The prediction of stress-relaxation of ligaments and tendons using the quasi-linear viscoelastic model

A finite porous-viscoelastic model capturing mechanical behavior of human cervix under multi-step spherical indentation

The cervix is a soft tissue exhibiting time-dependent behavior under mechanical loads. The cervix is a vital mechanical barrier to protect the growing fetus. The remodeling of the cervical tissue, characterized by an increase in time-dependent material properties, is necessary for a safe parturition. The failure of its mechanical function and accelerated tissue remodeling is hypothesized to lead to preterm birth, which is birth before 37 weeks of gestation. To understand the mechanism of the time-dependent behavior of the cervix under compressive states, we employ a porous-viscoelastic material model to describe a set of spherical indentation tests performed on nonpregnant and term pregnant tissue. A genetic algorithm-based inverse finite element analysis is used to fit the force-relaxation data by optimizing the material parameters, and the statistical analysis of the optimized material parameters is conducted on different sample groups. The force response is captured well using the porous-viscoelastic model. The indentation force-relaxation of the cervix is explained by the porous effects and the intrinsic viscoelastic properties of the extracellular matrix (ECM) microstructure. The hydraulic permeability obtained from the inverse finite element analysis agrees with the trend of the value directly measured previously by our group. The nonpregnant samples are found significantly more permeable than the pregnant samples. Within nonpregnant samples, the posterior internal os is found significantly less permeable than the anterior and posterior external os. The proposed model exhibits the superior capability to capture the force-relaxation response of the cervix under indentation, as compared to the conventional quasi-linear viscoelastic framework (range of r2 of the porous-viscoelastic model 0.88-0.98 vs. quasi-linear model: 0.67-0.89). As a constitutive model with a relatively simple form, the porous-viscoelastic framework has the potential to be used to understand disease mechanisms of premature cervical remodeling, model contact of the cervix with biomedical devices, and interpret force readings from novel in-vivo measurement tools such as an aspiration device.

Estimation of forces on anterior cruciate ligament in dynamic activities

In this work, a nonlinear strain rate dependent plugin developed for the OpenSim® platform was used to estimate the instantaneous strain rate (ISR) and the forces on the ACL's anteromedial (aACL) and posterolateral (pACL) bundles during walking and sudden change of direction of running termed as 'plant-and-cut' (PC). The authors obtained the kinematics data for walking via optical motion capture. PC movements, along with running kinematics, were obtained from the literature. A nonlinear plugin developed for ligaments was interfaced with OpenSim® platform to simulate walking and PC motions with a flexed knee and an extended knee. PC phase is sandwiched between an approach phase and take-off phase and was studied at various event velocities (1.8, 3, and 4.2 m s^-1), and angles of PC (23°, 34°, and 45°) as encountered in adult ball games. In both cases of PC-with-extended knee and PC-with-flexed-knee, the maximum forces on both the ACL bundles were observed after the take-off phase. A maximum force of ~ 35 N kg^-1 of body weight (BW) was observed on aACL after the take-off phase for an event velocity of 4.2 m s^-1. In the posterolateral bundle (pACL), the maximum forces (~ 40 N kg^-1 of BW) were observed towards the end of the mid-swing phase (after the take-off phase) for the various combinations of the parameters studied. The forces observed in the simulation of PC-with-flexed-knee and PC-with-extended-knee has resulted in magnitude higher than sustainable by the adults. This study is novel in attempting to incorporate differing rates-of-strain that have been shown to alter soft tissue properties into the OpenSim® musculoskeletal model. The proposed model can be used by researchers to predict the forces during various kinematic activities for other soft tissues.

The differences in viscoelastic properties of subtendons result from the anatomical tripartite structure of human Achilles tendon - ex vivo experimental study and modeling

The human Achilles tendon (AT) is a hierarchical structure macroscopically composed of three subtendons originating from the soleus (SOL) and gastrocnemius (GL, GM) muscles. According to recent reports, the divisible structure of the AT together with diverse material properties of its subtendons are suspected as a probable cause of non-homogeneous stress and strain distribution occurring in loaded AT. Despite numerous investigations on human AT, there is still relatively little knowledge regarding mechanical properties of subtendon-level hierarchy, which is crucial in fully understanding the multiscale relationship which governs tendon mechanics. In this paper we present the first ex vivo study conducted on SOL, GL, and GM subtendons of human AT. We investigate differences in viscoelastic properties of SOL, GM, and GL subtendons in terms of tensile modulus, mechanical hysteresis as well as stress relaxation observed at two different values of strain. Our results show that the most significant differences in mechanical properties exist between subtendon attached to the soleus muscle (SOL) and subtendons originating from the two heads of the gastrocnemius muscle (GM and GL). We used our experimental results to calibrate three different constitutive models: the hyperelastic Yeoh model with power-law flow, the microstructurally motivated Holzapfel-Gasser-Ogden model enhanced with strain-dependent Berström-Boyce flow and the phenomenological elasto-viscoplastic Arruda-Boyce-based model with strain-dependent Berström-Boyce flow supplemented with component representing matrix response. All calibrated models may be applied to commercial FEA software as a sufficient solution for rapid mechanical response modeling of human AT subtendons or for the purpose of future development of comprehensive patient-specific models of human lower limbs. STATEMENT OF SIGNIFICANCE: The divisible structure of the Achilles tendon together with diverse material properties of its subtendons are suspected as a probable cause of non-homogeneous stress and strain distribution occurring in loaded Achilles tendon. Despite numerous investigations on mechanical properties of Achilles tendon, there is still relatively little knowledge regarding mechanical properties of subtendon-level hierarchy, which is crucial in fully understanding the multiscale relationship which governs tendon mechanics. This study is the first reported ex vivo investigation conducted on SOL, GL, and GM human Achilles subtendons. We investigate differences in the viscoelastic properties of individual subtendons and demonstrate that the observed differences should be considered as muscle-dependent. Our experimental research is supported with a modeling study in which we calibrate three different constitutive models.

The rate of tendon failure in a collagen fibre recruitment-based model

Accurate characterization of the mechanical response of collagenous tissues is critical for investigations into mechanisms of soft tissue injury. These tissues are inherently viscoelastic, exhibiting strain-rate dependent stiffnesses, creep, and stress-relaxation. The strain-rate features of the failure portion of the stress-strain curve are less well developed. Collagen-distribution based models are improving and capable of reproducing the non-linear aspects of the elastic response of soft tissues, but still require parameterization of failure regions. Therefore, the purpose of this investigation, was to determine whether the parameters characterizing the rate of damage accumulation in a collagen-distribution model are proportional to strain rate. Fifty rat tail tendons were subjected to one of five different strain rates (0.01, 0.05, 0.1, 0.15, 0.20 s^-1) until failure in an uni-axial strain test. To test the hypothesis that the parameters associated with damage rate are proportional to strain rate, a collagen distribution model was employed with the parameters describing the rate of fibre damage being obtained by least-squares and regressed against the strain rate. The breaking function was found to be proportional to strain rate, with a proportionality constant of 60.7 s^-1. Properties characterizing the failure portion of the stress-strain curves for rat tail tendons are also reported. The Young's Modulus did not vary with strain rate and was found to be 103.3 ± 49.5 MPa. Similarly, failure stresses and strains did not vary across the strain rates tested, and were 15.6 ± 6.1 MPa and 32.2 ± 9.1%, respectively.

A one-dimensional collagen-based biomechanical model of passive soft tissue with viscoelasticity and failure

INTRODUCTION

Strains and sprains of soft tissues, including tendons and ligaments, are frequently occurring injuries. Musculoskeletal models show great promise in prediction and prevention of these injuries. However, these models rarely account for the viscoelastic properties of ligaments and tendons, much less their failure properties. The purpose of this project was to develop, simplify, and analyze a collagen-distribution model to address these limitations.

MODEL DEVELOPMENT

A distribution-moment approximation was applied to an existing partial differential equation model to reduce its computational complexity. The resulting model was equipped with a Voigt model in series, which endowed it with viscoelastic properties in addition to failure properties.

RESULTS

The model was able to reproduce the characteristic toe, linear, and failure regions ubiquitous throughout in-vitro tests on tissue specimens. In addition, it was able to reproduce a tri-phasic creep test consisting of an initial deformation, a steady-state, and failure. Stress-relaxation and hysteresis were also reproducible by the model.

DISCUSSION AND CONCLUSION

The ability to reproduce so many characteristics of biological tissues suggests more bio-fidelity was achieved by the reduced model was other currently available models. Future work to further improve its bio-fidelity is proposed for specific tendons and ligaments.

A Recruitment Model of Tendon Viscoelasticity That Incorporates Fibril Creep and Explains Strain-Dependent Relaxation

Soft tissues exhibit complex viscoelastic behavior, including strain-rate dependence, hysteresis, and strain-dependent relaxation. In this paper, a model for soft tissue viscoelasticity is developed that captures all of these features and is based upon collagen recruitment, whereby fibrils contribute to tissue stiffness only when taut. We build upon existing recruitment models by additionally accounting for fibril creep and by explicitly modeling the contribution of the matrix to the overall tissue viscoelasticity. The fibrils and matrix are modeled as linear viscoelastic and each fibril has an associated critical strain (corresponding to its length) at which it becomes taut. The model is used to fit relaxation tests on three rat tail tendon fascicles and predict their response to cyclic loading. It is shown that all of these mechanical tests can be reproduced accurately with a single set of constitutive parameters, the only difference between each fascicle being the distribution of their fibril crimp lengths. By accounting for fibril creep, we are able to predict how the fibril length distribution of a fascicle changes over time under a given deformation. Furthermore, the phenomenon of strain-dependent relaxation is explained as arising from the competition between the fibril and matrix relaxation functions.

Mechanobiological modelling of tendons: Review and future opportunities

Tendons are adapted to carry large, repeated loads and are clinically important for the maintenance of musculoskeletal health in an increasing, actively ageing population, as well as in elite athletes. Tendons are known to adapt to mechanical loading. Also, their healing and disease processes are highly sensitive to mechanical load. Computational modelling approaches developed to capture this mechanobiological adaptation in tendons and other tissues have successfully addressed many important scientific and clinical issues. The aim of this review is to identify techniques and approaches that could be further developed to address tendon-related problems. Biomechanical models are identified that capture the multi-level aspects of tendon mechanics. Continuum whole tendon models, both phenomenological and microstructurally motivated, are important to estimate forces during locomotion activities. Fibril-level microstructural models are documented that can use these estimated forces to detail local mechanical parameters relevant to cell mechanotransduction. Cell-level models able to predict the response to such parameters are also described. A selection of updatable mechanobiological models is presented. These use mechanical signals, often continuum tissue level, along with rules for tissue change and have been applied successfully in many tissues to predict in vivo and in vitro outcomes. Signals may include scalars derived from the stress or strain tensors, or in poroelasticity also fluid velocity, while adaptation may be represented by changes to elastic modulus, permeability, fibril density or orientation. So far, only simple analytical approaches have been applied to tendon mechanobiology. With the development of sophisticated computational mechanobiological models in parallel with reporting more quantitative data from in vivo or clinical mechanobiological studies, for example, appropriate imaging, biochemical and histological data, this field offers huge potential for future development towards clinical applications.

Modelling approaches for evaluating multiscale tendon mechanics

Tendon exhibits anisotropic, inhomogeneous and viscoelastic mechanical properties that are determined by its complicated hierarchical structure and varying amounts/organization of different tissue constituents. Although extensive research has been conducted to use modelling approaches to interpret tendon structure-function relationships in combination with experimental data, many issues remain unclear (i.e. the role of minor components such as decorin, aggrecan and elastin), and the integration of mechanical analysis across different length scales has not been well applied to explore stress or strain transfer from macro- to microscale. This review outlines mathematical and computational models that have been used to understand tendon mechanics at different scales of the hierarchical organization. Model representations at the molecular, fibril and tissue levels are discussed, including formulations that follow phenomenological and microstructural approaches (which include evaluations of crimp, helical structure and the interaction between collagen fibrils and proteoglycans). Multiscale modelling approaches incorporating tendon features are suggested to be an advantageous methodology to understand further the physiological mechanical response of tendon and corresponding adaptation of properties owing to unique in vivo loading environments.

Visco-hyperelastic constitutive modeling of soft tissues based on short and long-term internal variables

Background

Differential-type and integral-type formulations are two common approaches in modeling viscoelastic materials. A differential-type theory is often derived from a Helmholtz free energy function and is usually more suitable for the prediction of strain-rate dependent mechanical behavior during rapid loading, while an integral-type theory usually captures stress relaxation more efficiently than a differential-type theory. A modeling approach is needed to predict the viscoelastic responses during both rapid loading and relaxation phases.

Methods

A constitutive modeling methodology based on the short and long-term internal variables was proposed in the present study in order to fully use the better features of the two types of theories. The short-term variables described the loading rate, while the long-term variables involving time constants characterized loading history and stress relaxation.

Results

The application of the methodology was demonstrated with particular formulations for ligament and articular cartilage. Model parameters were calibrated for both tissues with experimental data from the literature. It was found that the proposed model could well predict a wide range of strain-rate dependent load responses during both loading and relaxation phases.

Conclusion

Introducing different internal variables in terms of their time scales reduced the difficulties in the material characterization process and enabled the model to predict the experimental data more accurately, in particular at high strain-rates.

Assessing viscoelastic properties of chitosan scaffolds and validation with cyclical tests

We evaluated and modeled the viscoelastic characteristics of chitosan and chitosan-gelatin scaffolds prepared using a freeze-drying technique. Chitosan and chitosan-gelatin solutions (0.5 and 2 wt.%) were frozen at -80°C and freeze-dried. Using the scaffolds, uniaxial tensile properties were evaluated under physiological conditions (hydrated in phosphate buffered saline at 37°C) at a cross-head speed of 0.17 mms(-1) (10 mm min(-1)). From the break strain, the limit of strain per ramp was calculated to be 5% and the samples were stretched at a strain rate of 2.5%s(-1). The ramp-and-hold type of stress-relaxation test was performed for five successive stages. Chitosan and chitosan-gelatin showed nearly 90% relaxation of stress after each stage. The relaxation behavior was independent of the concentration of chitosan and gelatin. Also, changes in the microstructure of the tested samples were evaluated using an inverted microscope. The micrographs acquired after relaxation experiments showed orientation of pores, suggesting the retention of the stretched state even after many hours of relaxation. Based on these observations, two models (i) containing a hyper-elastic spring (containing two parameters) and (ii) retaining pseudo-components (containing three parameters) were developed in Visual Basic Applications accessed through MS Excel. The models were used to fit the experimental stress-relaxation data and the parameters obtained from modeling were used to predict their respective cyclic behaviors, which were compared with cyclical experimental results. These results showed that the model could be used to predict the cyclical behavior under the tested strain rates. The model predictions were also tested using cyclic properties at a lower strain rate of 0.0867%s(-1) (5%min(-1)) for 0.5 wt.% scaffolds but the model could not predict cyclical behavior at a very slow rate. In summary, the pseudo-component modeling approach can be used to model the sequential strain-and-hold stage and predict cyclical properties for the same strain rate.

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.

Viscoelastic properties of the ovine posterior spinal ligaments are strain dependent

BACKGROUND

The biomechanical role of the posterior spinal ligaments for spinal stability has been stated in previous studies. The investigation of the viscoelastic properties of human lumbar spinal ligaments is essential for the understanding of physiological differences between healthy and degenerated tissues. The stress-relaxation behavior of biological tissues is commonly described with the quasi-linear viscoelastic model of Fung, which assumes that the stress-relaxation response is independent of the applied strain. The goal of this study was to investigate the stress-relaxation response of ovine posterior spinal ligaments at different elongations to verify the above-mentioned hypothesis.

METHODS

Twenty-four ovine lumbar spinal segments, consisting of only the supraspinous and interspinous ligaments and adjoining spinous processes, were elongated uniaxially to different strain levels within the physiological elastic region (5-20%). The experimental data were described with a non-linear viscoelastic model: the modified superposition method of Findley.

FINDINGS

A linear dependency of the relaxation rate to the applied strains was observed on intact segments, when both ligaments were considered, as well as on each individual ligament. This result can be applied to the human spinal ligaments, due to similarities observed between the sheep and human spinal segment under physiological loading.

INTERPRETATION

The non-linear viscoelastic modified superposition method of Findley is an appropriate model for describing the viscoelastic properties of lumbar spinal ligaments in vitro due to its ability to address variation in applied strain during the force relaxation measurements.

The viscoelastic properties of passive eye muscle in primates. II: testing the quasi-linear theory

We have extensively investigated the mechanical properties of passive eye muscles, in vivo, in anesthetized and paralyzed monkeys. The complexity inherent in rheological measurements makes it desirable to present the results in terms of a mathematical model. Because Fung's quasi-linear viscoelastic (QLV) model has been particularly successful in capturing the viscoelastic properties of passive biological tissues, here we analyze this dataset within the framework of Fung's theory.We found that the basic properties assumed under the QLV theory (separability and superposition) are not typical of passive eye muscles. We show that some recent extensions of Fung's model can deal successfully with the lack of separability, but fail to reproduce the deviation from superposition.While appealing for their elegance, the QLV model and its descendants are not able to capture the complex mechanical properties of passive eye muscles. In particular, our measurements suggest that in a passive extraocular muscle the force does not depend on the entire length history, but to a great extent is only a function of the last elongation to which it has been subjected. It is currently unknown whether other passive biological tissues behave similarly.

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 degrees C and 37 degrees 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 degrees 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.

Characterization of the mechanical properties of tough biopolymer fibres from the mussel byssus of Aulacomya ater

Byssus fibres are tough biopolymer fibres produced by mussels to attach themselves to rocks. In this communication, we present the mechanical properties of the byssus from the South American mussel Aulacomya ater which have not been previously reported in the literature. The mechanical properties of the whole threads were assessed by uniaxial tensile tests of dry and hydrated specimens. Elastoplastic and elastomeric stress-strain curves were found for byssal threads from A. ater in the dry and hydrated state, respectively. The results obtained from mechanical tests were modelled using linear, power-law-type and Mooney-Rivlin relationships. These methods for dealing with tensile measurements of mussel byssus have the potential to be used with other stretchy biomaterials.