric axisymmetric deformation. A comparison with the experimental data proves that, unlike the quasi-linear viscoelastic model (Fung, Biomechanics: Mechanics of Living Tissues. Springer, New York, 1993) the constitutive law can capture the observed nonlinearities in the stress relaxation response of rat tail tendon fascicles.     

Fluid pressure driven fibril reinforcement in creep and relaxation tests of articular cartilage

Biological tissues exhibit diverse mechanical behaviors because of complex material properties. As has been shown for ligaments and intervertebral discs, mathematical models often appear to well predict load responses individually by adjusting model parameters, but likely fail to describe several different load responses simultaneously using the same model parameters. In the present study, we attempted to describe and explain both creep and relaxation responses of articular cartilage using a fibril-reinforced model, which has been successfully used to account for the load response of the relaxation tests of articular cartilage. Experiments were performed on bovine articular cartilage disks (n=8) using multi-step loading protocols, involving both creep and relaxation in each protocol. The experimental results indicated that mechanical changes, such as fiber recruitment in collagen network during stretch, recovered fully upon unloading. Creep loading did not affect relaxation properties, and vice versa. Relaxation proceeded much faster than creep, because of different fluid pressure profiles. The load sharing among the proteoglycan matrix, collagen network and fluid pressurization was predicted to differ for the creep and relaxation testing. The experimentally observed strong creep and relaxation responses in unconfined compression could not be predicted if either fibril reinforcement or fluid pressurization were neglected. It was essential to consider the interplay between nonlinear fibril reinforcement and fluid pressurization for the transient response (this interplay may be best termed as fluid pressure driven fibril reinforcement). Fibril reinforcement played a relatively insignificant role in the compressive load response at equilibrium, in agreement with previous findings for cartilage stress relaxation testing.     

Preparation and strength of poly(ethylene terephthalate) fiber bundles for model synthetic tendons

Poly(ethylene terephthalate) fibers suited for model synthetic tendons were prepared on a pilot-plant scale by additional drawing (postdrawing) of commercial textile fibers texturized by false twist. The modified fibers have a tensile modulus of 7-9 GPa, a one-minute creep modulus of 5-6 GPa, tensile strength 0.55-0.63 GPa, and strain at break of 16-19%. It is essential for their application in synthetic tendons that plastic deformation was reduced to about 0.5%, which could be completely eliminated by subsequent mechanical conditioning of fibers. Since the strength of model tendons consisting of 20 vol % of fibers and of a hydrogel matrix is primarily determined by the strength of the fibers, some aspects of the theory of strength of fiber bundles are briefly recapitulated. The prediction is compared with earlier experimental results obtained for the synthetic tendons. Such an approach, taking into account the fiber-strength distribution in the bundle, predicts a strength of model tendons lower by 20-40% than does the rule of mixtures.     

The role of viscoelasticity of collagen fibers in articular cartilage: axial tension versus compression

The role of viscoelasticity of collagen fibers in bovine articular cartilage was examined in compression and tension using stress relaxation measurements in the axial direction (normal to the articular surface). Experimentally, for a given axial strain, both peak and equilibrium loads were higher in tension than in compression, whereas stress relaxation was stronger in compression, as indicated by the higher peak-to-equilibrium ratios. A viscoelastic fibril-reinforced model including fluid flow was used for analysis of the experimental data. The collagen fibrillar matrix was assumed to be viscoelastic with a strain-dependent tensile modulus, and the nonfibrillar matrix was modeled as linearly elastic. For axial tension, collagen viscoelasticity was found to account for most of the stress relaxation, while the effects of fluid pressurization on the tensile stress were negligible. In contrast, for axial compression, the dominant mechanism for stress relaxation arose from fluid pressurization, while the associated relaxation in collagen fibers mainly resulted in an increase in radial strain. The effective Poisson's ratio, defined as the ratio of the radial and axial strains, was generally smaller in compression than in tension, and deviated from the true Poisson's ratio in tensile tests because of the frictional contacts between the specimen and the loading platens. Furthermore, lower collagen elasticity in the axial direction was observed than in the radial direction. This study illustrates the essential role of collagen viscoelasticity and interstitial fluid pressurization in the mechanical response of articular cartilage.     

Influence of physical exercise on aging rats. III. Life-long exercise modifies the aging changes of the mechanical properties of limb muscle tendons

We have previously shown that long-term regular physical exercise has a systemic influence on the rat by slowing the aging of its connective tissues, measured as thermal stability and biomechanical properties of tail tendons. This paper analyses whether the properties of limb muscle tendons are influenced not only by the aging process and the systemic effects of exercise but also from direct mechanical stimuli from long-term physical exercise. Male Sprague-Dawley rats were trained in a treadmill from the age of 5 to 23 months. The effects of training on muscle tendons were analyzed with respect to biomechanical properties. Also, the viscoelastic activation energies for interactions between collagen and the proteoglycan gel as well as between collagen fibrils were measured. Finally the asymptotes from the creep curves were calculated in order to estimate the magnitude of the viscoelastic creep. The effects of aging were analyzed with respect to the same parameters by comparing the group of 23-month-old sedentary rats with a 5-month-old baseline group. The biomechanical parameters did not change significantly with physical exercise. Neither did the activation energies change, but the asymptotes of the creep curves decreased, showing that there was less viscoelastic creep. Aging rendered the tendons significantly stronger and stiffer, increased the energy-absorbing capacity and decreased the strain values. The activation energies did not change with aging, but the high creep curve asymptote for the flexor tendons decreased. We conclude that aging rendered both types of tendons stiffer, and decreased their strain values at breaking point. Aging also increased the stress value, the energy absorption and the dry weight for the flexor tendon. Further, while physical exercise has a systemic delaying effect on age changes in connective tissues, in tendons subjected to substantial mechanical loads this effect as measured with biomechanical methods is counteracted by the optimization process elicited by the same physical exercise.     

Quantification of collagen organization using fractal dimensions and Fourier transforms

Collagen fibers and fibrils that comprise tendons and ligaments are disrupted or damaged during injury. Fibrillogenesis during healing produces a matrix that is initially quite disorganized, but remodels over time to resemble, but not replicate, the original roughly parallel microstructure. Quantification of these changes is traditionally a laborious and subjective task. In this work we applied two automated techniques, fast Fourier transformation (FFT) and fractal dimension analysis (FA) to quantify the organization of collagen fibers or fibrils. Using multi-photon images of collagen fibers obtained from rat ligament we showed that for healing ligaments, FA differentiates more clearly between the different time-points during healing. Using scanning electron microscopy images of overstretched porcine flexor tendon, we showed that combining FFT and FA measures distinguishes the damaged and undamaged groups more clearly than either method separately.     

骨性关节炎动物模型内侧副韧带的应力松弛

背景：韧带和其他生物软组织一样,具有黏弹性特性,其应力松弛和蠕变黏弹性特性是为适应人的生理功能需要而存在的。研究说明骨性关节炎对韧带力学性能具有一定影响。 目的：比较正常和骨性关节炎动物模型内侧副韧带的应力松弛流变特性,确定骨性关节炎对内侧副韧带应力松弛特性的影响。 方法：以闭合关节刻痕法复制骨性关节炎动物模型,取正常和骨性关节炎动物模型大鼠膝关节内侧副韧带各10个试样,进行应力松弛实验。应力松弛实验的应变增加速度为5%/s。设定实验时间为7 200 s,采集100个数据,观察应力松弛数据和曲线、应力与时间的变化规律,以一元线性回归分析的方法处理实验数据。 结果与结论：正常组和病态组试样应力松弛最初600 s变化较快,之后应力缓慢下降,正常组7 200 s应力松弛量0.47 MPa,病态组7 200 s应力松弛量0.29 MPa。模型组7 200 s应力松弛量显著低于正常组(P < 0.05),且应力松弛曲线是以对数关系变化的。说明骨性关节炎时可以使膝关节对内侧副韧带应力松弛量降低,对膝关节应力松弛特性具有一定影响。     

Mechanical characterization of collagen fibers and scaffolds for tissue engineering

Engineered tissues must utilize scaffolding biomaterials that support desired cellular functions and possess or can develop appropriate mechanical characteristics. This study assessed properties of collagen as a scaffolding biomaterial for ligament replacements. Mechanical properties of extruded bovine achilles tendon collagen fibers were significantly affected by fiber diameter, with smaller fibers displaying higher tangent moduli and peak stresses. Mechanical properties of 125 micrometer-diameter extruded fibers (tangent modulus of 359.6+/-28.4MPa; peak stress of 36.0+/-5.4MPa) were similar to properties reported for human ligaments. Scaffolds of extruded fibers did not exhibit viscoelastic creep properties similar to natural ligaments. Collagen fibers from rat tail tendon (a well-studied comparison material) displayed characteristic strain-softening behavior, and scaffolds of rat tail fibers demonstrated a non-intuitive relationship between tangent modulus and specimen length. Composite scaffolds (extruded collagen fibers cast within a gel of Type I rat tail tendon collagen) were maintained with and without fibroblasts under standard culture conditions for 25 days; cell-incorporated scaffolds displayed significantly higher tangent moduli and peak stresses than those without cells. Because tissue-engineered products must possess appropriate mechanical as well as biological/chemical properties, data from this study should help enable the development of improved tissue analogues.     

A viscoelastic poromechanical model of the knee joint in large compression

The elastic response of the knee joint in various loading and pathological conditions has been investigated using anatomically accurate geometry. However, it is still challenging to predict the poromechanical response of the knee in realistic loading conditions. In the present study, a viscoelastic, poromechanical model of the knee joint was developed for soft tissues undergoing large deformation. Cartilages and menisci were modeled as fibril-reinforced porous materials and ligaments were considered as fibril-reinforced hyperelastic solids. Quasi-linear viscoelasticty was formulated for the collagen network of these tissues and nearly incompressible Neo-Hookean hyperelasticity was used for the non-fibrillar matrix. The constitutive model was coded with a user defined FORTRAN subroutine, in order to use ABAQUS for the finite element analysis. Creep and stress relaxation were investigated with large compression of the knee in full extension. The contact pressure distributions were found similar in creep and stress relaxation. However, the load transfer in the joint was completely different in these two loading scenarios. During creep, the contact pressure between cartilages decreased but the pressure between cartilage and meniscus increased with time. This led to a gradual transfer of some loading from the central part of cartilages to menisci. During stress relaxation, however, both contact pressures decreased monotonically.     

Effect of Decorin and Dermatan Sulfate on the Mechanical Properties of a Neocartilage

Decorin is known to influence the size of collagen fibrils in ligaments and tendons and it has been hypothesized to provide a structural link between collagen fibrils in connective tissues, including cartilage. Coincidently, mechanical properties of skin, ligament, and tendons are altered in decorin knockout mice, suggesting it may influence the structural properties of tissue or tissue matrix organization. To further examine the role of decorin in the extracellular matrix development and subsequent material properties of cartilage, tissue (neocartilage) was grown in a 3D culture model using a pure population of genetically modified chondrocytes stably overexpressing decorin (DCN) or decorin lacking dermatan sulfate (MDCN). An empty vector (CON) served as a virus control. Following generation of the cartilage-like tissues, mechanical properties in tension and compression, collagen fibril diameter, matrix organization, and biochemistry of the tissue were determined. There were no differences between CON and DCN tissues in any parameter measured. In contrast, tissue generated in MDCN cultures was thinner, had higher collagen density, and higher elastic moduli as compared to both CON and DCN tissues. Considering there was no difference in stiffness between CON and DCN tissues, the notion that decorin contributes to the mechanical properties via load transfer was refuted in this model. However, contrasts in the mechanical properties of the MDCN tissue suggest that the dermatan sulfate chains on decorin influences the organization/maturation and resultant mechanical properties of the matrix by as an yet-unidentified regulatory mechanism.     

Biocompatibility of NDGA-polymerized collagen fibers. II. Attachment, proliferation, and migration of tendon fibroblasts in vitro

The material properties of collagen fibers polymerized with nordihydroguaiaretic acid (NDGA) are equivalent to native tendon, suggesting that NDGA crosslinking may provide a viable approach to stabilizing collagenous materials for use in repairing ruptured, lacerated, or surgically transected fibrous tissues, such as tendons and ligaments (Koob & Hernandez, Biomaterials, in press). The present study evaluated the biocompatibility of these fibers with cultured bovine tendon fibroblasts. Fibroblast attachment, migration, and proliferation on NDGA-crosslinked materials were compared to those on prepolymerized type I tendon collagen constructs as well as on tissue-culture-treated plastic. Fibroblast attachment on NDGA-crosslinked collagen fibrils was equivalent to attachment on plates coated with collagen alone. Over a period of 8 days in culture, attached fibroblasts proliferated on NDGA-crosslinked collagen at a rate identical to that of fibroblasts attached to native collagen. In order for the biomaterial effectively to bridge gaps in fibrous tissues, fibroblasts must be able to migrate and replicate on the bridging fiber. Control and crosslinked fibers were inserted in calf tendon explants, with a portion of the fiber extending out of the sectioned end of the tendon. Explants were cultured for 9 weeks, and the number of cells was measured at weekly intervals. Cells appeared on the fibers after 1 week of culture. By 2 weeks, cells had colonized the entire fiber. The number of cells continued to increase throughout the 9 weeks in culture, forming a layer several cells thick. Histologic analysis indicated that the fibroblasts populating the fibers appeared to originate in the epitenon. There was no difference in the rate of fibroblast migration and replication, nor in the ultimate number of colonizing cells, between control collagen fibers and NDGA-crosslinked fibers. NDGA-crosslinked fibers may provide a means of bridging gaps in ruptured, lacerated, or surgically transected tendons by providing a mechanically competent scaffold on which tendon fibroblasts can migrate, attach, and proliferate.     

Effect of X-ray irradiation on the elastic strain evolution in the mineral phase of bovine bone under creep and load-free conditions

Both the load partitioning between hydroxyapatite (HAP) and collagen during compressive creep deformation of bone and the HAP residual strain in unloaded bone have been shown in previous synchrotron X-ray diffraction studies to be affected by the X-ray irradiation dose. Here, through detailed analysis of the X-ray diffraction patterns of bovine bone, the effect of X-ray dose on (i) the rate of HAP elastic strain accumulation/shedding under creep conditions and (ii) the HAP lattice spacing and average root mean square (RMS) strain under load-free conditions are examined. These strain measurements exhibit three stages in response to increasing X-ray dose. Up to ～75 kGy (stage I) no effect of dose is observed, indicating a threshold behavior. Between ～75 and ～300 kGy (stage II) in unloaded bone the HAP d-spacing increases and the RMS strain decreases with dose, indicating strain relaxation of HAP. Furthermore, under constant compressive load creep conditions, the rate of compressive elastic strain accumulation in HAP decreases with increasing dose until, at ～115 kGy, it changes sign, indicating that the HAP phase is shedding load during creep deformation. These stage II behaviors are consistent with HAP–collagen interfacial damage, which allows the HAP elastic strain to relax within both the loaded and unloaded samples. Finally, for doses in excess of ～300 kGy (stage III, measured up to 7771 kGy) the HAP lattice spacing and RMS strain for load-free samples and the rate of HAP elastic strain shedding for crept samples remain independent of dose, suggesting a saturation of damage and/or stiffening of the collagen matrix due to intermolecular cross-linking.     

Novel system for engineering bioartificial tendons and application of mechanical load

Cells cultured in three-dimensional collagen gels express a more native state phenotype because they form a syncytial network that can be mechanically loaded. Moreover, cells remodel their matrix by eliminating water, and by reorganizing and aligning the collagen fibrils. Last, the ability to subject cells to mechanical loading in a native matrix is desirable because cells, in tissues as well as the matrix, bear strains and alter their expression profile consistent with either immobilization, moderate activity, or repetitive loading. This is the first report of a model bioreactor system to fabricate and culture tendon cell-populated, linear, tethered matrix constructs that can be mechanically loaded by a computer-driven, pressure-controlled system. Bioartificial tissues (BATs) as tendon constructs were molded in a novel, rubber bottom Tissue Train culture plate bearing nonwoven nylon mesh anchors at the east and west poles of each culture well. Mechanical loading was achieved by placing an Arctangle loading post (an Arctangle is a rectangle with curved short ends) beneath each well of the six-well culture plate and using vacuum to displace the flexible membrane downward, resulting in uniaxial strain on the BAT. BATs populated with avian flexor tendon cells expressed collagen genes I, III, and XII as well as aggrecan, fibronectin, prolyl hydroxylase, and tenascin, consistent with expression levels of cells grown on collagen-bonded two-dimensional surfaces or in native, whole, avian flexor tendon. Likewise, cells in BATs established a morphology of linearly arranged cells aligned with the principal strain direction as in fasicles of whole tendons. Last, BATs that were mechanically loaded had an ultimate tensile strength that was nearly 3-fold greater than that of nonloaded BATs in the first week of culture. Taken together, these results indicate that tendon cells fabricated in a mechanically loaded, linear collagen gel construct assume a phenotype that is similar to that of a native tendon in terms of appearance and expression and are stronger than nonexercised counterparts yet far weaker than native adult tendons. This technique represents a novel approach to culturing cells in a mechanically active, three-dimensional culture environment that can be readily used for the fabrication of tissue simulates for drug testing or tissue engineering.     

Viscoelasticity imaging using ultrasound: parameters and error analysis

Techniques are being developed to image viscoelastic features of soft tissues from time-varying strain. A compress-hold-release stress stimulus commonly used in creep-recovery measurements is applied to samples to form images of elastic strain and strain retardance times. While the intended application is diagnostic breast imaging, results in gelatin hydrogels are presented to demonstrate the techniques. The spatiotemporal behaviour of gelatin is described by linear viscoelastic theory formulated for polymeric solids. Measured creep responses of polymers are frequently modelled as sums of exponentials whose time constants describe the delay or retardation of the full strain response. We found the spectrum of retardation times tau to be continuous and bimodal, where the amplitude at each tau represents the relative number of molecular bonds with a given strength and conformation. Such spectra indicate that the molecular weight of the polymer fibres between bonding points is large. Imaging parameters are found by summarizing these complex spectral distributions at each location in the medium with a second-order Voigt rheological model. This simplification reduces the dimensionality of the data for selecting imaging parameters while preserving essential information on how the creeping deformation describes fluid flow and collagen matrix restructuring in the medium. The focus of this paper is on imaging parameter estimation from ultrasonic echo data, and how jitter from hand-held force applicators used for clinical applications propagate through the imaging chain to generate image noise.     

A Linear,Biphasic Model Incorporating a Brinkman Term to Describe the Mechanics of Cell-Seeded Collagen Hydrogels

Protein-based hydrogels are commonly used as in vitro models of native tissues because they can mimic specific aspects of the three-dimensional extracellular matrix present in vivo. Bulk mechanical stimulation is often applied to these gels to determine the response of embedded cells to biomechanical factors such as stress and strain. This study develops and applies a linear, biphasic formulation of hydrogel mechanics that includes a Brinkman term to account for viscous effects. The model is used to predict fluid pressure, relative velocity, and estimated shear stress exerted on cells seeded within a cyclically strained collagen hydrogel with and without imposed cross flow. The model was validated using a confined compression creep test of a cardiac fibroblast-seeded collagen type I hydrogel, and the effect of the added Brinkman term was assessed. The model indicated that the effects of strain and interstitial fluid flow are strongly interdependent in the collagen hydrogel. Our results suggest that the contribution of the Brinkman term is greater in protein hydrogels than in native tissues, and that studies that apply cyclic strain to cell-seeded hydrogels should account for the induced interstitial fluid flow. This study, therefore, has relevance to the increasing number of studies that examine cellular responses to mechanical stresses using in vitro hydrogel models.     

Interleukin-1beta increases elasticity of human bioartificial tendons

Stiffness is an important mechanical property of connective tissues, especially for tissues subjected to cyclic strain in vivo, such as tendons. Therefore, modulation of material properties of native or engineered tissues is an important consideration for tissue repair. Interleukin 1-beta (IL-1beta) is a cytokine most often associated in connective tissues with induction of matrix metalloproteinases and matrix destruction. However, IL-1beta may also be involved in constructive remodeling and confer a cell survival value to tenocytes. In this study, we investigated the effects of IL-1beta on the properties of human tenocyte-populated bioartificial tendons (BATs) fabricated in a novel three-dimensional (3D) culture system. IL-1beta treatment reduced the ultimate tensile strength and elastic modulus of BATs and increased the maximum strain. IL-1beta at low doses (1, 10 pM) upregulated elastin expression and at a high dose (100 pM) downregulated type I collagen expression. Matrix metalloproteinases, which are involved in matrix remodeling, were also upregulated by IL-1beta. The increased elasticity prevented BATs from rupture caused by applied strain. The results in this study suggest that IL-1beta may act as a defense/survival factor in response to applied mechanical loading. The balance between cell intrinsic strain and external matrix strain is important for maintaining the integrity of tendons.     

Homologous structure-function relationships between native fibrocartilage and tissue engineered from MSC-seeded nanofibrous scaffolds

Understanding the interplay of composition, organization and mechanical function in load-bearing tissues is a prerequisite in the successful engineering of tissues to replace diseased ones. Mesenchymal stem cells (MSCs) seeded on electrospun scaffolds have been successfully used to generate organized tissues that mimic fibrocartilages such as the knee meniscus and the annulus fibrosus of the intervertebral disc. While matrix deposition has been observed in parallel with improved mechanical properties, how composition, organization, and mechanical function are related is not known. Moreover, how this relationship compares to that of native fibrocartilage is unclear. Therefore, in the present work, functional fibrocartilage constructs were formed from MSC-seeded nanofibrous scaffolds, and the roles of collagen and glycosaminoglycan (GAG) in compressive and tensile properties were determined. MSCs deposited abundant collagen and GAG over 120 days of culture, and these extracellular molecules were organized in such a way that they performed similar mechanical functions to their native roles: collagen dominated the tensile response while GAG was important for compressive properties. GAG removal resulted in significant stiffening in tension. A similar stiffening response was observed when GAG was removed from native inner annulus fibrosus, suggesting an interaction between collagen fibers and their surrounding extrafibrillar matrix that is shared by both engineered and native fibrocartilages. These findings strongly support the use of electrospun scaffolds and MSCs for fibrocartilage tissue engineering, and provide insight on the structure-function relations of both engineered and native biomaterials.