A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fascicle microstructure

The alteration of the structure and macroscopic mechanical response of porcine patellar tendon by elastase digestion

Background: The treatment of patellar tendon injury has always been an unsolved problem, and mechanical characterization is very important for its repair and reconstruction. Elastin is a contributor to mechanics, but it is not clear how it affects the elasticity, viscoelastic properties, and structure of patellar tendon. Methods: The patellar tendons from six fresh adult experimental pigs were used in this study and they were made into 77 samples. The patellar tendon was specifically degraded by elastase, and the regional mechanical response and structural changes were investigated by: (1) Based on the previous study of elastase treatment conditions, the biochemical quantification of collagen, glycosaminoglycan and total protein was carried out; (2) The patellar tendon was divided into the proximal, central, and distal regions, and then the axial tensile test and stress relaxation test were performed before and after phosphate-buffered saline (PBS) or elastase treatment; (3) The dynamic constitutive model was established by the obtained mechanical data; (4) The structural relationship between elastin and collagen fibers was analyzed by two-photon microscopy and histology. Results: There was no statistical difference in mechanics between patellar tendon regions. Compared with those before elastase treatment, the low tensile modulus decreased by 75%-80%, the high tensile modulus decreased by 38%-47%, and the transition strain was prolonged after treatment. For viscoelastic behavior, the stress relaxation increased, the initial slope increased by 55%, the saturation slope increased by 44%, and the transition time increased by 25% after enzyme treatment. Elastin degradation made the collagen fibers of patellar tendon become disordered and looser, and the fiber wavelength increased significantly. Conclusion: The results of this study show that elastin plays an important role in the mechanical properties and fiber structure stability of patellar tendon, which supplements the structure-function relationship information of patellar tendon. The established constitutive model is of great significance to the prediction, repair and replacement of patellar tendon injury. In addition, human patellar tendon has a higher elastin content, so the results of this study can provide supporting information on the natural properties of tendon elastin degradation and guide the development of artificial patellar tendon biomaterials.

Shape or size matters? Towards standard reporting of tensile testing parameters for human soft tissues: systematic review and finite element analysis

Material properties of soft-tissue samples are often derived through uniaxial tensile testing. For engineering materials, testing parameters (e.g., sample geometries and clamping conditions) are described by international standards; for biological tissues, such standards do not exist. To investigate what testing parameters have been reported for tensile testing of human soft-tissue samples, a systematic review of the literature was performed using PRISMA guidelines. Soft tissues are described as anisotropic and/or hyperelastic. Thus, we explored how the retrieved parameters compared against standards for engineering materials of similar characteristics. All research articles published in English, with an Abstract, and before 1 January 2023 were retrieved from databases of PubMed, Web of Science, and BASE. After screening of articles based on search terms and exclusion criteria, a total 1,096 articles were assessed for eligibility, from which 361 studies were retrieved and included in this review. We found that a non-tapered shape is most common (209 of 361), followed by a tapered sample shape (92 of 361). However, clamping conditions varied and were underreported (156 of 361). As a preliminary attempt to explore how the retrieved parameters might influence the stress distribution under tensile loading, a pilot study was performed using finite element analysis (FEA) and constitutive modeling for a clamped sample of little or no fiber dispersion. The preliminary FE simulation results might suggest the hypothesis that different sample geometries could have a profound influence on the stress-distribution under tensile loading. However, no conclusions can be drawn from these simulations, and future studies should involve exploring different sample geometries under different computational models and sample parameters (such as fiber dispersion and clamping effects). Taken together, reporting and choice of testing parameters remain as challenges, and as such, recommendations towards standard reporting of uniaxial tensile testing parameters for human soft tissues are proposed.

Continuum soft tissue models from upscaling of arrays of hyperelastic cells

Constitutive models for soft tissue mechanics are typically constructed by fitting phenomenological models to experimental measurements. However, a significant challenge is to rationally construct soft tissue models that encode the properties of the constituent cells and their extracellular matrix. This work presents a framework to derive multiscale soft tissue models that incorporate properties of individual cells without assuming homogeneity or periodicity at the cell level. We consider a viscoelastic model for each cell (which can deform, grow and divide), that we couple to form a network description of a one-dimensional line of cells. We use a discrete-to-continuum approach to form (nonlinear) continuum partial differential equation models for the tissue. These models elucidate the contrasting role of the two forms of dissipation: substrate dissipation localizes the deformation to the neighbourhood of the free boundary and inhibits morphoelastic growth, whereas internal cell dissipation promotes spatial uniformity and does not influence the elongation length. Furthermore, cell division is shown to increase the rate of elongation of the array compared with growth alone, provided the substrate dissipation is proportional to the cell surface area.

One step further in biomechanical models in palaeontology: a nonlinear finite element analysis review

Finite element analysis (FEA) is no longer a new technique in the fields of palaeontology, anthropology, and evolutionary biology. It is nowadays a well-established technique within the virtual functional-morphology toolkit. However, almost all the works published in these fields have only applied the most basic FEA tools i.e., linear materials in static structural problems. Linear and static approximations are commonly used because they are computationally less expensive, and the error associated with these assumptions can be accepted. Nonetheless, nonlinearities are natural to be used in biomechanical models especially when modelling soft tissues, establish contacts between separated bones or the inclusion of buckling results. The aim of this review is to, firstly, highlight the usefulness of non-linearities and secondly, showcase these FEA tool to researchers that work in functional morphology and biomechanics, as non-linearities can improve their FEA models by widening the possible applications and topics that currently are not used in palaeontology and anthropology.

Bayesian inference on a microstructural, hyperelastic model of tendon deformation

Microstructural models of soft-tissue deformation are important in applications including artificial tissue design and surgical planning. The basis of these models, and their advantage over their phenomenological counterparts, is that they incorporate parameters that are directly linked to the tissue's microscale structure and constitutive behaviour and can therefore be used to predict the effects of structural changes to the tissue. Although studies have attempted to determine such parameters using diverse, state-of-the-art, experimental techniques, values ranging over several orders of magnitude have been reported, leading to uncertainty in the true parameter values and creating a need for models that can handle such uncertainty. We derive a new microstructural, hyperelastic model for transversely isotropic soft tissues and use it to model the mechanical behaviour of tendons. To account for parameter uncertainty, we employ a Bayesian approach and apply an adaptive Markov chain Monte Carlo algorithm to determine posterior probability distributions for the model parameters. The obtained posterior distributions are consistent with parameter measurements previously reported and enable us to quantify the uncertainty in their values for each tendon sample that was modelled. This approach could serve as a prototype for quantifying parameter uncertainty in other soft tissues.

A hyperelastic-damage model to study the anisotropic mechanical behavior of coral-hydrogel bio-composite

In this research, a hyperelastic strain energy function was developed to investigate the anisotropic mechanical behavior of a soft bio-composite containing the coral collagen fibers and alginate hydrogel matrix. This hyperelastic function considers the matrix isotropic response, anisotropic behavior of soft crimpled fibers, and matrix-fiber interaction. The material parameters of the model were assumed as a function of the volume fraction of fibers to consider the fiber content. The published experimental data of matrix and bio-composites with different volume fractions of fibers in different directions were used to find the material parameters. A damage model was developed to take into account the damage of matrix, fibers, and fiber-matrix interaction. Results showed that the interaction contribution increases by increasing the volume fraction of fibers due to the crosslinks between the matrix and fibers. Moreover, by exceeding the fibers volume fraction from 20%, the anisotropic stiffening dominates the interaction stiffening in the longitudinal test owing to the load-bearing of soft fibers. Simulation results exhibited that cross-plied bio-composites show more uniform deformations than angle-plied bio-composites. Moreover, the damage results showed that the matrix plays a significant role in the failure of bio-composites.

Collagen fibril assembly: New approaches to unanswered questions

Collagen fibrils are essential for metazoan life. They are the largest, most abundant, and most versatile protein polymers in animals, where they occur in the extracellular matrix to form the structural basis of tissues and organs. Collagen fibrils were first observed at the turn of the 20th century. During the last 40 years, the genes that encode the family of collagens have been identified, the structure of the collagen triple helix has been solved, the many enzymes involved in the post-translational modifications of collagens have been identified, mutations in the genes encoding collagen and collagen-associated proteins have been linked to heritable disorders, and changes in collagen levels have been associated with a wide range of diseases, including cancer. Yet despite extensive research, a full understanding of how cells assemble collagen fibrils remains elusive. Here, we review current models of collagen fibril self-assembly, and how cells might exert control over the self-assembly process to define the number, length and organisation of fibrils in tissues.

A microstructural model of tendon failure

Collagen fibrils are the most important structural component of tendons. Their crimped structure and parallel arrangement within the tendon lead to a distinctive non-linear stress-strain curve when a tendon is stretched. Microstructural models can be used to relate microscale collagen fibril mechanics to macroscale tendon mechanics, allowing us to identify the mechanisms behind each feature present in the stress-strain curve. Most models in the literature focus on the elastic behaviour of the tendon, and there are few which model beyond the elastic limit without introducing phenomenological parameters. We develop a model, built upon a collagen recruitment approach, that only contains microstructural parameters. We split the stress in the fibrils into elastic and plastic parts, and assume that the fibril yield stretch and rupture stretch are each described by a distribution function, rather than being single-valued. By changing the shapes of the distributions and their regions of overlap, we can produce macroscale tendon stress-strain curves that generate the full range of features observed experimentally, including those that could not be explained using existing models. These features include second linear regions occurring after the tendon has yielded, and step-like failure behaviour present after the stress has peaked. When we compare with an existing model, we find that our model reduces the average root mean squared error from 4.53MPa to 2.29MPa, and the resulting parameter values are closer to those found experimentally. Since our model contains only parameters that have a direct physical interpretation, it can be used to predict how processes such as ageing, disease, and injury affect the mechanical behaviour of tendons, provided we can quantify the effects of these processes on the microstructure.

Shear wave speeds in nearly-incompressible fibrous materials with two fiber families

An analytical and numerical investigation of shear wave behavior in nearly-incompressible soft materials with two fiber families was performed, focusing on the effects of material parameters and imposed pre-deformations on wave speed. This theoretical study is motivated by the emerging ability to image shear waves in soft biological tissues by magnetic resonance elastography (MRE). In MRE, the relationships between wave behavior and mechanical properties can be used to characterize tissue properties non-invasively. We demonstrate these principles in two material models, each with two fiber families. One model is a nearly-incompressible linear elastic model that exhibits both shear and tensile anisotropy; the other is a two-fiber-family version of the widely-used Holzapfel-Gasser-Ogden (HGO) model, which is nonlinear. Shear waves can be used to probe nonlinear material behavior using infinitesimal dynamic deformations superimposed on larger, quasi-static "pre-deformations." In this study, closed-form expressions for shear wave speeds in the HGO model are obtained in terms of the model parameters and imposed pre-deformations. Analytical expressions for wave speeds are confirmed by finite element simulations of shear waves with various polarizations and propagation directions. These studies support the feasibility of estimating the parameters of an HGO material model noninvasively from measured shear wave speeds.

Geometrically nonlinear modelling of pre-stressed viscoelastic fibre-reinforced composites with application to arteries

Mechanical behaviour of pre-stressed fibre-reinforced composites is modelled in a geometrically exact setting. A general approach which includes two different reference configurations is employed: one configuration corresponds to the load-free state of the structure and another one to the stress-free state of each material particle. The applicability of the approach is demonstrated in terms of a viscoelastic material model; both the matrix and the fibre are modelled using a multiplicative split of the deformation gradient tensor; a transformation rule for initial conditions is elaborated and specified. Apart from its simplicity, an important advantage of the approach is that well-established numerical algorithms can be used for pre-stressed inelastic structures. The interrelation between the advocated approach and the widely used "opening angle" approach is clarified. A full-scale FEM simulation confirms the main predictions of the "opening angle" approach. A locking effect is discovered: in some cases the opening angle of the composite is essentially smaller than the opening angles of its individual layers. Thus, the standard cutting test typically used to analyse pre-stresses does not carry enough information and more refined experimental techniques are needed.

Characterization of hyperelastic and damage behavior of tendons

In this paper, we characterized the hyperelastic and damage behavior of the Extensor Digitorum Longus (EDL) human tendon under loading conditions. The study was conducted in both categories of models, phenomenological and physically motivated, to allow the prediction and the macroscopic response of the tendon under specific loading conditions, assuming that its response follows a hyperelastic anisotropic model in conjunction with damage law. We benchmarked multiple hyperelastic and damage models to fit the response of the tendons in uniaxial tensile loading conditions, and by employing a genetic algorithm, we obtained the material parameters for both elastic and damage models. The objective of this study was to explore different mathematical models to determine which would be the best option to predict the behavior of tendons and ligaments in complex biological systems using Finite Elements (FE) models. Therefore, we took into account accuracy as well as computational features. We considered the model proposed by Shearer and coupled it with a sigmoid function, which governs the evolution of damage in tendons, as the most appropriate for the fitting of the experimental data. The achieved solution shows to be of high interest attributable to the simplicity of the damage law function and its low computational cost.

Shear Wave Propagation and Estimation of Material Parameters in a Nonlinear, Fibrous Material

This paper describes the propagation of shear waves in a Holzapfel-Gasser-Ogden (HGO) material and investigates the potential of magnetic resonance elastography (MRE) for estimating parameters of the HGO material model from experimental data. In most MRE studies the behavior of the material is assumed to be governed by linear, isotropic elasticity or viscoelasticity. In contrast, biological tissue is often nonlinear and anisotropic with a fibrous structure. In such materials, application of a quasi-static deformation (predeformation) plays an important role in shear wave propagation. Closed form expressions for shear wave speeds in an HGO material with a single family of fibers were found in a reference (undeformed) configuration and after imposed predeformations. These analytical expressions show that shear wave speeds are affected by the parameters (μ0, k1, k2, κ) of the HGO model and by the direction and amplitude of the predeformations. Simulations of corresponding finite element (FE) models confirm the predicted influence of HGO model parameters on speeds of shear waves with specific polarization and propagation directions. Importantly, the dependence of wave speeds on the parameters of the HGO model and imposed deformations could ultimately allow the noninvasive estimation of material parameters in vivo from experimental shear wave image data.

A Phenomenological Approach to Study Mechanical Properties of Polymeric Porous Structures Processed Using Supercritical CO₂

This work proposes a modeling of the mechanical properties of porous polymers processed by scCO₂, using a phenomenological approach. Tensile and compression tests of alginate/gelatin and cellulose acetate/graphene oxide were modeled using three hyperelastic equations, derived from strain energy functions. The proposed hyperelastic equations provide a fair good fit for mechanical behavior of the nanofibrous system alginate/gelatin (deviations lower than 10%); whereas, due to the presence of the solid in the polymer network, a four-parameter model must be used to fit the composite cellulose acetate/graphene oxide behavior. Larger deviations from the experimental data were observed for the system cellulose acetate/graphene oxide because of its microporous structure. A finite element method was, then, proposed to model both systems; it allowed a realistic description of observable displacements and effective stresses. The results indicate that materials processed using scCO₂, when submitted to large stresses, do not obey Hooke´s law and must be considered as hyperelastic.

An Iterative Method for Estimating Nonlinear Elastic Constants of Tumor in Soft Tissue from Approximate Displacement Measurements

Objectives

Various elastography techniques have been proffered based on linear or nonlinear constitutive models with the aim of detecting and classifying pathologies in soft tissues accurately and noninvasively. Biological soft tissues demonstrate behaviors which conform to nonlinear constitutive models, in particular the hyperelastic ones. In this paper, we represent the results of our steps towards implementing ultrasound elastography to extract hyperelastic constants of a tumor inside soft tissue.

Methods

Hyperelastic parameters of the unknown tissue have been estimated by applying the iterative method founded on the relation between stress, strain, and the parameters of a hyperelastic model after (a) simulating the medium's response to a sinusoidal load and extracting the tissue displacement fields in some instants and (b) estimating the tissue displacement fields from the recorded/simulated ultrasound radio frequency signals and images using the cross correlation-based technique.

Results

Our results indicate that hyperelastic parameters of an unidentified tissue could be precisely estimated even in the conditions where there is no prior knowledge of the tissue, or the displacement fields have been approximately calculated using the data recorded by a clinical ultrasound system.

Conclusions

The accurate estimation of nonlinear elastic constants yields to the correct cognizance of pathologies in soft tissues.

Four-Dimensional Imaging of Soft Tissue and Implanted Biomaterial Mechanics: A Barbed Suture Case Study for Tendon Repair

Timely, recent developments in X-ray microcomputed tomography (XμCT) imaging such as increased resolution and improved sample preparation enable nondestructive time-lapse imaging of polymeric biomaterials when implanted in soft tissue, which we demonstrate herein. Imaging the full three-dimensional (3D) structure of an implanted biomaterial provides new opportunities to assess the micromechanics of the interface between the implant and tissues and how this changes over time as force is applied in load-bearing musculoskeletal applications. In this paper, we present a case study demonstrating in situ XμCT and finite element analysis, using a dynamically loaded barbed suture repair for its novel use in tendon tissue. The aim of this study was to identify the distribution of stress in the suture and tendon as load is applied. The data gained demonstrate a clear 3D visualization of microscale features in both the tissue and implant in wet conditions. XμCT imaging has revealed, for the first time, pores around the suture, preventing full engagement of all the barbs with the tendon tissue. Subsequent finite element analysis reveals the localized stress and strain, which are not evenly distributed along the suture, or throughout the tissue. This case study demonstrates for the first time a powerful in situ mechanical imaging tool, which could be readily adapted by other laboratories to interrogate and optimize the interface between the implanted biomaterials and the soft tissue.

The relative compliance of energy-storing tendons may be due to the helical fibril arrangement of their fascicles

A nonlinear elastic microstructural model is used to investigate the relationship between structure and function in energy-storing and positional tendons. The model is used to fit mechanical tension test data from the equine common digital extensor tendon (CDET) and superficial digital flexor tendon (SDFT), which are used as archetypes of positional and energy-storing tendons, respectively. The fibril crimp and fascicle helix angles of the two tendon types are used as fitting parameters in the mathematical model to predict their values. The outer fibril crimp angles were predicted to be 15.1° ± 2.3° in the CDET and 15.8° ± 4.1° in the SDFT, and the average crimp angles were predicted to be 10.0° ± 1.5° in the CDET and 10.5° ± 2.7° in the SDFT. The crimp angles were not found to be statistically significantly different between the two tendon types (p = 0.572). By contrast, the fascicle helix angles were predicted to be 7.9° ± 9.3° in the CDET and 29.1° ± 10.3° in the SDFT and were found to be statistically highly significantly different between the two tendon types (p < 0.001). This supports previous qualitative observations that helical substructures are more likely to be found in energy-storing tendons than in positional tendons and suggests that the relative compliance of energy-storing tendons may be directly caused by these helical substructures.

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.

Histology-based homogenization analysis of soft tissue: application to prostate cancer

It is well known that the changes in tissue microstructure associated with certain pathophysiological conditions can influence its mechanical properties. Quantitatively relating the tissue microstructure to the macroscopic mechanical properties could lead to significant improvements in clinical diagnosis, especially when the mechanical properties of the tissue are used as diagnostic indices such as in digital rectal examination and elastography. In this study, a novel method of imposing periodic boundary conditions in non-periodic finite-element meshes is presented. This method is used to develop quantitative relationships between tissue microstructure and its apparent mechanical properties for benign and malignant tissue at various length scales. Finally, the inter-patient variation in the tissue properties is also investigated. Results show significant changes in the statistical distribution of the mechanical properties at different length scales. More importantly the loss of the normal differentiation of glandular structure of cancerous tissue has been demonstrated to lead to changes in mechanical properties and anisotropy. The proposed methodology is not limited to a particular tissue or material and the example used could help better understand how changes in the tissue microstructure caused by pathological conditions influence the mechanical properties, ultimately leading to more sensitive and accurate diagnostic technologies.

Optimal Contrast Agent Staining of Ligaments and Tendons for X-Ray Computed Tomography

X-ray computed tomography has become an important tool for studying the microstructures of biological soft tissues, such as ligaments and tendons. Due to the low X-ray attenuation of such tissues, chemical contrast agents are often necessary to enhance contrast during scanning. In this article, the effects of using three different contrast agents—iodine potassium iodide solution, phosphotungstic acid and phosphomolybdic acid—are evaluated and compared. Porcine anterior cruciate ligaments, patellar tendons, medial collateral ligaments and lateral collateral ligaments were used as the basis of the study. Three samples of each of the four ligament/tendon types were each assigned a different contrast agent (giving a total of twelve samples), and the progression of that agent through the tissue was monitored by performing a scan every day for a total period of five days (giving a total of sixty scans). Since the samples were unstained on day one, they had been stained for a total of four days by the time of the final scans. The relative contrast enhancement and tissue deformation were measured. It was observed that the iodine potassium iodide solution penetrated the samples fastest and caused the least sample shrinkage on average (although significant deformation was observed by the time of the final scans), whereas the phosphomolybdic acid caused the greatest sample shrinkage. Equations describing the observed behaviour of the contrast agents, which can be used to predict optimal staining times for ligament and tendon X-ray computed tomography, are presented.