Anisotropic elasto-damage constitutive model for the biomechanical analysis of tendons

A computational study of cell membrane damage and intracellular delivery in a cross-slot microchannel

We propose a three-dimensional computational framework to simulate the flow-induced cell membrane damage and the resulting enhanced intracellular mass transport in a cross-slot microchannel. We model the cell as a liquid droplet enclosed by a viscoelastic membrane and solve the cell deformation using a well-tested immersed-boundary lattice-Boltzmann method. The cell membrane damage, which is directly related to the membrane permeability, is considered using continuum damage mechanics. The transport of the diffusive solute into the cell is solved by a lattice-Boltzmann model. After validating the computational framework against several benchmark cases, we consider a cell flowing through a cross-slot microchannel, focusing on the effects of the flow strength, channel fluid viscosity and cell membrane viscosity on the membrane damage and enhanced intracellular transport. Interestingly, we find that under a comparable pressure drop across the device, for cells with low membrane viscosity, the inertial flow regime, which can be achieved by driving a low-viscosity liquid at a high speed, often leads to much larger membrane damage, compared with the high-viscosity low-speed viscous flow regime. However, the enhancement can be significantly reduced or even reversed by an increase of the cell membrane viscosity, which limits cell deformation, particularly in the inertial flow regime. Our computational framework and simulation results may guide the design and optimisation of microfluidic devices, which use cross-slot geometry to disrupt cell membranes to enhance intracellular delivery of solutes.

Nonlocal damage evaluation of a sigmoid-based damage model for fibrous biological soft tissues

The comprehension and modeling of the mechanical behavior of soft biological tissues are essential due to their clinical applications. This knowledge is essential for predicting tissue responses accurately and enhancing our ability to compute the behavior of biological structures and bio-prosthetic devices under specific loading conditions. The current research is centered on modeling the initiation and progression of soft tissues damage, which typically exhibit intricate anisotropic and nonlinear elastic characteristics. For this purpose, the following study presents a comparative analysis of the computational performance of two distinct damage modeling techniques. The first technique employs a well-established damage model, based on a piece-wise exponential damage function as proposed by Calvo et al. (Int J Numer Methods Eng 69:2036-2057, 2007. https://doi.org/10.1002/nme.1825 ). The second approach adopts a sigmoid function, as proposed by López-Campos et al. (Comput Methods Biomech Biomed Eng 23(6):213-223. https://doi.org/10.1080/10255842.2019.1710742 ). The aim of this study is to verify the validity of the López-Campos sigmoid-based damage model to be used in finite element simulation, the implementation of which is unknown. For this proposal, both models were implemented within a commercial Finite Element software package, and their responses to local and non-local damage algorithms were assessed in depth through two standard benchmark tests: a plate with a hole and a ball burst. The results of this study indicate that, for a wide range of cases, such as in-plane stresses, out-plane stresses, stress concentration and contact, all over large displacement conditions, the López-Campos damage model shows a good response to non-local algorithms achieving mesh independence and convergence in all these cases. The results obtained are in line with those obtained for the Calvo's damage model, showing, in addition, larger deformations under in-plane stress and stress concentration conditions and a lower number of iterations under out-plane stress and contact conditions. Consequently, the López-Campos' damage model emerges as a valuable and useful tool in the field of mechanical damage research in biological systems.

Finite element modeling of meniscal tears using continuum damage mechanics and digital image correlation

Meniscal tears are a common, painful, and debilitating knee injury with limited treatment options. Computational models that predict meniscal tears may help advance injury prevention and repair, but first these models must be validated using experimental data. Here we simulated meniscal tears with finite element analysis using continuum damage mechanics (CDM) in a transversely isotropic hyperelastic material. Finite element models were built to recreate the coupon geometry and loading conditions of forty uniaxial tensile experiments of human meniscus that were pulled to failure either parallel or perpendicular to the preferred fiber orientation. Two damage criteria were evaluated for all experiments: von Mises stress and maximum normal Lagrange strain. After we successfully fit all models to experimental force-displacement curves (grip-to-grip), we compared model predicted strains in the tear region at ultimate tensile strength to the strains measured experimentally with digital image correlation (DIC). In general, the damage models underpredicted the strains measured in the tear region, but models using von Mises stress damage criterion had better overall predictions and more accurately simulated experimental tear patterns. For the first time, this study has used DIC to expose strengths and weaknesses of using CDM to model failure behavior in soft fibrous tissue.

Predicting the Effect of Localized ACL Damage on Neighbor Ligament Mechanics via Finite Element Modeling

The anterior cruciate ligament (ACL) plays a pivotal role in support of the knee under loading. When damaged, it is known that substantial changes in the mechanics of the neighboring ligaments can be observed. However, a localized damage approach to investigating how ACL deficiency influences the neighboring ligaments has not been carried out. To do this, a finite element model, incorporating a continuum damage material model of the ACL, was implemented. Localized ACL damage was induced using high quadriceps force loading. Once damaged, anterior shear forces or tibial torque loadings were applied to the knee joint. The relative changes in stress contour and average mid-substance stress were examined for each of the neighboring ligaments following localized ACL damage. It was observed that localized ACL damage could produce notable changes in the mechanics of the neighboring knee ligaments, with non-homogenous stress contour shape changes and average stress magnitude being observed to increase in most cases, with a notable exception occurring in the MCL for both loading modes. In addition, the ligament bearing the most loading also changed with ACL deficiency. These changes carry implications as to morphological effects that may be induced following localized ACL damage, indicating that early diagnosis of ACL injury may be helpful in mitigating other complications post injury.

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.

On a phase-field approach to model fracture of small intestine walls

We address anisotropic elasticity and fracture in small intestine walls (SIWs) with both experimental and computational methods. Uniaxial tension experiments are performed on porcine SIW samples with varying alignments and quantify their nonlinear elastic anisotropic behavior. Fracture experiments on notched SIW strips reveal a high sensitivity of the crack propagation direction and the failure stress on the tissue orientation. From a modeling point of view, the observed anisotropic elastic response is studied with a continuum mechanical model stemming from a strain energy density with a neo-Hookean component and an anisotropic component with four families of fibers. Fracture is addressed with the phase-field approach, featuring two-fold anisotropy in the fracture toughness. Elastic and fracture model parameters are calibrated based on the experimental data, using the maximum and minimum limits of the experimental stress-stretch data set. A very good agreement between experimental data and computational results is obtained, the role of anisotropy being effectively captured by the proposed model in both the elastic and the fracture behavior. STATEMENT OF SIGNIFICANCE: This article reports a comprehensive experimental data set on the mechanical failure behavior of small intestinal tissue, and presents the corresponding protocols for preparing and testing the samples. On the one hand, the results of this study contribute to the understanding of small intestine mechanics and thus to understanding of load transfer mechanisms inside the tissue. On the other hand, these results are used as input for a phase-field modelling approach, presented in this article. The presented model can reproduce the mechanical failure behavior of the small intestine in an excellent way and is thus a promising tool for the future mechanical description of diseased small intestinal tissue.

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.

Modelling of damage of a liquid-core microcapsule in simple shear flow until rupture

Capsules, composed of a liquid core protected by a thin deformable membrane, offer high-potential applications in many fields of industry such as bioengineering. One of their limitations comes from the absence of models of capsule damage and/or rupture when they are subjected to an external flow. To assess when rupture is initiated, we develop a fluid-structure interaction (FSI) numerical model of a capsule in Stokes flow that accounts for potential damage of the capsule membrane. We consider the framework of Continuum Damage Mechanics and model the membrane with an isotropic brittle damage model, in which the membrane damage state depends on the history of loading. The FSI problem is solved by coupling the finite element method, to solve for the membrane deformation, with the boundary integral method, to solve for the inner and outer fluid flows. The model is applied to an initially spherical capsule subjected to a simple shear flow. Damage initiates at a critical value of the capillary number, ratio of the fluid viscous forces to the membrane elastic forces, and rupture at a higher capillary number, when it reaches a threshold value. The material parameters introduced in the damage model do not influence the mode of damage but only the values of the critical and threshold capillary numbers. When the capillary number is larger than the critical value, damage develops in the two symmetric central regions containing the vorticity axis. It is indeed in these regions that the internal tensions are the highest on the membrane.

Shear-deformation based continuum-damage constitutive modeling of brain tissue

Traumatic brain injury (TBI) is a leading cause of death in the United States. Depending on the severity of injury, complications such as memory loss and emotional changes are common. While the exact mechanisms are still unclear, these cognitive deficiencies are thought to arise from microstructural damages to the brain tissue, such as in diffuse-axonal injury where neuron fibers are sheared. Constitutive models can predict such damage at a microstructural level and allow for insight into the mechanisms of injury initiating at lower length scales. In this study, we developed a continuum damage model of brain tissue that is validated by experimental quasi-static stress-strain tests in tension, compression, and shear. The present work shows that damage is most present in the shear stress state, making the tissue suitable for damage modeling via shear interaction terms. Using this model, new insights into microstructural breakdown due to shear stresses and strains can be gained by application to simulations.

A knowledge map analysis of brain biomechanics: Current evidence and future directions

Although brain, one of the most complex organs in the mammalian body, has been subjected to many studies from physiological and pathological points of view, there remain significant gaps in the available knowledge regarding its biomechanics. This article reviews the research trends in brain biomechanics with a focus on injury. We used published scientific articles indexed by Web of Science database over the past 40 years and tried to address the gaps that still exist in this field. We analyzed the data using VOSviewer, which is a software tool designed for scientometric studies. The results of this study showed that the response of brain tissue to external forces has been one of the significant research topics among biomechanicians. These studies have addressed the effects of mechanical forces on the brain and mechanisms of traumatic brain injury, as well as characterized changes in tissue behavior under trauma and other neurological diseases to provide new diagnostic and monitoring methods. In this study, some challenges in the field of brain injury biomechanics have been identified and new directions toward understanding the gaps in this field are suggested.

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.

Implementation of a new constitutive model for abdominal muscles

BACKGROUND AND OBJECTIVE

Abdominal hernia repair is one of the most often performed surgical procedures worldwide. Numerical simulations of the abdominal wall mechanics can be a valuable tool to devise actions aimed at preventing hernia formation. A first step towards this goal is the development of consistent constitutive models for the tissues that form the human abdominal wall. In this study we propose, for each of the tissues involved, a new formulation of the so-called transversely isotropic hyperelastic model (TIHM).

METHODS

We propose a new TIHM for the human abdominal wall tissues and we present a systemic view of the methodology that we have implemented in the present study. First we consider the mathematical background of the TIHM. The novelty of our formulation is that both the isotropic and the fiber contributions to the strain energy function are characterized exclusively by polynomial convex functions of certain invariant quantities. Then, we provide a detailed description on how the constitutive model is implemented into an open source finite element (FE) software. In our approach we use the specific interface provided by the MFront software to incorporate our TIHM formulation into the Code Aster FE solver. For each of the tissues considered, the values of the TIHM constants are adjusted by means of a numerical simulation of previous experimental data from tensile tests.

RESULTS

We studied the following abdominal wall tissues: linea alba, rectus sheath, external oblique muscle, internal oblique muscle, transversus abdominis muscle and rectus abdominis muscle. Our formulation closely reproduces tensile test data for each tissue in the corresponding FE numerical simulation.

CONCLUSIONS

The new TIHM formulation is suitable for a future numerical investigation of the abdominal wall, which will in turn help us to assess the best zone to practice a colostomy. The methodology implemented in the present study can be easily extended in the future to develop and implement a TIHM for active muscles and/or a different type of constitutive model which might be suitable to characterize other tissues of biomedical interest.

Biomechanical evaluation of patellar tendon repair using Krackow suture technique

Background

Patellar tendon rupture is a potentially devastating injury. Surgical repair is the primary treatment recommended for the patients with patellar tendon ruptures. Given the tendon properties, the suture technique is critical for proper tissue repair. Providing adequate loading during early mobilization is essential to prevent tendon suture repair failure. Therefore, the current study evaluated the mechanical characteristics of various applied loadings on patellar tendon repair using Krackow suture via a porcine model.

Methods

Twelve fresh porcine hindlimbs with patellar tendon rupture were repaired by Krackow method using synthetic and non-absorbable No. 5 Ethibond sutures. Loadings of 100 and 200 N were applied during the cyclic loading test. A three-dimensional optical motion capture system was used to record the gap formation at the initial, 50th, 100th, 150th, 200th, 250th, 500th, 750th, and 1000th cycle. After cyclic loading, the specimen was loaded to failure under displacement control at a rate of 1 mm/s.

Results

Suture breakage was the primary failure mode in both loading conditions. After 1000 cyclic loadings of 100 N, the ultimate failure strength was 243.6 ± 25.8 N. However, the specimens tested under 200 N of loading failed before reaching 200 cycles. Under the 100 N loading, the largest gap deformation (1.89 ± 0.23 mm) and residual deformation (0.213 ± 0.183 mm) were found in the initial cycle. The average cumulative displacement was 5.13 mm from the initial cycle to the 100th cycle and 4.5 mm from the 250th to the 1000th cycle.

Conclusions

Our findings can serve as reference values for further comparisons with various repair techniques or materials. This study suggests that the initially applied load after patellar tendon repair is an important risk factor of re-rupture.

Molecular-level collagen damage explains softening and failure of arterial tissues: A quantitative interpretation of CHP data with a novel elasto-damage model

The present experimental-modelling study provides a quantitative interpretation of mechanical data and damage measurements obtained from collagen hybridizing peptide (CHP) techniques on overstretched sheep cerebral arterial tissues. To this aim, a structurally-motivated constitutive model is developed in the framework of continuum damage mechanics. The model includes two internal variables for describing the effects of collagen triple-helical unfolding via interstrand delamination: one governs plastic mechanisms in collagen fibers, leading to a stress softening response of the tissue at the macroscale; the other one describes the loss of fiber structural integrity, leading to tissue final failure. The proposed model is calibrated using the obtained mechanical experimental data, showing excellent fitting capabilities. The predicted evolution of internal variables agree well with independent measurements of molecular-level CHP-based damage data, obtaining an independent a posteriori validation of damage predictions. Moreover, available data on inelastic tissue elongation following supraphysiological loads are successfully reproduced. These outcomes further the hypothesis that the accumulation of interstrand delamination is a primary cause for the evolution of inelastic mechanisms in tissues, and in particular of stress softening up to failure.

Evaluating Plastic Deformation and Damage as Potential Mechanisms for Tendon Inelasticity using a Reactive Modeling Framework

Inelastic behaviors, such as softening, a progressive decrease in modulus before failure, occur in tendon and are important aspect in degeneration and tendinopathy. These inelastic behaviors are generally attributed to two potential mechanisms: plastic deformation and damage. However, it is not clear which is primarily responsible. In this study, we evaluated these potential mechanisms of tendon inelasticity by using a recently developed reactive inelasticity model (RIE), which is a structurally-inspired continuum mechanics framework that models tissue inelasticity based on the molecular bond kinetics. Using RIE, we formulated two material models, one specific to plastic deformation and the other to damage. The models were independently fit to published experimental tensile tests of rat tail tendons. We quantified the inelastic effects and compared the performance of the two models in fitting the mechanical response during loading, relaxation, unloading, and reloading phases. Additionally, we validated the models by using the resulting fit parameters to predict an independent set of experimental stress-strain curves from ramp-to-failure tests. Overall, the models were both successful in fitting the experiments and predicting the validation data. However, the results did not strongly favor one mechanism over the other. As a result, to distinguish between plastic deformation and damage, different experimental protocols will be needed. Nevertheless, these findings suggest the potential of RIE as a comprehensive framework for studying tendon inelastic behaviors.

A Reactive Inelasticity Theoretical Framework for Modeling Viscoelasticity, Plastic Deformation, and Damage in Fibrous Soft Tissue

Fibrous soft tissues are biopolymeric materials that are made of extracellular proteins, such as different types of collagen and proteoglycans, and have a high water content. These tissues have nonlinear, anisotropic, and inelastic mechanical behaviors that are often categorized into viscoelastic behavior, plastic deformation, and damage. While tissue's elastic and viscoelastic mechanical properties have been measured for decades, there is no comprehensive theoretical framework for modeling inelastic behaviors of these tissues that is based on their structure. To model the three major inelastic mechanical behaviors of tissue's fibrous matrix, we formulated a structurally inspired continuum mechanics framework based on the energy of molecular bonds that break and reform in response to external loading (reactive bonds). In this framework, we employed the theory of internal state variables (ISV) and kinetics of molecular bonds. The number fraction of bonds, their reference deformation gradient, and damage parameter were used as state variables that allowed for consistent modeling of all three of the inelastic behaviors of tissue by using the same sets of constitutive relations. Several numerical examples are provided that address practical problems in tissue mechanics, including the difference between plastic deformation and damage. This model can be used to identify relationships between tissue's mechanical response to external loading and its biopolymeric structure.

Isotropic Failure Criteria Are Not Appropriate for Anisotropic Fibrous Biological Tissues

The von Mises (VM) stress is a common stress measure for finite element models of tissue mechanics. The VM failure criterion, however, is inherently isotropic, and therefore may yield incorrect results for anisotropic tissues, and the relevance of the VM stress to anisotropic materials is not clear. We explored the application of a well-studied anisotropic failure criterion, the Tsai–Hill (TH) theory, to the mechanically anisotropic porcine aorta. Uniaxial dogbones were cut at different angles and stretched to failure. The tissue was anisotropic, with the circumferential failure stress nearly twice the axial (2.67 ± 0.67 MPa compared to 1.46 ± 0.59 MPa). The VM failure criterion did not capture the anisotropic tissue response, but the TH criterion fit the data well (R2 = 0.986). Shear lap samples were also tested to study the efficacy of each criterion in predicting tissue failure. Two-dimensional failure propagation simulations showed that the VM failure criterion did not capture the failure type, location, or propagation direction nearly as well as the TH criterion. Over the range of loading conditions and tissue geometries studied, we found that problematic results that arise when applying the VM failure criterion to an anisotropic tissue. In contrast, the TH failure criterion, though simplistic and clearly unable to capture all aspects of tissue failure, performed much better. Ultimately, isotropic failure criteria are not appropriate for anisotropic tissues, and the use of the VM stress as a metric of mechanical state should be reconsidered when dealing with anisotropic tissues.

Polydispersity controls the strength of semi-flexible polymer networks

The classical theory of polymer elasticity is built upon the assumption of network monodispersity-the premise that polymer networks are comprised of sub-chains of equal length. The crosslinking of biopolymers, however, is a random process and the resultant networks are likely to be polydisperse. The effect of structural polydispersity on the mechanical behavior of biopolymer networks is not well understood. The purpose of this contribution is to show how network polydispersity controls mechanical behavior and the ultimate properties of crosslinked semi-flexible filaments at finite deformations. The proposed micromechanical continuum model is based on the force-elongation relation of individual chains of different lengths. It is shown that the mechanical strength of the network is controlled by the finite-extensibility of filaments and the degradation of shorter filaments at relatively small stretches. The progressive failure of filaments continues and eventually determines the ultimate strength of the network. The predicted stress-stretch behaviors are in reasonable agreement with the experimental data for connective tissues.

Constitutive behaviour of paddlefish (Polyodon spathula) cartilage

This study explored the constitutive behaviour of the midline cartilage of the paddlefish (Polyodon spathula) rostrum and examined the properties of the cartilage as a function of stress state and strain rate. The cartilage structure of the paddlefish contributes to the physical characteristics of the unique rostrum, and is of much interest as an inspiration for engineered structures. Low-strain rate testing (∼0·01/s) results indicated an average compressive yield stress of 11·34 ± 3·10 MPa and an average tensile yield stress of 3·62 ± 0·62 MPa. High-strain rate testing (∼100/s) results showed an average compressive yield stress of 25·07 ± 10·86 MPa. A histological study showed that the material is a type of hyaline cartilage. An Ogden hyperelastic model was used for finite-element analysis and fit very well with experimental data. This paper contributes to an overall study examining the electrosensory capabilities, hydrodynamics and structure–property relationships of the rostrum.

Investigating mechanisms of tendon damage by measuring multi-scale recovery following tensile loading

Tendon pathology is associated with damage. While tendon damage is likely initiated by mechanical loading, little is known about the specific etiology. Damage is defined as an irreversible change in the microstructure that alters the macroscopic mechanical parameters. In tendon, the link between mechanical loading and microstructural damage, resulting in macroscopic changes, is not fully elucidated. In addition, tendon damage at the macroscale has been proposed to initiate when tendon is loaded beyond a strain threshold, yet the metrics to define the damage threshold are not determined. We conducted multi-scale mechanical testing to investigate the mechanism of tendon damage by simultaneously quantifying macroscale mechanical and microstructural changes. At the microscale, we observe full recovery of the fibril strain and only partial recovery of the interfibrillar sliding, indicating that the damage initiates at the interfibrillar structures. We show that non-recoverable sliding is a mechanism for tendon damage and is responsible for the macroscale decreased linear modulus and elongated toe-region observed at the fascicle-level, and these macroscale properties are appropriate metrics that reflect tendon damage. We concluded that the inflection point of the stress-strain curve represents the damage threshold and, therefore, may be a useful parameter for future studies. Establishing the mechanism of damage at multiple length scales can improve prevention and rehabilitation strategies for tendon pathology.

STATEMENT OF SIGNIFICANCE

Tendon pathology is associated with mechanically induced damage. Damage, as defined in engineering, is an irreversible change in microstructure that alters the macroscopic mechanical properties. Although microstructural damage and changes to macroscale mechanics are likely, this link to microstructural change was not yet established. We conducted multiscale mechanical testing to investigate the mechanism of tendon damage by simultaneously quantifying macroscale mechanical and microstructural changes. We showed that non-recoverable sliding between collagen fibrils is a mechanism for tendon damage. Establishing the mechanism of damage at multiple length scales can improve prevention and rehabilitation strategies for tendon pathology.

Fatigue damage of collagenous tissues: experiment, modeling and simulation studies

Mechanical fatigue damage is a critical issue for soft tissues and tissue-derived materials, particularly for musculoskeletal and cardiovascular applications; yet, our understanding of the fatigue damage process is incomplete. Soft tissue fatigue experiments are often difficult and time-consuming to perform, which has hindered progress in this area. However, the recent development of soft-tissue fatigue-damage constitutive models has enabled simulation-based fatigue analyses of tissues under various conditions. Computational simulations facilitate highly controlled and quantitative analyses to study the distinct effects of various loading conditions and design features on tissue durability; thus, they are advantageous over complex fatigue experiments. Although significant work to calibrate the constitutive models from fatigue experiments and to validate predictability remains, further development in these areas will add to our knowledge of soft-tissue fatigue damage and will facilitate the design of durable treatments and devices. In this review, the experimental, modeling, and simulation efforts to study collagenous tissue fatigue damage are summarized and critically assessed.

Damage Models for Soft Tissues: A Survey

Damage to soft tissues in the human body has been investigated for applications in healthcare, sports, and biomedical engineering. This paper reviews and classifies damage models for soft tissues to summarize achievements, identify new directions, and facilitate finite element analysis. The main ideas of damage modeling methods are illustrated and interpreted. A few key issues related to damage models, such as experimental data curve-fitting, computational effort, connection between damage and fractures/cracks, damage model applications, and fracture/crack extension simulation, are discussed. Several new challenges in the field are identified and outlined. This review can be useful for developing more advanced damage models and extending damage modeling methods to a variety of soft tissues.

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.

Analysis of the structural behaviour of colonic segments by inflation tests: Experimental activity and physio-mechanical model

A coupled experimental and computational approach is provided for the identification of the structural behaviour of gastrointestinal regions, accounting for both elastic and visco-elastic properties. The developed procedure is applied to characterize the mechanics of gastrointestinal samples from pig colons. Experimental data about the structural behaviour of colonic segments are provided by inflation tests. Different inflation processes are performed according to progressively increasing top pressure conditions. Each inflation test consists of an air in-flow, according to an almost constant increasing pressure rate, such as 3.5 mmHg/s, up to a prescribed top pressure, which is held constant for about 300 s to allow the development of creep phenomena. Different tests are interspersed by 600 s of rest to allow the recovery of the tissues' mechanical condition. Data from structural tests are post-processed by a physio-mechanical model in order to identify the mechanical parameters that interpret both the non-linear elastic behaviour of the sample, as the instantaneous pressure-stretch trend, and the time-dependent response, as the stretch increase during the creep processes. The parameters are identified by minimizing the discrepancy between experimental and model results. Different sets of parameters are evaluated for different specimens from different pigs. A statistical analysis is performed to evaluate the distribution of the parameters and to assess the reliability of the experimental and computational activities.

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.

A damage model for the percutaneous triple hemisection technique for tendo-achilles lengthening

A full understanding of the mechanisms of action in the percutaneous triple hemisection technique for tendo-achilles lengthening has yet to be acquired and therefore, an accurate prediction of the amount of lengthening that occurs is difficult to make. The purpose of this research was to develop a phenomenological damage model that utilizes both matrix and fiber damage and replicates the observed behavior of the tendon tissue during the lengthening process. Matrix damage was triggered and evolved relative to shear strain and the fiber damage was triggered and evolved relative to fiber stretch. Three examples are given to show the effectiveness of the model. Implementation of the damage model provides a tool for studying this common procedure, and may allow for numerical investigation of alternative surgical approaches that could reduce the incidence rates of severe over-lengthening.

Failure Properties and Damage of Cervical Spine Ligaments, Experiments and Modeling

Cervical spine ligaments have an important role in providing spinal cord stability and restricting excessive movements. Therefore, it is of great importance to study the mechanical properties and model the response of these ligaments. The aim of this study is to characterize the aging effects on the failure properties and model the damage of three cervical spine ligaments: the anterior and the posterior longitudinal ligament and the ligamentum flavum. A total of 46 samples of human cadaveric ligaments removed within 24–48 h after death have been tested. Uniaxial tension tests along the fiber direction were performed in physiological conditions. The results showed that aging decreased the failure properties of all three ligaments (failure load, failure elongation). Furthermore, the reported nonlinear response of cervical ligaments has been modeled with a combination of the previously reported hyperelastic and damage model. The model predicted a nonlinear response and damage region. The model fittings are in agreement with the experimental data and the quality of agreement is represented with the values of the coefficient of determination close to 1.

Visualizing tendon elasticity in an ex vivo partial tear model

Supersonic shear imaging (SSI) is evaluated as a means of visualizing changes in regional tendon elasticity caused by partial tears in a porcine model. Thirty digital flexor tendons were cut to 25% (n = 10), 50% (n = 10) and 75% (n = 10) of the tendon thickness along the deep surface. Tendon elasticity was mapped left of, centered on and right of the tear site before and after tearing from 0% to 2% strain. Shear wave speed increased at 1% (p < 0.05) and 2% (p < 0.001) strain for all regions. Deep surface shear wave speed decreased in the 25%, 50% and 75% tears (p < 0.05 and p < 0.001). Computational tendon tear models were also created to investigate regional changes in strain resulting from a tear. In the computational models, strain on the deep surface decreased progressively with increasing tear size. Visualization of tendon shear wave speed was achieved in normal and partially torn tendons, indicating the potential of SSI to add tendon shear wave speed to traditional morphologic assessment of partial tears, which may improve assessment of tendon health.

Prediction of the elastic strain limit of tendons

The elastic strain limit (ESL) of tendons is the point where maximum elastic modulus is reached, after which micro-damage starts. Study of damage progression in tendons under repetitive (fatigue) loading requires a priori knowledge about ESL. In this study, we propose three different approaches for predicting ESL. First, one single value is assumed to represent the ESL of all tendon specimens. Second, different extrapolation curves are used for extrapolating the initial part of the stress-strain curve. Third, a method based on comparing the shape of the initial part of the stress-strain curve of specimens with a database of stress-strain curves is used. A large number of porcine tendon explants (97) were tested to examine the above-mentioned approaches. The variants of the third approach yielded significantly (p<0.05) smaller error values as compared to the other approaches. The mean absolute percentage error of the best-performing variant of the shape-based comparison was between 8.14±6.44% and 9.96±9.99% depending on the size of the initial part of the stress-strain curves. Interspecies generalizability of the best performing method was also studied by applying it for prediction of the ESL of horse tendons. The ESL of horse tendons was predicted with mean absolute percentage errors ranging between 10.53±7.6% and 19.16±14.31% depending on the size of the initial part of the stress-strain curves and the type of normalization. The results of this study suggest that both ESL and the shape of stress-strain curves may be highly different between different individuals and different anatomical locations.

A conceptual framework for computational models of Achilles tendon homeostasis

Computational modeling of tendon lags the development of computational models for other tissues. A major bottleneck in the development of realistic computational models for Achilles tendon is the absence of detailed conceptual and theoretical models as to how the tissue actually functions. Without the conceptual models to provide a theoretical framework to guide the development and integration of multiscale computational models, modeling of the Achilles tendon to date has tended to be piecemeal and focused on specific mechanical or biochemical issues. In this paper, we present a new conceptual model of Achilles tendon tissue homeostasis, and discuss this model in terms of existing computational models of tendon. This approach has the benefits of structuring the research on relevant computational modeling to date, while allowing us to identify new computational models requiring development. The critically important functional issue for tendon is that it is continually damaged during use and so has to be repaired. From this follows the centrally important issue of homeostasis of the load carrying collagen fibrils within the collagen fibers of the Achilles tendon. Collagen fibrils may be damaged mechanically-by loading, or damaged biochemically-by proteases. Upon reviewing existing computational models within this conceptual framework of the Achilles tendon structure and function, we demonstrate that a great deal of theoretical and experimental research remains to be done before there are reliably predictive multiscale computational model of Achilles tendon in health and disease.

Constitutive formulations for the mechanical investigation of colonic tissues

A constitutive framework is provided for the characterization of the mechanical behavior of colonic tissues, as a fundamental tool for the development of numerical models of the colonic structures. The constitutive analysis is performed by a multidisciplinary approach that requires the cooperation between experimental and computational competences. The preliminary investigation pertains to the review of the tissues histology. The complex structural configuration of the tissues and the specific distributions of fibrous elements entail the nonlinear mechanical behavior and the anisotropic response. The identification of the mechanical properties requires to perform mechanical tests according to different loading situations, as different loading directions. Because of the typical functionality of colon structures, the tissues mechanics is investigated by tensile tests, which are performed on taenia coli and haustra specimens from fresh pig colons. Accounting for the histological investigation and the results from the mechanical tests, a specific hyperelastic framework is provided within the theory of fiber-reinforced composite materials. Preliminary analytical formulations are defined to identify the constitutive parameters by the inverse analysis of the experimental tests. Finite element models of the specimens are developed accounting for the actual configuration of the colon structures to verify the quality of the results. The good agreement between experimental and numerical model results suggests the reliability of the constitutive formulations and parameters. Finally, the developed constitutive analysis makes it possible to identify the mechanical behavior and properties of the different colonic tissues.

Validation of an empirical damage model for aging and in vivo injury of the murine patellar tendon

While useful models have been proposed to predict the mechanical impact of damage in tendon and other soft tissues, the applicability of these models for describing in vivo injury and age-related degeneration has not been investigated. Therefore, the objective of this study was to develop and validate a simple damage model to predict mechanical alterations in mouse patellar tendons after aging, injury, or healing. To characterize baseline properties, uninjured controls at age 150 days were cyclically loaded across three strain levels and five frequencies. For comparison, damage was induced in mature (120 day-old) mice through either injury or aging. Injured mice were sacrificed at three or six weeks after surgery, while aged mice were sacrificed at either 300 or 570 days old. Changes in mechanical properties (relative to baseline) in the three week post-injury group were assessed and used to develop an empirical damage model based on a simple damage parameter related to the equilibrium stress at a prescribed strain (6%). From the derived model, the viscoelastic properties of the 300 day-old, 570 day-old, and six week post-injury groups were accurately predicted. Across testing conditions, nearly all correlations between predicted and measured parameters were statistically significant and coefficients of determination ranged from R² = 0.25 to 0.97. Results suggest that the proposed damage model could exploit simple in vivo mechanical measurements to predict how an injured or aged tendon will respond to complex physiological loading regimens.

Investigation of the biomechanical behaviour of hindfoot ligaments

The aim of this work is to provide a computational tool for the mechanical characterization of the hindfoot ligaments. The investigation is performed by a coupled numerical and experimental approach. For this purpose, a numerical model that represents the complex structural configuration of the hindfoot and the typical features of the mechanical behaviour of the ligament tissue is developed. The geometrical analysis of the anatomical site is performed starting from the processing of computed tomography and magnetic resonance images. Accounting for morphometric measurements, the virtual solid model provides an averaged configuration of the hindfoot structure. In order to specify the mechanical behaviour of the ligament tissue, a fibre-reinforced visco-hyperelastic model is adopted. The formulation accounts for the anisotropic configuration, geometric non-linearity, non-linear elasticity and time-dependent phenomena. Numerical analyses are performed to evaluate the biological tissues and structure mechanics with regard to physiological boundary conditions, accounting for dorsiflexion and plantarflexion movements. In order to evaluate the reliability of the numerical model developed, the experimental data are compared with the numerical results. The numerical results are in agreement with the range of values obtained by experimental test confirming the accuracy of the procedure adopted.

A simple constitutive model for the Scapholunate ligament

A simple constitutive equation is presented to describe the non-linear load-displacement behavior of the Scapholunate ligament. The model is based on an elastic large strain probabilistic constitutive equation and has been formulated offering three characteristic constants. Experimental data from five human cadaveric Scapholunate ligaments were used to corroborate the accuracy of the model. Good correlation (r(2)=0.90-0.98) was found with the experimental data. Representative characteristic constants for the human Scapholunate ligament based on the model were determined.

Damage mechanics of porcine flexor tendon: mechanical evaluation and modeling

Porcine flexor tendons underwent cyclic and stress relaxation testing before and after strain exceeding elastic limit ("overstretch") to examine which mechanical parameters undergo changes following subfailure damage. From these data, we developed an "effective strain" damage model (in which the tendon is modeled as if being pulled to a lower strain). Damage was induced at three strain levels to determine the extent to which post-damage parameter changes were affected by overstretch strain level. We found that diffuse damage induced by overstretch decreased elastic and viscoelastic parameters obtained during testing. The stress response of tendon to strain is therefore altered following damage. We next compared the strain-dependent parameter behavior to damage-dependent behavior to determine the effective strain for each parameter. Effects of damage became more pronounced as strain during overstretch increased; following overstretch to 6.5, 9, or 13% strain, effective strain was 2.43 ± 0.33, 1.98 ± 0.3, or 0.88 ± 0.43% strain, respectively. By determining the effective strain and using it to calculate predicted values of post-damage mechanical parameters, it was possible to predict the stress relaxation behavior of tendons with Schapery's nonlinear viscoelastic model. Using this approach, a single parameter predicts both elastic and viscoelastic compromise from known, strain-dependent behaviors.

Constitutive modeling of the non-linear visco-elasticity of the periodontal ligament

A non-linear visco-elastic constitutive model is adopted to describe the relaxation phenomena of the periodontal ligament (PDL). The introduction of a non-linear formulation of visco-elasticity is necessary because experimental data from the literature referring to animal models show that the relaxation rate depends on the level of strain applied. In particular, the percentage of relaxation increases with decrease of the applied strain. The constitutive model is consistent with the non-linear elastic behavior of the PDL in the case of high rate loading and large strains attained by the tissue. A hyperelastic formulation is adopted for the elastic behavior of the PDL and this formulation is developed adopting suitable measures of stress and strain. The anisotropy of the tissue induced by specific spatial orientation of collagen fibers is included in the model. With respect to recent numerical formulation proposed to describe the non-linear visco-elasticity of the PDL, the proposed model has the advantage of being more consistent with the micro-structural configuration of the tissue and the large strains it can undergo. The results obtained show that a reasonable description of the PDL relaxation phenomena can be obtained by assuming that relaxation times are independent of strain, whereas the relative stiffness results are dependent on strain applied through an exponential function.

Biomechanical behaviour of heel pad tissue experimental testing, constitutive formulation, and numerical modelling

This paper deals with the constitutive formulation of heel pad tissue and presents a procedure for identifying constitutive parameters using experimental data, with the aim of developing a computational approach for investigating the actual biomechanical response. The preliminary definition of constitutive parameters was developed using a visco-hyperelastic formulation, considering experimental data from in vitro compression tests on specimens of fat pad tissue and data from in vivo tests to identify the actual trend of tissue stiffness. The discrepancy between model results and experimental data was evaluated on the basis of a specific cost function, adopting a stochastic/deterministic procedure. The parameter evaluation was upgraded by considering experimental tests performed on the fat pad tissues of a cadaveric foot using in situ indentation tests at 0.01 and 350 mm/s strain rates. The constitutive formulation was implemented in a numerical model. The comparison of data from in situ tests and numerical results led to an optimal domain of parameters based on an admissible discrepancy criterion. Numerical results evaluated for different sets of parameters inside the domain are reported and compared with experimental data for a reliability evaluation of the proposed procedure.

A constitutive model for the mechanical characterization of the plantar fascia

A constitutive model is proposed to describe the mechanical behavior of the plantar fascia. The mechanical characterization of the plantar fascia regards the role in the foot biomechanics and it is involved in many alterations of its functional behavior, both of mechanical and nonmechanical origin. The structural conformation of the plantar fascia in its middle part is characterized by the presence of collagen fibers reinforcing the tissue along a preferential orientation, which is that supporting the major loading. According to this anatomical evidence, the tissue is described by developing an isotropic fiber-reinforced constitutive model and since the elastic response of the fascia is here considered, the constitutive model is based on the theory of hyperelasticity. The model is consistent with a kinematical description of large strains mechanical behavior, which is typical of soft tissues. A fitting procedure of the constitutive model is implemented making use of experimental curves taken from the literature and referring to specimens of human plantar fascia. A satisfactory fitting of the tensile behavior of the plantar fascia has been performed, showing that the model correctly interprets the mechanical behavior of the tissue in the light of comparison to experimental data at disposal. A critical analysis of the model with respect to the problem of the identification of the constitutive parameters is proposed as the basis for planning a future experimental investigation of mechanical behavior of the plantar fascia.

Automated image analysis method for quantifying damage accumulation in tendon

Tendon pathology is frequently sub-clinical prior to frank rupture, denoting the need for non-destructive methods of assessing disease presence and progression. Despite the lack of clinical presentation, previous studies have observed that distinct changes to the tendon microstructure are present, and that such qualitative changes have a dose-response relationship with the level of damage accumulated. These initial findings suggest that there is value in investigating the physical nature of damage within tendon, not only to better understand the pathological process, but also to gain insight into reparative processes and develop treatments. However, a necessary first step towards carrying out these avenues of research is to develop diagnostic tools for quantitatively assessing the level of damage present. In this study, we established a dose-response relationship between a quantitative measure of structural damage and the level of global damage induced. Furthermore, we developed and validated an automated technique for quantifying matrix disorganization (damage), which correlates and co-localizes strongly with manual quantification. In combination, these findings allow us to measure the amount of damage accumulation of a region of tendon on a clinical scale, rapidly and accurately.