A single integral finite strain viscoelastic model of ligaments and tendons

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.

Fiber Rearrangement and Matrix Compression in Soft Tissues: Multiscale Hypoelasticity and Application to Tendon

It is widely accepted that the nonlinear macroscopic mechanical behavior of soft tissue is governed by fiber straightening and re-orientation. Here, we provide a quantitative assessment of this phenomenon, by means of a continuum micromechanics approach. Given the negligibly small bending stiffness of crimped fibers, the latter are represented through a number of hypoelastic straight fiber phases with different orientations, being embedded into a hypoelastic matrix phase. The corresponding representative volume element (RVE) hosting these phases is subjected to “macroscopic” strain rates, which are downscaled to fiber and matrix strain rates on the one hand, and to fiber spins on the other hand. This gives quantitative access to the fiber decrimping (or straightening) phenomenon under non-affine conditions, i.e. in the case where the fiber orientations cannot be simply linked to the macroscopic strain state. In the case of tendinous tissue, such an RVE relates to the fascicle material with 50 μm characteristic length, made up of crimped collagen bundles and a gel-type matrix in-between. The fascicles themselves act as parallel fibers in a similar matrix at the scale of a tissue-related RVE with 500 μm characteristic length. As evidenced by a sensitivity analysis and confirmed by various mechanical tests, it is the initial crimping angle which drives both the degree of straightening and the shape of the macroscopic stress-strain curve, while the final linear portion of this curve depends almost exclusively on the collagen bundle elasticity. Our model also reveals the mechanical cooperation of the tissue’s key microstructural components: while the fibers carry tensile forces, the matrices undergo hydrostatic pressure.

Biplanar High-Speed Fluoroscopy of Pony Superficial Digital Flexor Tendon (SDFT)-An In Vivo Pilot Study

The superficial digital flexor tendon (SDFT) is the most frequently injured structure of the musculoskeletal system in sport horses and a common cause for early retirement. This project's aim was to visualize and measure the strain of the sound, injured, and healing SDFTs in a pony during walk and trot. For this purpose, biplanar high-speed fluoroscopic kinematography (FluoKin), as a high precision X-ray movement analysis tool, was used for the first time in vivo with equine tendons. The strain in the metacarpal region of the sound SDFT was 2.86% during walk and 6.78% during trot. When injured, the strain increased to 3.38% during walk and decreased to 5.96% during trot. The baseline strain in the mid-metacarpal region was 3.13% during walk and 6.06% during trot and, when injured, decreased to 2.98% and increased to 7.61%, respectively. Following tendon injury, the mid-metacarpal region contributed less to the overall strain during walk but showed increased contribution during trot. Using this marker-based FluoKin technique, direct, high-precision, and long-term strain measurements in the same individual are possible. We conclude that FluoKin is a powerful tool for gaining deeper insight into equine tendon biomechanics.

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.

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.

The inflation of viscoelastic balloons and hollow viscera

For the first time, the problem of the inflation of a nonlinear viscoelastic thick-walled spherical shell is considered. Specifically, the wall has quasilinear viscoelastic constitutive behaviour, which is of fundamental importance in a wide range of applications, particularly in the context of biological systems such as hollow viscera, including the lungs and bladder. Experiments are performed to demonstrate the efficacy of the model in fitting relaxation tests associated with the volumetric inflation of murine bladders . While the associated nonlinear elastic problem of inflation of a balloon has been studied extensively, there is a paucity of studies considering the equivalent nonlinear viscoelastic case. We show that, in contrast to the elastic scenario, the peak pressure associated with the inflation of a neo-Hookean balloon is not independent of the shear modulus of the medium. Moreover, a novel numerical technique is described in order to solve the nonlinear Volterra integral equation in space and time originating from the fundamental problem of inflation and deflation of a thick-walled nonlinear viscoelastic shell under imposed pressure.

Quantitative Constitutive Behaviour and Viscoelastic Properties of Fresh Flexor Tendons

The objective of this study was to obtain detailed high quality experimental data under well-controlled test conditions in order to quantify tendon viscoelastic behaviour and provide an experimental basis for large deformation mathematical modelling. Eighty-six fresh chicken flexor digitorum profundus (FDP) tendons were mechanically tested using an Instron 5565 universal testing machine and a Bioplus bath containing physiological saline solution. At low strain rates (≤ 0.001s−1), no strain rate dependence was found and the value of the elastic modulus was 324±17 MPa. At medium strain rates (0.003s−1 – 0.1s−1), the elastic stiffness increased with increasing strain rate. For example, the values of the elastic modulus were 427±10, 653±21 and 837±11 MPa for strain rates of 0.006s−1, 0.012s−1 and 0.05s−1 respectively. A series of stress relaxation experiments at different levels of strain were conducted. The higher the initial stress level, the faster the stress relaxed. For all stress relaxation tests, the relationship of normalised relaxation function G(t) against log time (ln(t)) was approximately linear (R2 = 0.935 ± 0.028) and this supports Fung's quasi-linear viscoelasticity (QLV) hypothesis. Some preliminary work on bovine flexor tendons using a video extensometer is also reported.

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.

Viscoelastic characterization of extraocular Z-myotomy

PURPOSE

Z-myotomy is an extraocular muscle (EOM) weakening procedure in which two incisions are made from longitudinally-separated, opposite EOM margins for treatment of strabismus. We examined the in vitro biomechanics of Z-myotomy using tensile loading.

METHODS

Fresh bovine rectus EOMs were reduced to 20 × 10 × 2-mm dimensions, and clamped in a microtensile load cell under physiological conditions. Extraocular muscles were elongated until failure following scissors incisions made from opposite sides, spaced 8 mm apart and each encompassing 0%, 40%, 50%, 60%, or 80% EOM width. Initial strain to 30% elongation was imposed at 100 mm/s, after which elongation was maintained for greater than 100 seconds during force recording at maintained deformation. Stress relaxation tests with nonincised specimens having widths ranging from 1 to 9 mm were conducted for viscoelastic characterization of corresponding equivalence to 20% to 80% Z-myotomy. Data were modeled using the Wiechert viscoelastic formulation.

RESULTS

There was progressively reduced EOM failure force to an asymptotic minimum at 60% or greater Z-myotomy. Each Z-myotomy specimen could be matched for equivalent failure force to a non-Z-myotomy specimen with a different width. Both tensile and stress relaxation data could be modeled accurately using the Wiechert viscoelastic formulation.

CONCLUSIONS

The parallel fiber structure results in low shear force transfer across EOM width, explaining the biomechanics of Z-myotomy. Z-myotomy progressively reduces force transmission to an asymptotic minimum for less than 60% surgical dose, with no further reduction for greater amounts of surgery. Equivalence to EOM specimens having regular cross-sections permits viscoelastic biomechanical characterization of Z-myotomy specimens with irregular cross-sections.

On nonlinear viscoelastic deformations: a reappraisal of Fung's quasi-linear viscoelastic model

This paper offers a reappraisal of Fung's model for quasi-linear viscoelasticity. It is shown that a number of negative features exhibited in other works, commonly attributed to the Fung approach, are merely a consequence of the way it has been applied. The approach outlined herein is shown to yield improved behaviour and offers a straightforward scheme for solving a wide range of models. Results from the new model are contrasted with those in the literature for the case of uniaxial elongation of a bar: for an imposed stretch of an incompressible bar and for an imposed load. In the latter case, a numerical solution to a Volterra integral equation is required to obtain the results. This is achieved by a high-order discretization scheme. Finally, the stretch of a compressible viscoelastic bar is determined for two distinct materials: Horgan–Murphy and Gent.

Biomechanics of superior oblique Z-tenotomy

BACKGROUND

A recent report suggests that 70%-80% Z-tenotomy of the superior oblique tendon is necessary to effectively treat A-pattern strabismus associated with over depression in adduction. To clarify the clinical effect, we compared the biomechanics of Z-tenotomy on the superior oblique tendon, superior rectus tendon, and isotropic latex material.

METHODS

Fresh bovine superior oblique tendons were trimmed to 20 mm × 10 mm dimensions similar to human superior oblique tendon and clamped in a microtensile load cell under physiological conditions of temperature and humidity. Minimal preload was applied to avoid slackness. Tendons were elongated until failure following Z-tenotomies, made from opposite tendon margins, spaced 8 mm apart and each encompassing 0%, 20%, 40%, 50%, 60%, or 80% tendon width. Digitally sampled failure force was monitored using a precision strain gauge. Control experiments were performed in similar-sized specimens of bovine superior rectus tendon and isotropic latex.

RESULTS

Progressively increasing Z-tenotomy of latex caused a linearly graded reduction in force. In contrast, Z-tenotomy of up to 50% in superior oblique and superior rectus tendons caused nonlinear reduction in force transmission that reached a negligible value at 50% tenotomy and greater.

CONCLUSIONS

Z-tenotomy up to 50% progressively reduces extraocular tendon force transmission, but Z-tenotomy of ≥50% is biomechanically equivalent in vitro to complete tenotomy.

How preconditioning affects the measurement of poro-viscoelastic mechanical properties in biological tissues

It is known that initial loading curves of soft biological tissues are substantially different from subsequent loadings. The later loading curves are generally used for assessing the mechanical properties of a tissue, and the first loading cycles, referred to as preconditioning, are omitted. However, slow viscoelastic phenomena related to fluid flow or collagen viscoelasticity are initiated during these first preconditioning loading cycles and may persist during the actual data collection. When these data are subsequently used for fitting of material properties, the viscoelastic phenomena that occurred during the initial cycles are not accounted for. The aim of the present study is to explore whether the above phenomena are significant for articular cartilage, by evaluating the effect of such time-dependent phenomena by means of computational modeling. Results show that under indentation, collagen viscoelasticity dominates the time-dependent behavior. Under UC, fluid-dependent effects are more important. Interestingly, viscoelastic and poroelastic effects may act in opposite directions and may cancel each other out in a stress-strain curve. Therefore, equilibrium may be apparent in a stress-strain relationship, even though internally the tissue is not in equilibrium. Also, the time-dependent effects of viscoelasticity and poroelasticity may reinforce each other, resulting in a sustained effect that lasts longer than suggested by their individual effects. Finally, the results illustrate that data collected from a mechanical test may depend on the preconditioning protocol. In conclusion, preconditioning influences the mechanical response of articular cartilage significantly and therefore cannot be neglected when determining the mechanical properties. To determine the full viscoelastic and poroelastic properties of articular cartilage requires fitting to both preconditioning and post-preconditioned loading cycles.

Recent advances in computational mechanics of the human knee joint

Computational mechanics has been advanced in every area of orthopedic biomechanics. The objective of this paper is to provide a general review of the computational models used in the analysis of the mechanical function of the knee joint in different loading and pathological conditions. Major review articles published in related areas are summarized first. The constitutive models for soft tissues of the knee are briefly discussed to facilitate understanding the joint modeling. A detailed review of the tibiofemoral joint models is presented thereafter. The geometry reconstruction procedures as well as some critical issues in finite element modeling are also discussed. Computational modeling can be a reliable and effective method for the study of mechanical behavior of the knee joint, if the model is constructed correctly. Single-phase material models have been used to predict the instantaneous load response for the healthy knees and repaired joints, such as total and partial meniscectomies, ACL and PCL reconstructions, and joint replacements. Recently, poromechanical models accounting for fluid pressurization in soft tissues have been proposed to study the viscoelastic response of the healthy and impaired knee joints. While the constitutive modeling has been considerably advanced at the tissue level, many challenges still exist in applying a good material model to three-dimensional joint simulations. A complete model validation at the joint level seems impossible presently, because only simple data can be obtained experimentally. Therefore, model validation may be concentrated on the constitutive laws using multiple mechanical tests of the tissues. Extensive model verifications at the joint level are still crucial for the accuracy of the modeling.

Viscoelastic properties of isolated collagen fibrils

Understanding the viscoelastic behavior of collagenous tissues with complex hierarchical structures requires knowledge of the properties at each structural level. Whole tissues have been studied extensively, but less is known about the mechanical behavior at the submicron, fibrillar level. Using a microelectromechanical systems platform, in vitro coupled creep and stress relaxation tests were performed on collagen fibrils isolated from the sea cucumber dermis. Stress-strain-time data indicate that isolated fibrils exhibit viscoelastic behavior that could be fitted using the Maxwell-Weichert model. The fibrils showed an elastic modulus of 123 ± 46 MPa. The time-dependent behavior was well fit using the two-time-constant Maxwell-Weichert model with a fast time response of 7 ± 2 s and a slow time response of 102 ± 5 s. The fibrillar relaxation time was smaller than literature values for tissue-level relaxation time, suggesting that tissue relaxation is dominated by noncollagenous components (e.g., proteoglycans). Each specimen was tested three times, and the only statistically significant difference found was that the elastic modulus is larger in the first test than in the subsequent two tests, indicating that viscous properties of collagen fibrils are not sensitive to the history of previous tests.

Viscoelastic properties of bovine orbital connective tissue and fat: constitutive models

Reported mechanical properties of orbital connective tissue and fat have been too sparse to model strain-stress relationships underlying biomechanical interactions in strabismus. We performed rheological tests to develop a multi-mode upper convected Maxwell (UCM) model of these tissues under shear loading. From 20 fresh bovine orbits, 30 samples of connective tissue were taken from rectus pulley regions and 30 samples of fatty tissues from the posterior orbit. Additional samples were defatted to determine connective tissue weight proportion, which was verified histologically. Mechanical testing in shear employed a triborheometer to perform: strain sweeps at 0.5-2.0 Hz; shear stress relaxation with 1% strain; viscometry at 0.01-0.5 s(-1) strain rate; and shear oscillation at 1% strain. Average connective tissue weight proportion was 98% for predominantly connective tissue and 76% for fatty tissue. Connective tissue specimens reached a long-term relaxation modulus of 668 Pa after 1,500 s, while corresponding values for fatty tissue specimens were 290 Pa and 1,100 s. Shear stress magnitude for connective tissue exceeded that of fatty tissue by five-fold. Based on these data, we developed a multi-mode UCM model with variable viscosities and time constants, and a damped hyperelastic response that accurately described measured properties of both connective and fatty tissues. Model parameters differed significantly between the two tissues. Viscoelastic properties of predominantly connective orbital tissues under shear loading differ markedly from properties of orbital fat, but both are accurately reflected using UCM models. These viscoelastic models will facilitate realistic global modeling of EOM behavior in binocular alignment and strabismus.

Nonlinear Model for Viscoelastic Behavior of Achilles Tendon

Although the mechanical properties of ligament and tendon are well documented in research literature, very few unified mechanical formulations can describe a wide range of different loadings. The aim of this study was to propose a new model, which can describe tendon responses to various solicitations such as cycles of loading, unloading, and reloading or successive relaxations at different strain levels. In this work, experiments with cycles of loading and reloading at increasing strain level and sequences of relaxation were performed on white New Zealand rabbit Achilles tendons. We presented a local formulation of thermodynamic evolution outside equilibrium at a representative element volume scale to describe the tendon’s macroscopic behavior based on the notion of relaxed stress. It was shown that the model corresponds quite well to the experimental data. This work concludes with the complexity of tendons’ mechanical properties due to various microphysical mechanisms of deformation involved in loading such as the recruitment of collagen fibers, the rearrangement of the microstructure (i.e., collagens type I and III, proteoglycans, and water), and the evolution of relaxed stress linked to these mechanisms.

Characterization of the Fatigue Behavior of the Medial Collateral Ligament Utilizing Traditional and Novel Mechanical Variables for the Assessment of Damage Accumulation

Ligaments are regularly subjected to repetitive loading in vivo. Typically, mechanical studies focus on repetitive loading protocols of short duration, while those characterizing damage accumulation over a longer duration (i.e., fatigue studies) are lacking. The aims of this study were as follows: (a) to demonstrate that damage does accumulate in ligament tissue subjected to repetitive loading and (b) to evaluate existing and new methods for characterizing fatigue damage accumulation. It was hypothesized that ligaments would accumulate damage with repetitive loading as evidenced by failure at stresses well below ultimate tensile strength, creep curve discontinuities, and by reductions in stiffness during loading. Eight normal medial collateral ligaments from female New Zealand white rabbits were cycled in tension, between 0 MPa and 28 MPa, to failure or until 259,200 cycles, whichever came first. Medial collateral ligaments that did not fail were subsequently loaded to failure. Displacement rates (dlmax/dt) as well as primary, secondary, and tertiary creeps were monitored as indices of damage accumulation and impending mechanical failure. Additionally, the relative utilities of tangent, secant, and chord stiffness parameters were critically evaluated. Finally, new uses for the second derivative of force-displacement data were explored. Three out of eight ligaments failed during testing, demonstrating that ligaments can fail in fatigue under moderate tensile stress in vitro. The evaluation of displacement rates (dlmax/dt), as well as primary through tertiary creep patterns, were not well suited to predicting failure in normal ligaments until rupture was all but imminent. Tangent stiffness, which was calculated from a mathematically defined start of the “linear region,” was surprisingly constant throughout testing. Secant stiffness dropped in a predictable fashion, providing a global indicator of tissue stiffness, but did not provide any insight into fiber mechanics. Chord stiffness, on the other hand, appeared to be sensitive to fiber recruitment patterns. The second derivative of force-displacement data proved to be a useful means of (a) objectively defining the start of the linear region and (b) inferring changes in fiber recruitment patterns within ligament tissue. Tangent, secant, and chord stiffnesses highlight different attributes of ligament responses to loading; hence these parameters cannot be used interchangeably. Additionally, the second derivative of the force-displacement curve was introduced as a useful descriptive and analytical tool.

Role of biomechanics in the understanding of normal, injured, and healing ligaments and tendons

Ligaments and tendons are soft connective tissues which serve essential roles for biomechanical function of the musculoskeletal system by stabilizing and guiding the motion of diarthrodial joints. Nevertheless, these tissues are frequently injured due to repetition and overuse as well as quick cutting motions that involve acceleration and deceleration. These injuries often upset this balance between mobility and stability of the joint which causes damage to other soft tissues manifested as pain and other morbidity, such as osteoarthritis.

The healing of ligament and tendon injuries varies from tissue to tissue. Tendinopathies are ubiquitous and can take up to 12 months for the pain to subside before one could return to normal activity. A ruptured medial collateral ligament (MCL) can generally heal spontaneously; however, its remodeling process takes years and its biomechanical properties remain inferior when compared to the normal MCL. It is also known that a midsubstance anterior cruciate ligament (ACL) tear has limited healing capability, and reconstruction by soft tissue grafts has been regularly performed to regain knee function. However, long term follow-up studies have revealed that 20–25% of patients experience unsatisfactory results. Thus, a better understanding of the function of ligaments and tendons, together with knowledge on their healing potential, may help investigators to develop novel strategies to accelerate and improve the healing process of ligaments and tendons.

With thousands of new papers published in the last ten years that involve biomechanics of ligaments and tendons, there is an increasing appreciation of this subject area. Such attention has positively impacted clinical practice. On the other hand, biomechanical data are complex in nature, and there is a danger of misinterpreting them. Thus, in these review, we will provide the readers with a brief overview of ligaments and tendons and refer them to appropriate methodologies used to obtain their biomechanical properties. Specifically, we hope the reader will pay attention to how the properties of these tissues can be altered due to various experimental and biologic factors. Following this background material, we will present how biomechanics can be applied to gain an understanding of the mechanisms as well as clinical management of various ligament and tendon ailments. To conclude, new technology, including imaging and robotics as well as functional tissue engineering, that could form novel treatment strategies to enhance healing of ligament and tendon are presented.

Quasilinear viscoelastic behavior of bovine extraocular muscle tissue

PURPOSE

Until now, there has been no comprehensive mathematical model of the nonlinear viscoelastic stress-strain behavior of extraocular muscles (EOMs). The present study describes, with the use of a quasilinear viscoelastic (QLV) model, the nonlinear, history-dependent viscoelastic properties and elastic stress-strain relationship of EOMs.

METHODS

Six oculorotary EOMs were obtained fresh from a local abattoir. Longitudinally oriented specimens were taken from different regions of the EOMs and subjected to uniaxial tensile, relaxation, and cyclic loading testing with the use of an automated load cell under temperature and humidity control. Twelve samples were subjected to uniaxial tensile loading with 1.7%/s strain rate until failure. Sixteen specimens were subjected to relaxation studies over 1500 seconds. Cyclic loading was performed to validate predictions of the QLV model characterized from uniaxial tensile loading and relaxation data.

RESULTS

Uniform and highly repeatable stress-strain behavior was observed for 12 specimens extracted from various regions of all EOMs. Results from 16 different relaxation trials illustrated that most stress relaxation occurred during the first 30 to 60 seconds for 30% extension. Elastic and reduced relaxation functions were fit to the data, from which a QLV model was assembled and compared with cyclic loading data. Predictions of the QLV model agreed with observed peak cyclic loading stress values to within 8% for all specimens and conditions.

CONCLUSIONS

Close agreement between the QLV model and the relaxation and cyclic loading data validates model quantification of EOM mechanical properties and will permit the development of accurate overall models of mechanics of ocular motility and strabismus.

In situ measurement and modeling of biomechanical response of human cadaveric soft tissues for physics-based surgical simulation

BACKGROUND

Development of a laparoscopic surgery simulator that delivers high-fidelity visual and haptic (force) feedback, based on the physical models of soft tissues, requires the use of empirical data on the mechanical behavior of intra-abdominal organs under the action of external forces. As experiments on live human patients present significant risks, the use of cadavers presents an alternative. We present techniques of measuring and modeling the mechanical response of human cadaveric tissue for the purpose of developing a realistic model. The major contribution of this paper is the development of physics-based models of soft tissues that range from linear elastic models to nonlinear viscoelastic models which are efficient for application within the framework of a real-time surgery simulator.

METHODS

To investigate the in situ mechanical, static, and dynamic properties of intra-abdominal organs, we have developed a high-precision instrument by retrofitting a robotic device from Sensable Technologies (position resolution of 0.03 mm) with a six-axis Nano 17 force-torque sensor from ATI Industrial Automation (force resolution of 1/1,280 N along each axis), and used it to apply precise displacement stimuli and record the force response of liver and stomach of ten fresh human cadavers.

RESULTS

The mean elastic modulus of liver and stomach is estimated as 5.9359 kPa and 1.9119 kPa, respectively over the range of indentation depths tested. We have also obtained the parameters of a quasilinear viscoelastic (QLV) model to represent the nonlinear viscoelastic behavior of the cadaver stomach and liver over a range of indentation depths and speeds. The models are found to have an excellent goodness of fit (with R (2) > 0.99).

CONCLUSIONS

The data and models presented in this paper together with additional ones based on the principles presented in this paper would result in realistic physics-based surgical simulators.

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.

Viscoelastic properties of the cervical spinal ligaments under fast strain-rate deformations

The mechanical response of ligaments under fast strain-rate deformations is a necessary input into computational models that are used for injury assessment. However, this information frequently is not available for the ligaments that are routinely injured in fast-rate loading scenarios. In the current study, experiments were conducted at fast strain rates for the cervical spinal ligaments: the anterior longitudinal ligament, the posterior longitudinal ligament and the ligamentum flavum. Bone-ligament-bone complexes at three spine levels were harvested for mechanical testing. Displacement-controlled sub-failure uniaxial tensile tests were performed in both load-relaxation and sinusoidal conditions. A nonlinear (separable) viscoelastic model was used to examine the experimental data. An unexpected result of the modeling was that the instantaneous elastic functions could be approximated as linear for these strain rates. A five-parameter model was sufficient to characterize the ligament viscoelastic responses and had good predictive capacity under different applied loading conditions.

A finite element model predicts the mechanotransduction response of tendon cells to cyclic tensile loading

The importance of fluid-flow-induced shear stress and matrix-induced cell deformation in transmitting the global tendon load into a cellular mechanotransduction response is yet to be determined. A multiscale computational tendon model composed of both matrix and fluid phases was created to examine how global tendon loading may affect fluid-flow-induced shear stresses and membrane strains at the cellular level. The model was then used to develop a quantitative experiment to help understand the roles of membrane strains and fluid-induced shear stresses on the biological response of individual cells. The model was able to predict the global response of tendon to applied strain (stress, fluid exudation), as well as the associated cellular response of increased fluid-flow-induced shear stress with strain rate and matrix-induced cell deformation with strain amplitude. The model analysis, combined with the experimental results, demonstrated that both strain rate and strain amplitude are able to independently alter rat interstitial collagenase gene expression through increases in fluid-flow-induced shear stress and matrix-induced cell deformation, respectively.

A simplified approach to quasi-linear viscoelastic modeling

The fitting of quasi-linear viscoelastic (QLV) constitutive models to material data often involves somewhat cumbersome numerical convolution. A new approach to treating quasi-linearity in 1-D is described and applied to characterize the behavior of reconstituted collagen. This approach is based on a new principle for including nonlinearity and requires considerably less computation than other comparable models for both model calibration and response prediction, especially for smoothly applied stretching. Additionally, the approach allows relaxation to adapt with the strain history. The modeling approach is demonstrated through tests on pure reconstituted collagen. Sequences of "ramp-and-hold" stretching tests were applied to rectangular collagen specimens. The relaxation force data from the "hold" was used to calibrate a new "adaptive QLV model" and several models from literature, and the force data from the "ramp" was used to check the accuracy of model predictions. Additionally, the ability of the models to predict the force response on a reloading of the specimen was assessed. The "adaptive QLV model" based on this new approach predicts collagen behavior comparably to or better than existing models, with much less computation.

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

Recent studies have questioned the ability of the quasi-linear viscoelastic (QLV) model to predict stresses and strains in response to loading conditions other than those used to fit the model. The objective of this study was to evaluate the ability of several models in the literature to predict the elastic stress response of ligament and tendon at strain levels higher than the levels used to fit the model. The constitutive models were then used to evaluate the ability of the QLV model to predict the overall stress response during stress relaxation. The models expressing stress as an exponential function of strain significantly overestimated stress when used at higher strain levels. The polynomial formulation of the Mooney-Rivlin model more accurately predicted the stress-strain behavior of ligament and tendon. The results demonstrate that the ability of the QLV model to accurately predict the stress-relaxation response is dependent in part on the accuracy of the function used to model the elastic response of the soft tissue.

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 1<or=t <or=500 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.

Fractional Calculus in Biomechanics: A 3D Viscoelastic Model Using Regularized Fractional Derivative Kernels with Application to the Human Calcaneal Fat Pad

A viscoelastic model of the K-BKZ (Kaye, Technical Report 134, College of Aeronautics, Cranfield 1962; Bernstein et al., Trans Soc Rheol 7: 391-410, 1963) type is developed for isotropic biological tissues and applied to the fat pad of the human heel. To facilitate this pursuit, a class of elastic solids is introduced through a novel strain-energy function whose elements possess strong ellipticity, and therefore lead to stable material models. This elastic potential - via the K-BKZ hypothesis - also produces the tensorial structure of the viscoelastic model. Candidate sets of functions are proposed for the elastic and viscoelastic material functions present in the model, including two functions whose origins lie in the fractional calculus. The Akaike information criterion is used to perform multi-model inference, enabling an objective selection to be made as to the best material function from within a candidate set.

A Constituent-Based Model for the Nonlinear Viscoelastic Behavior of Ligaments

This paper presents a constitutive model for predicting the nonlinear viscoelastic behavior of soft biological tissues and in particular of ligaments. The constitutive law is a generalization of the well-known quasi-linear viscoelastic theory (QLV) in which the elastic response of the tissue and the time-dependent properties are independently modeled and combined into a convolution time integral. The elastic behavior, based on the definition of anisotropic strain energy function, is extended to the time-dependent regime by means of a suitably developed time discretization scheme. The time-dependent constitutive law is based on the postulate that a constituent-based relaxation behavior may be defined through two different stress relaxation functions: one for the isotropic matrix and one for the reinforcing (collagen) fibers. The constitutive parameters of the viscoelastic model have been estimated by curve fitting the stress relaxation experiments conducted on medial collateral ligaments (MCLs) taken from the literature, whereas the predictive capability of the model was assessed by simulating experimental tests different from those used for the parameter estimation. In particular, creep tests at different maximum stresses have been successfully simulated. The proposed nonlinear viscoelastic model is able to predict the time-dependent response of ligaments described in experimental works (Bonifasi-Lista et al., 2005, J. Orthopaed. Res., 23, pp. 67–76;Hingorani et al., 2004, Ann. Biomed. Eng., 32, pp. 306–312;Provenzano et al., 2001, Ann. Biomed. Eng., 29, pp. 908–214;Weiss et al., 2002, J. Biomech., 35, pp. 943–950). In particular, the nonlinear viscoelastic response which implies different relaxation rates for different applied strains, as well as different creep rates for different applied stresses and direction-dependent relaxation behavior, can be described.

An inelastic multi-mechanism constitutive equation for cerebral arterial tissue

Intracranial aneurysms (ICA) are abnormal saccular dilations of cerebral arteries, commonly found at apices of arterial bifurcations and outer walls of curved arterial segments. Histological evidence suggests the stages in ICA development include the deformation of a segment of arterial wall into a "bleb" with no identifiable neck region followed by the development of an aneurysm with a clear neck. Afterwards, the aneurysm may undergo further enlargement, possibly with significant biological response including calcification and thrombosis. Past studies of the biomechanics of cerebral aneurysm tissue have been directed at modeling elastic deformations of pre-existing aneurysms. Taking this approach, the aneurysm wall is treated as a different entity than the arterial tissue from which it developed. In the current work, a nonlinear, inelastic, dual-mechanism constitutive equation for cerebral arterial tissue is developed. It is the first to model the recruitment of collagen fibers and degradation of the internal elastic lamina, two important characteristics of early stage aneurysm formation.

Pedestrian injuries: viscoelastic properties of human knee ligaments at high loading rates

OBJECTIVE

Accidents involving pedestrians are very common, and often lead to severe injuries to the lower extremities. In a large portion of pedestrian-automobile collisions, knee ligament injuries are sustained. In this study, the viscoelastic properties of the four major human knee ligaments were investigated at loading rates representative for pedestrian-automobile collisions.

METHODS

Bone-ligament-bone specimens were tested in knee distraction loading. The collateral ligaments and the separate functional bundles of the cruciate ligaments were tested in the anatomical position corresponding to a fully extended knee. A series of step-and-hold tests and ramp tests at different rates were conducted to characterize the time-dependent behavior of the knee ligaments for deformation rates associated with the pedestrian impact loading environment. The quasi linear viscoelastic (QLV) theory was used to describe the structural response of the knee ligaments and averaged parameters for this model were determined.

RESULTS

The QLV theory was found to be applicable for the time range that is relevant for pedestrian-automobile collisions. The structural behavior of the knee ligaments was found to be particularly rate-sensitive for high elongation rates, as occur during these collisions. The ligament stiffness was found to increase with age for both the collateral ligaments and with weight for the medial collateral ligament.

CONCLUSIONS

For the loading conditions that are relevant for pedestrian-automobile collisions, the use of the QLV model for the description of the mechanical behavior of knee ligaments is appropriate. The rate-sensitivity is particularly important for these extreme loading conditions. The relaxation behavior was found to be consistent between different ligament types and samples. Variations due to donor anthropometry were found predominantly for the instantaneous elastic behavior.

Pedestrian lower limb injury criteria evaluation: a finite element approach

OBJECTIVE

In pedestrian traumas, lower limb injuries occur under lateral shearing and bending at the knee joint level. One way to improve injury mechanisms description and consequently knee joint safety is to evaluate the ultimate shearing and bending levels at which ligaments start being injured.

METHODS

As such data cannot easily and accurately be recorded clinically or during experiments, we show in this article how numerical simulation can be used to estimate such thresholds. This work was performed with the Lower Limb Model for Safety (LLMS) in pure lateral bending and shearing conditions, with an extended range of impact velocities.

RESULTS

One result concerns the ultimate knee lateral bending angle and shearing displacement measurements for potential failure of ligaments (posterior cruciate, medial collateral, anterior cruciates and tibial collateral). They were evaluated to be close to 16 degrees and 15 mm, respectively.

CONCLUSION

The lower leg model used in this study is an advanced FE model of the lower limb, validated under various situations. Its accurate anatomical description allows a wide range of applications. According to the validity domain of the model, it offered a valuable tool for the numerical evaluation of potential injuries and the definition of injury risk criterion for knee joint.

Mechanical and functional properties of the equine superficial digital flexor tendon

The in vitro and in vivo mechanical properties of the superficial digital flexor tendon have been described. To date the focus has been on single load to failure testing, however refined in vivo methods may prove useful to evaluate the effects of treatment and exercise on tendons. During maximal exercise, the adult superficial digital flexor tendon operates close to its functional limits with a narrow biomechanical safety margin. This combined with exercise and age associated microdamage, and a limited adaptive ability may increase the risk of fatigue failure. Studies evaluating treatment regimens for tendonitis have focused on repair and regeneration and yielded varying results. It would appear that the superficial digital flexor tendon has a limited ability if any to adapt positively to exercise after maturity. In contrast, the foal's superficial digital flexor tendon may have a greater adaptive ability and may respond to an appropriate exercise regimen to produce a more functionally adapted tendon. Recent studies have shown that foals allowed free pasture exercise develop a larger, stronger, more elastic tendon compared to foals that were confined or subjected to a training program. Effects on the non-collagenous matrix appear to be responsible for these differences. In contrast, training or excess exercise may have permanent detrimental effects on the biomechanical and functional properties of the superficial digital flexor tendon in the foal. The implication is that the determination of optimum exercise intensity and timing, and the role of the non-collagenous matrix in tendon physiology in the young horse may hold the key to developing tendons more capable of resisting injury.

Three-dimensional finite element modeling of ligaments: technical aspects

The objective of this paper is to describe strategies for addressing technical aspects of the computational modeling of ligaments with the finite element (FE) method. Strategies for FE modeling of ligament mechanics are described, differentiating between whole-joint models and models of individual ligaments. Common approaches to obtain three-dimensional ligament geometry are reviewed, with an emphasis on techniques that rely on volumetric medical image data. Considerations for the three-dimensional constitutive modeling of ligaments are reviewed in the context of ligament composition and structure. A novel approach to apply in situ strain to FE models of ligaments is described, and test problems are presented that demonstrate the efficacy of the approach. Approaches for the verification and validation of ligament FE models are outlined. The paper concludes with a discussion of future research directions.

Heat-Induced Changes in the Finite Strain Viscoelastic Behavior of a Collaagenous Tissue

Supra-physiological temperatures are increasingly being used to treat many different soft tissue diseases and injuries. To identify improved clinical treatments, however, there is a need for better information on the effect of the mechanics on the thermal damage process as well as the effect of the incurred damage on the subsequent mechanical properties. In this paper, we report the first biaxial data on the stress relaxation behavior of a collagenous tissue before and after thermal damage. Based on a two-dimensional finite strain viscoelastic model, which incorporates an exponential elastic response, it is shown that the thermal damage can significantly decrease the characteristic time for stress relaxation and the stress residual.

Biomechanics of knee ligaments: injury, healing, and repair

Knee ligament injuries are common, particularly in sports and sports related activities. Rupture of these ligaments upsets the balance between knee mobility and stability, resulting in abnormal knee kinematics and damage to other tissues in and around the joint that lead to morbidity and pain. During the past three decades, significant advances have been made in characterizing the biomechanical and biochemical properties of knee ligaments as an individual component as well as their contribution to joint function. Further, significant knowledge on the healing process and replacement of ligaments after rupture have helped to evaluate the effectiveness of various treatment procedures. This review paper provides an overview of the current biological and biomechanical knowledge on normal knee ligaments, as well as ligament healing and reconstruction following injury. Further, it deals with new and exciting functional tissue engineering approaches (ex. growth factors, gene transfer and gene therapy, cell therapy, mechanical factors, and the use of scaffolding materials) aimed at improving the healing of ligaments as well as the interface between a replacement graft and bone. In addition, it explores the anatomical, biological and functional perspectives of current reconstruction procedures. Through the utilization of robotics technology and computational modeling, there is a better understanding of the kinematics of the knee and the in situ forces in knee ligaments and replacement grafts. The research summarized here is multidisciplinary and cutting edge that will ultimately help improve the treatment of ligament injuries. The material presented should serve as an inspiration to future investigators.

A discrete-time approach to the formulation of constitutive models for viscoelastic soft tissues

This paper presents a novel approach to constitutive modeling of viscoelastic soft tissues. This formulation combines an anisotropic strain energy function, accounting for preferred material directions, to define the elastic stress-strain relationship, and a discrete time black-box dynamic model, borrowed from the theory of system identification, to describe the time-dependent behavior. This discrete time formulation is straightforwardly oriented to the development of a recursive time integration scheme that calculates the current stress state by using strain and stress values stored at a limited number of previous time instants. The viscoelastic model and the numerical procedure are assessed by implementing two numerical examples, the simulation of a uniaxial tensile test and the inflation of a thin tube. Both simulations are performed using parameter values based on previous experiments on preserved bovine pericardium. Parameters are then adjusted to investigate the sensitivity of the model. The hypotheses the model relies upon are discussed and the main limitations are stated.

Nonlinear Viscoelastic Material Property Estimation of Lower Extremity Residual Limb Tissues

Axisymmetric nonlinear finite-element analysis was used to simulate force-relaxation and creep data obtained during in vivo indentation of the residual limb soft tissues of six individuals with trans-tibial amputation [1]. The finite-element models facilitated estimation of an appropriate set of nonlinear viscoelastic material coefficients of extended James-Green-Simpson material formulation for bulk soft tissue at discrete, clinically relevant test locations. The results indicate that over 90% of the experimental data can be simulated using the two-term viscoelastic Prony series extension of James-Green-Simpson material formulation. This phenomenological material formulation could not, however, predict the creep response from relaxation experiments, nor the relaxation response from creep experiments [2–5]. The estimated material coefficients varied with test location and subject indicating that these coefficients cannot be readily extrapolated to other sites or individuals.

Injury and repair of ligaments and tendons

In this chapter, biomechanical methods used to analyze healing and repair of ligaments and tendons are initially described such that the tensile properties of these soft tissues as well as their contribution to joint motion can be determined. The focus then turns to the important mechanical and biological factors that improve the healing process of ligaments. The biomechanics of surgical reconstruction of the anterior cruciate ligament and the key surgical parameters that affect the performance of the replacement grafts are subsequently reviewed. Finally, injury mechanisms and the biomechanical analysis of various treatment techniques for various types of tendon injuries are described.