Nonlinear Viscoelasticity in Rabbit Medial Collateral Ligament

The goal of this study was to characterize the viscoelastic behavior of the rabbit medial collateral ligament (MCL) at multiple levels of strain (between 0% and approximately 5%) and their corresponding stresses (between 0 and approximately 55 MPa) for stress relaxation and creep, respectively. We hypothesized that in the rabbit MCL the rate of stress relaxation would be strain dependent and the rate of creep would be stress dependent. Thirty MCLs from 15 rabbits were tested ex vivo for this study. Results show that within the physiologically relevant region of ligament behavior, the rate of stress relaxation is strain dependent in the rabbit MCL, with the rate of relaxation decreasing with increasing tissue strain. The rate of creep is stress dependent in the rabbit MCL, with the rate of creep decreasing with increasing stress. These results support our hypothesis, with the greatest nonlinearities in a physiologically relevant region of loading. As such, these nonlinearities should be considered when quantifying ligament viscoelastic behavior with a rabbit model.     

Viscoelastic Relaxation and Recovery of Tendon

Tendons exhibit complex viscoelastic behaviors during relaxation and recovery. Recovery is critical to predicting behavior in subsequent loading, yet is not well studied. Our goal is to explore time-dependent recovery of these tendons after loading. As a prerequisite, their strain-dependent viscoelastic behaviors during relaxation were also characterized. The porcine digital flexor tendon was used as a model of tendon behavior. Strain-dependent relaxation was observed in tests at 1, 2, 3, 4, 5, and 6% strain. Recovery behavior of the tendon was examined by performing relaxation tests at 6%, then dropping to a low but nonzero strain level. Results show that the rate of relaxation in tendon is indeed a function of strain. Unlike previously reported tests on the medial collateral ligament (MCL), the relaxation rate of tendons increased with increased levels of strain. This strain-dependent relaxation contrasts with quasilinear viscoelasticity (QLV), which predicts equal time dependence across various strains. Also, the tendons did not recover to predicted levels by nonlinear superposition models or QLV, though they did recover partially. This recovery behavior and behavior during subsequent loadings will then become problematic for both quasilinear and nonlinear models to correctly predict.     

An Evaluation of the Quasi-Linear Viscoelastic Properties of the Healing Medial Collateral Ligament in a Goat Model

The viscoelastic properties of the healing medial collateral ligament (MCL) at 12 weeks after isolated injury were investigated in a goat model. The stress-strain relationships, static and cyclic stress-relaxation behaviors of the healing MCL up to 5% strain were determined experimentally using a femur-MCL-tibia complex. These experimental data were used in combination with the quasi-linear viscoelastic (QLV) theory of Fung (1972) to characterize the reduced relaxation function, G(t) (described by constants C, tau1, and tau2) and the elastic response, sigmae(epsilon) (described by constants A and B) of this tissue. It was found that the percentage of stress relaxation for the healing MCLs was significantly greater than those for sham-operated controls (49.0 +/- 12.1% vs. 26.5 +/- 8.1%, respectively; p < 0.05). The product of constants A x B, i.e. the initial slope of the stress-strain curves, was found to be significantly lower for healing MCLs compared to those for sham-operated controls (32.9 +/- 15.8 MPa vs. 118.8 +/- 48.3 MPa; p < 0.05). The dimensionless constant C, i.e. the magnitude of the viscous response, was nearly three times greater for healing MCLs, while constant tau1 was found to be similar between the two groups (0.80 +/- 0.43 s vs. 0.89 +/- 0.52 s, respectively). Constant tau2 for the healing MCL was significantly less than the controls (1269 +/- 38 s vs. 1845 +/- 431 s; p < 0.05) indicating that the stress relaxation reached a plateau earlier. These constants of the QLV theory used to describe the healing MCL were validated for the strain level utilized in this experiment (approximately equal to 4.5%) by predicting the peak stresses during a cyclic stress-relaxation experiment. The theoretically determined values closely matched the experimentally measured values. Thus, this study demonstrates that the QLV theory could be successfully used to describe the viscoelastic behavior of the MCL during the early phases of healing.     

Viscoelastic effects during loading play an integral role in soft tissue mechanics

Viscoelastic relaxation during tensioning is an intrinsic protective mechanism of biological soft tissues. However, current viscoelastic characterization methodologies for these tissues either negate this important behavior or provide correction methods that are severely restricted to a specific viscoelastic formulation and/or assume an a priori (linear) strain ramp history. In order to address these shortcomings, we present a novel finite ramp time correction method for stress relaxation experiments (to incorporate relaxation manifested during loading) that is independent of a specific viscoelastic formulation and can accommodate an arbitrary strain ramp history. We demonstrate transferability of our correction method between viscoelastic formulations by applying it to quasi-linear viscoelastic (QLV) and fully nonlinear viscoelastic constitutive equations. The errors associated with currently accepted methodologies for QLV and fully nonlinear viscoelastic formulations are elucidated. Our correction method is validated by demonstrating the ability of its fitted parameters to predict an independent cyclic experiment across multiple strain amplitudes and frequencies. The results presented herein: (i) indicate that our correction method significantly reduces the errors associated with previous methodologies; and (ii) demonstrate the necessity for the use of a fully nonlinear viscoelastic formulation, which incorporates relaxation manifested during loading, to model the viscoelastic behavior of biological soft tissues.     

Nonlinear Viscoelastic Behavior of Human Knee Ligaments Subjected to Complex Loading Histories

The nonlinear viscoelastic structural response of the major human knee ligaments when subjected to complex loading histories is investigated, with emphasis on the collateral ligaments. Bone-ligament-bone specimens are tested in knee distraction loading, where the ligaments are in the anatomical position corresponding to a fully extended knee. Temporal nonlinearities for time scales in the range of s are characterized with a dedicated series of loading histories. In particular, the response to several complex sequences of step-and-hold tests and loading-unloading cycles is investigated. The separability of the time and deformation dependent behavior, as assumed for the often used quasi linear viscoelastic (QLV) theory, is found to be insufficient for describing the response in the time range considered. Non-recoverable inelastic flow is observed in this time range. A phenomenological 1-dimensional nonlinear viscoelastic model that qualitatively describes the experimentally observed inelastic phenomena is presented.     

Fatigue is More Damaging than Creep in Ligament Revealed by Modulus Reduction and Residual Strength

Following injury of a complementary joint restraint, ligaments can be subjected to higher than normal stresses. Normal ligaments are exposed to static (creep) and cyclic (fatigue) loading from which damage can accumulate at these higher than normal stresses. This study tracked damage accumulation during creep and fatigue loading of normal rabbit medial collateral ligaments (MCLs) over a range of stresses, using modulus reduction as a marker of damage. Creep tests were interrupted occasionally with unloading/reloading cycles to measure modulus. Test stresses were normalized to ultimate tensile strength (UTS): 60%, 30%, and 15% UTS. Not all creep and fatigues tests progressed until rupture but were stopped and followed by an assessment of the residual strength of that partially damaged ligament using a monotonic failure test. Fatigue loading caused earlier modulus reduction than creep. Modulus reduction occurred at lower increases in strain (strain relative to initial strain) for fatigue than creep. In other words, at the same time or increase in strain, fatigue is more damaging than creep because the modulus ratio reduction is greater. These findings suggest that creep and fatigue have different strain and damage mechanisms. Ligaments exposed to creep or fatigue loading which produced a modulus reduction had decreased residual strength and increased toe-region strain in a subsequent monotonic failure test. This finding confirmed that modulus reduction during creep and fatigue is a suitable marker of partial damage in ligament. Cyclic loading caused damage earlier than static loading, likely an important consideration when ligaments are loaded to higher than normal magnitudes following injury of a complementary joint restraint.     

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。     

A Progressive Rupture Model of Soft Tissue Stress Relaxation

A striking feature of stress relaxation in biological soft tissue is that it frequently follows a power law in time with an exponent that is independent of strain even when the elastic properties of the tissue are highly nonlinear. This kind of behavior is an example of quasi-linear viscoelasticity, and is usually modeled in a purely empirical fashion. The goal of the present study was to account for quasi-linear viscoelasticity in mechanistic terms based on our previously developed hypothesis that it arises as a result of isolated micro-yield events occurring in sequence throughout the tissue, each event passing the stress it was sustaining on to other regions of the tissue until they themselves yield. We modeled stress relaxation computationally in a collection of stress-bearing elements. Each element experiences a stochastic sequence of either increases in elastic equilibrium length or decreases in stiffness according to the stress imposed upon it. This successfully predicts quasi-linear viscoelastic behavior, and in addition predicts power-law stress relaxation that proceeds at the same slow rate as observed in real biological soft tissue.     

Microstructural morphology in the transition region between scar and intact residual segments of a healing rat medial collateral ligament.

This study used a rat model to investigate the microstructural organization of collagen through the transition from scar to intact residual segments of a healing medial collateral ligament (MCL). Twenty-two male retired breeder Sprague-Dawley rats were randomly separated into two groups. Eleven underwent surgical transections of both MCLs and were allowed unrestricted cage activity until euthanized two weeks post surgery. The remaining eleven rats were used as normal controls. All 44 MCLs were harvested including intact femoral and tibial insertions and prepared for scanning electron microscopy (SEM) imaging. At harvest the scar region in the healing ligaments was more translucent than the normal tissue. Ligaments were viewed from femoral to tibial insertions at magnifications of 100X through 20,000X. Tissue away from the scar region in the transected MCLs was indistinguishable from normal tissue in uninjured ligaments. Collagen fibers and fibrils in these tissues were more aligned along the longitudinal axis of the ligament than in the scar tissue. Continuity of collagen fibers and fibrils were consistently observed from the residual portions of the transected ligament through the scar region. Bifurcations/fusions, but no anastomoses, in fibers and fibrils were observed in both normal and scar tissues of ligaments. Qualitatively, bifurcations were encountered more frequently in scar tissue. In the transition region, larger diameter fibers from the residual tissue bifurcated into smaller diameter fibrils in the scar. This connection between larger diameter and smaller diameter fibers and fibrils indicates that bifurcations/fusions are likely to be the dominant way in which force is transmitted from a region with larger fibrils (residual ligament) into and through a region with smaller fibrils (scar).     

Microstructural Morphology in the Transition Region Between Scar and Intact Residual Segments of a Healing Rat Medial Collateral Ligament

This study used a rat model to investigate the microstructural organization of collagen through the transition from scar to intact residual segments of a healing medial collateral ligament (MCL). Twenty-two male retired breeder Sprague-Dawley rats were randomly separated into two groups. Eleven underwent surgical transections of both MCLs and were allowed unrestricted cage activity until euthanized two weeks post surgery. The remaining eleven rats were used as normal controls. All 44 MCLs were harvested including intact femoral and tibial insertions and prepared for scanning electron microscopy (SEM) imaging. At harvest the scar region in the healing ligaments was more translucent than the normal tissue. Ligaments were viewed from femoral to tibial insertions at magnifications of 100X through 20,000X. Tissue away from the scar region in the transected MCLs was indistinguishable from normal tissue in uninjured ligaments. Collagen fibers and fibrils in these tissues were more aligned along the longitudinal axis of the ligament than in the scar tissue. Continuity of collagen fibers and fibrils were consistently observed from the residual portions of the transected ligament through the scar region. Bifurcations/fusions, but no anastomoses, in fibers and fibrils were observed in both normal and scar tissues of ligaments. Qualitatively, bifurcations were encountered more frequently in scar tissue. In the transition region, larger diameter fibers from the residual tissue bifurcated into smaller diameter fibrils in the scar. This connection between larger diameter and smaller diameter fibers and fibrils indicates that bifurcations/fusions are likely to be the dominant way in which force is transmitted from a region with larger fibrils (residual ligament) into and through a region with smaller fibrils (scar).     

Combined treatment of therapeutic laser and herbal application improves the strength of repairing ligament

The present study investigated the effects of combined therapeutic laser and herbal medication protocols on injured medial collateral ligaments (MCLs) of rat knees. Fully 36 rats were evenly divided into 9 groups. Right MCLs of groups 1 to 6 and 8 were transected, while that of groups 7 and 9 remained intact. After surgery, group 1 was treated with 1 session of high-dosed laser; group 2 with 9 sessions of low-dosed laser; group 3 with an herbal plaster; groups 4 and 5 received combined treatments of groups 1 and ss and 2, and 3 respectively; groups 6 and 7 received only bandaging; groups 8 and 9 received placebo laser and no treatment, respectively. All MCLs were subjected to biomechanical testing at 3 weeks postsurgery. Results revealed significant differences among groups in ultimate tensile strength (UTS) and stiffness (p < 0.01). Combination of multiple low-dosed laser treatment with herbal treatment (group 5) resulted in higher UTS than either no treatment (groups 6 and 8), single high-dosed laser treatment (group 1), multiple low-dosed laser treatment (group 2), or herbal treatment (group 2) alone. We concluded that combined applications of laser and herb can enhance further biomechanical properties of repairing rat MCLs than separate applications at 3 weeks postinjury.     

Collagen fibril diameters in the healing adult rabbit medial collateral ligament.

This study was carried out to test the hypothesis that improvements in ligament scar mechanical behavior during healing may be related, in part, to increases in collagen fibril diameters. Forty-eight adult female New Zealand White rabbits had standardized midsubstance gap injuries created in their right medial collateral ligaments (MCLs) and were allowed normal cage activity until sacrifice in groups of 12 at 3, 6, 14 or 40 weeks post-injury. Eight animals in each group had both MCLs tested biomechanically while 4 animals had transmission EM investigation of midsubstance collagen fibril diameters by a standardized protocol. Results of mechanical tests showed a three- to fourfold increase in scar strength and stiffness over the intervals of healing studied while there was no change in collagen mean fibril minimum diameters. These results demonstrate no correlation between material or structural properties of scar and collagen fibril diameters in this model of healing and suggest that other mechanisms for scar mechanical improvement under these conditions must be investigated.     

Combined Treatment of Therapeutic Laser and Herbal Application Improves the Strength of Repairing Ligament

The present study investigated the effects of combined therapeutic laser and herbal medication protocols on injured medial collateral ligaments (MCLs) of rat knees. Fully 36 rats were evenly divided into 9 groups. Right MCLs of groups 1 to 6 and 8 were transected, while that of groups 7 and 9 remained intact. After surgery, group 1 was treated with 1 session of high-dosed laser; group 2 with 9 sessions of low-dosed laser; group 3 with an herbal plaster; groups 4 and 5 received combined treatments of groups 1 and ß and 2, and 3 respectively; groups 6 and 7 received only bandaging; groups 8 and 9 received placebo laser and no treatment, respectively. All MCLs were subjected to biomechanical testing at 3 weeks postsurgery. Results revealed significant differences among groups in ultimate tensile strength (UTS) and stiffness (p < 0.01). Combination of multiple low-dosed laser treatment with herbal treatment (group 5) resulted in higher UTS than either no treatment (groups 6 and 8), single high-dosed laser treatment (group 1), multiple low-dosed laser treatment (group 2), or herbal treatment (group 2) alone. We concluded that combined applications of laser and herb can enhance further biomechanical properties of repairing rat MCLs than separate applications at 3 weeks postinjury.     

Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries

Viscoelastic models can be used to better understand arterial wall mechanics in physiological and pathological conditions. The arterial wall reveals very slow time-dependent decays in uniaxial stress-relaxation experiments, coherent with weak power-law functions. Quasi-linear viscoelastic (QLV) theory was successfully applied to modeling such responses, but an accurate estimation of the reduced relaxation function parameters can be very difficult. In this work, an alternative relaxation function based on fractional calculus theory is proposed to describe stress relaxation experiments in strips cut from healthy human aortas. Stress relaxation (1 h) was registered at three incremental stress levels. The novel relaxation function with three parameters was integrated into the QLV theory to fit experimental data. It was based in a modified Voigt model, including a fractional element of order alpha, called spring-pot. The stress-relaxation prediction was accurate and fast. Sensitivity plots for each parameter presented a minimum near their optimal values. Least-squares errors remained below 2%. Values of order alpha = 0.1-0.3 confirmed a predominant elastic behavior. The other two parameters of the model can be associated to elastic and viscous constants that explain the time course of the observed relaxation function. The fractional-order model integrated into the QLV theory proved to capture the essential features of the arterial wall mechanical response.     

The stress relaxation characteristics of composite matrices etched to produce nanoscale surface features

Many synthetic and xenogenic natural matrices have been explored in tissue regeneration, however, they lack either mechanical strength or cell colonization characteristics found in natural tissue. Moreover natural matrices such as small intestinal submucosa (SIS) lack sample to sample homogeneity, leading to unpredictable clinical outcomes. This work explored a novel fabrication technique by blending together the useful characteristics of synthetic and natural polymers to form a composite structure by using a NaOH etching process that produces nanoscale surface features. The composite scaffold was formed by sandwiching a thin layer of PLGA between porous layers of gelatin–chitosan. The etching process increased the surface roughness of PLGA membrane, allowing easy spreading of the hydrophilic gelatin–chitosan solution on its hydrophobic surface and reducing the scaffold thickness by nearly 50% than otherwise. The viscoelastic properties of the scaffold, an area of mechanical analysis which remains largely unexplored in tissue regeneration was assessed. Stress relaxation experiments of the “ramp and hold” type performed at variable ranges of temperature (25 °C and 37 °C), loading rates (3.125% s^?1 and 12.5% s^?1) and relaxation times (60 s, 100 s and 200 s) found stress relaxation to be sensitive to temperature and the loading rate but less dependent on the relaxation time. Stress relaxation behavior of the composite matrix was compared with SIS structures at 25 °C (hydrated), 3.125% s^?1 loading rate and 100 s relaxation time which showed that the synthetic matrix was found to be strain softening as compared to the strain hardening behavior exhibited by SIS. Popularly used quasi-linear viscoelastic (QLV) model to describe biomechanics of soft tissues was utilized. The QLV model predicted the loading behavior with an average error of 3%. The parameters of the QLV model predicted using nonlinear regression analysis appear to be in concurrence with soft tissues.     

Using Uniaxial Pseudorandom Stress Stimuli to Develop Soft Tissue Constitutive Equations

A nonlinear systems identification method was used to develop constitutive equations for soft tissue specimens under uniaxial tension. The constitutive equations are developed from a single test by applying a pseudorandom Gaussian (PGN) stress input to the specimen, measuring the resulting strain, and calculating the Volterra–Wiener kernels. First and second order kernels were developed for two tissues with widely different properties, rat medial collateral knee ligaments, and rat skin. These kernels were used to predict the strain response to a variety of sinusoidal stress inputs. These predicted strains were compared with the measured strain response using the normalized mean squared error (NMSE). Results showed NMSEs in the range of 0.01–0.08 provided that the magnitudes of the applied stresses were present in the original PGN stress input. Overall, the method provides a means to develop soft tissue constitutive equations that can predict both nonlinear and viscoelastic behavior over a wide range of stress inputs. © 2002 Biomedical Engineering Society. PAC02: 8719Rr     

Treatment with bioscaffold enhances the the fibril morphology and the collagen composition of healing medial collateral ligament in rabbits

Porcine small intestinal submucosa (SIS) was shown to be an effective bioscaffold in enhancing the mechanical properties of healing medial collateral ligaments (MCL). The purpose of this study was to investigate whether there are corresponding improvements in morphology and tissue compositions. Fourteen rabbits were equally divided into two groups. In the SIS-treated group, a 6 mm gap was surgically created in the right MCL and a layer of SIS was sutured covering the gap. For the nontreated group, the gap-injured MCLs remained untreated. All the left MCLs were sham operated and used as controls. At 12 weeks, the status of collagen types I and V was evaluated with immunofluorescent staining. The collagen type V/I ratios were obtained using SDS-PAGE. Collagen fibril diameters were calculated from the transmission electron micrographs. The results revealed that in the SIS-treated group, the collagen fibers were more regularly aligned as were the cell nuclei. The collagen fibril diameters were 22.2% larger and the ratio of collagen type V/I was 28.4% lower than those for the nontreated group (p < 0.05). These improvements in the morphological characteristics and biochemical constituents of healing MCLs following SIS treatment are the likely reasons for improved mechanical properties.     

The quasi-linear viscoelastic properties of diabetic and non-diabetic plantar soft tissue

The purpose of this study was to characterize the viscoelastic behavior of diabetic and non-diabetic plantar soft tissue at six ulcer-prone/load-bearing locations beneath the foot to determine any changes that may play a role in diabetic ulcer formation and subsequent amputation in this predisposed population. Four older diabetic and four control fresh frozen cadaveric feet were each dissected to isolate plantar tissue specimens from the hallux, first, third, and fifth metatarsals, lateral midfoot, and calcaneus. Stress relaxation experiments were used to quantify the viscoelastic tissue properties by fitting the data to the quasi-linear viscoelastic (QLV) theory using two methods, a traditional frequency-insensitive approach and an indirect frequency-sensitive approach, and by measuring several additional parameters from the raw data including the rate and amount of overall relaxation. The stress relaxation response of both diabetic and non-diabetic specimens was unexpectedly similar and accordingly few of the QLV parameters for either fit approach and none of raw data parameters differed. Likewise, no differences were found between plantar locations. The accuracy of both fit methods was comparable, however, neither approach predicted the ramp behavior. Further, fit coefficients varied considerably from one method to the other, making it hard to discern meaningful trends. Future testing using alternate loading modes and intact feet may provide more insight into the role that time-dependent properties play in diabetic foot ulceration.     

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.