A novel microstructural approach in tendon viscoelastic modelling at the fibrillar level

Biomechanical Properties of Knee Medial Collateral Ligament Compared to Palmaris Longus for Ulnar Collateral Ligament Reconstruction

Ulnar collateral ligament reconstruction (UCLR) is frequently performed among injured overhead-throwing athletes. One of the most common graft choices when performing a UCLR is the ipsilateral palmaris longus tendon (PL). The purpose of this study was to investigate the material properties of aseptically processed cadaveric knee collateral ligaments (kMCL) as a potential graft source for UCLR and compare them to the gold standard PL autograft. Each PL and kMCL cadaveric sample was subjected to cyclic preconditioning, stress relaxation, and load-to-failure testing, and the mechanical properties were recorded. PL samples exhibited a greater average decrease in stress compared to the kMCL samples during the stress-relaxation test (p < 0.0001). PL samples also demonstrated a greater average Young's modulus in the linear region of the stress-strain curve compared to the kMCL samples (p < 0.01). The average yield strain and maximum strain of kMCL samples were significantly greater than the PL, p = 0.03 and 0.02, respectively. Both graft materials had comparable maximum toughness and demonstrated a similar ability to deform plastically without rupture. The clinical significance of our result is that prepared knee medial collateral ligament allografts may provide a viable graft material for use in the reconstruction of elbow ligaments.

Biomechanics of the Human Middle Ear with Viscoelasticity of the Maxwell and the Kelvin-Voigt Type and Relaxation Effect

The middle ear is one of the smallest biomechanical systems in the human body and is responsible for the hearing process. Hearing is modelled in different ways and by various methods. In this paper, three-degree-of-freedom models of the human middle ear with different viscoelastic properties are proposed. Model 1 uses the Maxwell type viscoelasticity, Model 2 is based on the Kelvin–Voigt viscoelasticity, and Model 3 uses the Kelvin–Voigt viscoelasticity with relaxation effect. The primary aim of the study is to compare the models and their dynamic responses to a voice excitation. The novelty of this study lies in using different models of viscoelasticity and relaxation effect that has been previously unstudied. First, mathematical models of the middle ear were built, then they were solved numerically by the Runge–Kutta procedure and finally, numerical results were compared with those obtained from experiments carried out on the temporal bone with the Laser Doppler Vibrometer. The models exhibit differences in the natural frequency and amplitudes near the second resonance. All analysed models can be used for modelling the rapidly changing processes that occur in the ear and to control active middle ear implants.

Energy dissipation in quasi-linear viscoelastic tissues, cells, and extracellular matrix

Characterizing how a tissue's constituents give rise to its viscoelasticity is important for uncovering how hidden timescales underlie multiscale biomechanics. These constituents are viscoelastic in nature, and their mechanics must typically be assessed from the uniaxial behavior of a tissue. Confounding the challenge is that tissue viscoelasticity is typically associated with nonlinear elastic responses. Here, we experimentally assessed how fibroblasts and extracellular matrix (ECM) within engineered tissue constructs give rise to the nonlinear viscoelastic responses of a tissue. We applied a constant strain rate, "triangular-wave" loading and interpreted responses using the Fung quasi-linear viscoelastic (QLV) material model. Although the Fung QLV model has several well-known weaknesses, it was well suited to the behaviors of the tissue constructs, cells, and ECM tested. Cells showed relatively high damping over certain loading frequency ranges. Analysis revealed that, even in cases where the Fung QLV model provided an excellent fit to data, the the time constant derived from the model was not in general a material parameter. Results have implications for design of protocols for the mechanical characterization of biological materials, and for the mechanobiology of cells within viscoelastic tissues.

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.

Comparison of structural anisotropic soft tissue models for simulating Achilles tendon tensile behaviour

The incidence of tendon injury (tendinopathy) has increased over the past decades due to greater participation in sports and recreational activities. But little is known about the aetiology of tendon injuries because of our limited knowledge in the complex structure-function relationship in tendons. Computer models can capture the biomechanical behaviour of tendons and its structural components, which is essential for understanding the underlying mechanisms of tendon injuries. This study compares three structural constitutive material models for the Achilles tendon and discusses their application on different biomechanical simulations. The models have been previously used to describe cardiovascular tissue and articular cartilage, and one model is novel to this study. All three constitutive models captured the tensile behaviour of rat Achilles tendon (root mean square errors between models and experimental data are 0.50-0.64). They further showed that collagen fibres are the main load-bearing component and that the non-collagenous matrix plays a minor role in tension. By introducing anisotropic behaviour also in the non-fibrillar matrix, the new biphasic structural model was also able to capture fluid exudation during tension and high values of Poisson׳s ratio that is reported in tendon experiments.

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.

A nonlinear-elastic constitutive model for soft connective tissue based on a histologic description: Application to female pelvic soft tissue

To understand the mechanical behavior of soft tissues, two fields of science are essential: biomechanics and histology. Nonetheless, those two fields have not yet been studied together often enough to be unified by a comprehensive model. This study attempts to produce such model. Biomechanical uniaxial tension tests were performed on vaginal tissues from 7 patients undergoing surgery. In parallel, vaginal tissue from the same patients was histologically assessed to determine the elastic fiber ratio. These observations demonstrated a relationship between the stiffness of tissue and its elastin content. To extend this study, a mechanical model, based on an histologic description, was developed to quantitatively correlate the mechanical behavior of vaginal tissue to its elastic fiber content. A satisfactory single-parameter model was developed assuming that the mechanical behavior of collagen and elastin was the same for all patients and that tissues are only composed of collagen and elastin. This single-parameter model showed good correlation with experimental results. The single-parameter mechanical model described here, based on histological description, could be very useful in helping to understand and better describe soft tissues with a view to their characterization. The mechanical behavior of a tissue can thus be determined thanks to its elastin content without introducing too many unidentified parameters.

Micromechanical poroelastic finite element and shear-lag models of tendon predict large strain dependent Poisson's ratios and fluid expulsion under tensile loading

As tendons are loaded, they reduce in volume and exude fluid to the surrounding medium. Experimental studies have shown that tendon stretching results in a Poisson's ratio greater than 0.5, with a maximum value at small strains followed by a nonlinear decay. Here we present a computational model that attributes this macroscopic observation to the microscopic mechanism of the load transfer between fibrils under stretch. We develop a finite element model based on the mechanical role of the interfibrillar-linking elements, such as thin fibrils that bridge the aligned fibrils or macromolecules such as glycosaminoglycans (GAGs) in the interfibrillar sliding and verify it with a theoretical shear-lag model. We showed the existence of a previously unappreciated structure-function mechanism whereby the Poisson's ratio in tendon is affected by the strain applied and interfibrillar-linker properties, and together these features predict tendon volume shrinkage under tensile loading. During loading, the interfibrillar-linkers pulled fibrils toward each other and squeezed the matrix, leading to the Poisson's ratio larger than 0.5 and fluid expulsion. In addition, the rotation of the interfibrillar-linkers with respect to the fibrils at large strains caused a reduction in the volume shrinkage and eventual nonlinear decay in Poisson's ratio at large strains. Our model also predicts a fluid flow that has a radial pattern toward the surrounding medium, with the larger fluid velocities in proportion to the interfibrillar sliding.

A fibre-reinforced poroviscoelastic model accurately describes the biomechanical behaviour of the rat Achilles tendon

Background

Computational models of Achilles tendons can help understanding how healthy tendons are affected by repetitive loading and how the different tissue constituents contribute to the tendon’s biomechanical response. However, available models of Achilles tendon are limited in their description of the hierarchical multi-structural composition of the tissue. This study hypothesised that a poroviscoelastic fibre-reinforced model, previously successful in capturing cartilage biomechanical behaviour, can depict the biomechanical behaviour of the rat Achilles tendon found experimentally.

Materials and Methods

We developed a new material model of the Achilles tendon, which considers the tendon’s main constituents namely: water, proteoglycan matrix and collagen fibres. A hyperelastic formulation of the proteoglycan matrix enabled computations of large deformations of the tendon, and collagen fibres were modelled as viscoelastic. Specimen-specific finite element models were created of 9 rat Achilles tendons from an animal experiment and simulations were carried out following a repetitive tensile loading protocol. The material model parameters were calibrated against data from the rats by minimising the root mean squared error (RMS) between experimental force data and model output.

Results and Conclusions

All specimen models were successfully fitted to experimental data with high accuracy (RMS 0.42-1.02). Additional simulations predicted more compliant and soft tendon behaviour at reduced strain-rates compared to higher strain-rates that produce a stiff and brittle tendon response. Stress-relaxation simulations exhibited strain-dependent stress-relaxation behaviour where larger strains produced slower relaxation rates compared to smaller strain levels. Our simulations showed that the collagen fibres in the Achilles tendon are the main load-bearing component during tensile loading, where the orientation of the collagen fibres plays an important role for the tendon’s viscoelastic response. In conclusion, this model can capture the repetitive loading and unloading behaviour of intact and healthy Achilles tendons, which is a critical first step towards understanding tendon homeostasis and function as this biomechanical response changes in diseased tendons.

Tendon mechanobiology: experimental models require mathematical underpinning

Mathematical and computational modeling is in demand to help address current challenges in mechanobiology of musculoskeletal tissues. In particular for tendon, the high clinical importance of the tissue, the huge mechanical demands placed on it and its ability to adapt to these demands, require coupled, multiscale models incorporating complex geometrical and microstructural information as well as time-based descriptions of cellular activity and response.This review introduces the information sources required to develop such multiscale models. It covers tissue structure and biomechanics, cell biomechanics, the current understanding of tendon's ability in health and disease to update its properties and structure and the few already existing multiscale mechanobiological models of the tissue. Finally, a sketch is provided of what such models could achieve ideally, pointing out where experimental data and knowledge are still missing.

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.

Non-uniform distribution of strain during stretch of relaxed skeletal muscle fibers from rat soleus muscle

Tension and regional average sarcomere length (L(s)) behavior were examined during repeated stretches of single, permeabilized, relaxed muscle fibers isolated from the soleus muscles of rats. We tested the hypothesis that during stretches of single permeabilized fibers, the global fiber strain is distributed non-uniformly along the length of a relaxed fiber in a repeatable pattern. Each fiber was subjected to eight constant-velocity stretch and release cycles with a strain of 32% and strain rate of 54% s(-1). Stretch-release cycles were separated by a 4.5 min interval. Throughout each stretch-release cycle, sarcomere lengths were measured using a laser diffraction technique in which 20 contiguous sectors along the entire length of a fiber segment were scanned within 2 ms. The results revealed that: (1) the imposed length change was not distributed uniformly along the fiber, (2) the first stretch-release cycle differed from subsequent cycles in passive tension and in the distribution of global fiber strain, and (3) a characteristic "signature" for the L(s) response emerged after cycle 3. The findings support the conclusions that longitudinal heterogeneity exists in the passive stiffness of individual muscle fibers and that preconditioning of fibers with stretch-release cycles produces a stable pattern of sarcomere strains.

Design and Preliminary Evaluation of a Novel Brace for Boutonniere Deformity

Boutonniere deformities are a common injury to the extensor mechanism of the finger. The deformity results in fixed contraction in the middle finger joint and is severely debilitating to functionality. Due to the complexity of the extensor mechanism, surgical repair is difficult, which usually requires multiple procedures, and in some cases is unsuccessful. Nonsurgical treatment of the deformity has not dramatically improved in many years and usually requires long-term use of braces and physical therapy. This work is focused on design and preliminary evaluation of an improved boutonniere brace to correct the deformity with emphasis on lower costs, integrating therapeutic techniques such as heat and motion to increase blood flow and patient comfort. A review of the current state of the art is presented along with the design approach used to develop an improved device. Experimental test results are also presented. This work demonstrates a new device and approach for treatment of boutonniere deformities that may translate to treatment of other conditions such as arthritis.

A unified multiscale mechanical model for soft collagenous tissues with regular fiber arrangement

In this paper the mechanical response of soft collagenous tissues with regular fiber arrangement (RSCTs) is described by means of a nanoscale model and a two-step micro-macro homogenization technique. The non-linear collagen constitutive behavior is modeled at the nanoscale by a novel approach accounting for entropic mechanisms as well as stretching effects occurring in collagen molecules. Crimped fibers are reduced to equivalent straight ones at the microscale and the constitutive response of RSCTs at the macroscale is formulated by homogenizing a fiber reinforced material. This approach has been applied to different RSCTs (tendon, periodontal ligament and aortic media), resulting effective and accurate as proved by the excellent agreement with available experimental data. The model is based on few parameters, directly related to histological and morphological evidences and whose sensitivity has been widely investigated. Applications to simulation of some physiopathological mechanisms are also proposed, providing confirmation of clinical evidences and quantitative indications helpful for clinical practice.

Hyperelastic Model of Anisotropic Fiber Reinforcements within Intestinal Walls for Applications in Medical Robotics

The development of an anatomically realistic model of intestinal tissue is essential for the progress of several clinical applications of medical robotics. A hyperelastic theory of the layered structure of the intestine is proposed in this paper to reproduce its purely elastic passive response from the structural organization of its main constituents. The hyperelastic strain energy function is decoupled into an isotropic term, describing the ground biological matrix, and an anisotropic term, describing the single contributions of the directional fiber-reinforcements. The response of the muscular coat layer has been modeled as a stiffening effect due to two longitudinal and circular muscular reinforcements. The contribution of the submucosa has been described from a uniform distribution of fibrillar collagen in a cross-ply arrangement. An experimental procedure has been proposed in order to characterize the passive response of porcine intestinal samples from planar uniaxial traction and shear tests. The experimental data have been non-linearly fitted in the least square sense with the results of the theoretical predictions. The mechanical parameters have been fitted with high accuracy (Rmin =0.9329, RMSEmax =0.01167), demonstrating the ability of the model to reproduce the mechanical coupling due to the presence of multiple directional reinforcements. The fundamental mechanical role of collagen morphology in the passive biomechanical behavior of intestinal wall is demonstrated. These results may drive a better understanding of the key factors in growth and remodeling of healthy and diseased tissue, together with numerous applications in robotic endoscopy, minimally invasive surgery, and biomedical research.

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.

A finite dissipative theory of temporary interfibrillar bridges in the extracellular matrix of ligaments and tendons

The structural integrity and the biomechanical characteristics of ligaments and tendons result from the interactions between collagenous and non-collagenous proteins (e.g. proteoglycans, PGs) in the extracellular matrix. In this paper, a dissipative theory of temporary interfibrillar bridges in the anisotropic network of collagen type I, embedded in a ground substance, is derived. The glycosaminoglycan chains of decorin are assumed to mediate interactions between fibrils, behaving as viscous structures that transmit deformations outside the collagen molecules. This approach takes into account the dissipative effects of the unfolding preceding fibrillar elongation, together with the slippage of entire fibrils and the strain-rate-dependent damage evolution of the interfibrillar bridges. Thermodynamic consistency is used to derive the constitutive equations, and the transition state theory is applied to model the rearranging properties of the interfibrillar bridges. The constitutive theory is applied to reproduce the hysteretic spectrum of the tissues, demonstrating how PGs determine damage evolution, softening and non-recoverable strains in their cyclic mechanical response. The theoretical predictions are compared with the experimental response of ligaments and tendons from referenced studies. The relevance of the proposed model in mechanobiology research is discussed, together with several applications from medical practice to bioengineering science.

Contribution of glycosaminoglycans to viscoelastic tensile behavior of human ligament

The viscoelastic properties of human ligament potentially guard against structural failure, yet the microstructural origins of these transient behaviors are unknown. Glycosaminoglycans (GAGs) are widely suspected to affect ligament viscoelasticity by forming molecular bridges between neighboring collagen fibrils. This study investigated whether GAGs directly affect viscoelastic material behavior in human medial collateral ligament (MCL) by using nondestructive tensile tests before and after degradation of GAGs with chondroitinase ABC (ChABC). Control and ChABC treatment (83% GAG removal) produced similar alterations to ligament viscoelasticity. This finding was consistent at different levels of collagen fiber stretch and tissue hydration. On average, stress relaxation increased after incubation by 2.2% (control) and 2.1% (ChABC), dynamic modulus increased after incubation by 3.6% (control) and 3.8% (ChABC), and phase shift increased after incubation by 8.5% (control) and 8.4% (ChABC). The changes in viscoelastic behavior after treatment were significantly more pronounced at lower clamp-to-clamp strain levels. A 10% difference in the water content of tested specimens had minor influence on ligament viscoelastic properties. The major finding of this study is that mechanical interactions between collagen fibrils and GAGs are unrelated to tissue-level viscoelastic mechanics in mature human MCL. These findings narrow the possible number of extracellular matrix molecules that have a direct contribution to ligament viscoelasticity.