A one-dimensional collagen-based biomechanical model of passive soft tissue with viscoelasticity and failure
Introduction
Tendons and ligaments are tensile load-bearing collagenous connective tissues, which transfer loads between muscles and bones, or between bones, respectively (Fratzl, 2008, Lin et al., 2004). Injuries to these tissues constitute the most common musculoskeletal complaints for patients seeking medical attention (Riley, 2008). Tendons, in particular, are exposed to tremendous mechanical loads on a regular basis. For example, the Achilles tendon has been shown to experience forces upwards of six times body weight during running (Almonroeder et al., 2013), and just under two times bodyweight during normal gait (Finni et al., 1998). Typically, tendon and ligament injuries are dichotomized into acute injuries, where a one-time loading event severs or otherwise damages the tissues; or chronic injuries, where cumulative micro-traumas culminate in a traumatic rupture or chronic state of tissue degeneration (McGill, 1997, Snedeker and Foolen, 2017). Characterizing these injuries requires first understanding the mechanical environment — stresses and strains — experienced by these tissues. This requires detailed biomechanical models, as directly measuring tendon or ligament forces in-vivo is currently not possible without invasive buckle transducer studies (Finni et al., 1998, Komi et al., 1987). Developing and validating these models first requires measuring the biomechanical properties of soft tissues from in-vitro studies (Fig. 1).
Soft tissues, like tendons and ligaments, portray a characteristic force-deflection, or stress-strain, curve when loaded in-vitro. Upon initial lengthening, the curve moves through a phase called the ‘toe region’, where there is a gradual increase in stiffness owing to the sequential uncrimping of the collagen fibres (Chazal et al., 1985, Frisén et al., 1969a, Frisén et al., 1969b, Rigby et al., 1959). Once all of the collagen fibres have been engaged, there is no subsequent increase in stiffness, and the curve becomes linear (Provenzano et al., 2002). Finally, as these fibres are strained to failure, the force abruptly decreases, and the curve moves through a phase called the failure region (Chazal et al., 1985, Knörzer et al., 1986, Mattucci et al., 2013). In addition to this complex hyperelastic mechanical response, these tissues exhibit some pervasive viscous behaviours, namely creep, stress-relaxation, and hysteresis (Bass et al., 2007, Gregory and Callaghan, 2010, Johnson et al., 1994, Keller et al., 1987, Lucas et al., 2009, Wren et al., 2001). Creep is typically characterized as a continued deformation in response to a constantly applied stress (Anssari-Benam et al., 2015, Provenzano et al., 2001), although it can also manifest from cyclic loading (Ekström et al., 1996). Stress-relaxation is conjugate to creep as it is a reduction in stress in response to a constantly applied strain (Fung, 1981, Nelson-Wong et al., 2018). Finally, hysteresis is defined from the dissipation of energy-typically as heat-as a tissue undergoes cyclic loading. Hysteresis is quantified as the difference in the area under a stress–strain (or force–deflection) curve between loading and unloading the tissue (Best, 1994, Möller et al., 1992, Yahia et al., 1991). This set of viscoelastic and failure properties, as demonstrated through a collection of in-vitro work, forms a basis of experimental observations on tendons and ligaments; they are phenomena any tissue model should aim to reproduce.
Musculoskeletal models are ubiquitous throughout biomechanics and have been used extensively to predict joint loading during gait (Callaghan et al., 1999, Lin et al., 2010, Rajagopal et al., 2016, Thelen and Anderson, 2006), quiet standing (Cholewicki et al., 1997), sitting (Callaghan and McGill, 2001), lifting (McGill and Norman, 1986), running (Almonroeder et al., 2013), and manual materials handling (Cudlip et al., 2015, Dickerson et al., 2007, Molinaro et al., 2020). They represent the current state-of-the-art in predicting tissue-level responses to in-vivo exposures, bypassing invasive experimental techniques like buckle transducers. Alternatively, they have been employed as forward models to predict tissue responses in vehicular impacts (de Jager, 1996, Fice et al., 2011, Kuo et al., 2019, Panzer et al., 2011, van Lopik and Acar, 2007), post-surgical adaptation (Hicks et al., 2011, Jalalian et al., 2013), athletic performance (Erdemir et al., 2007, Laschowski et al., 2018, McLean et al., 2003), and sit-to-stand motor control (Norman-Gerum and McPhee, 2018). Ultimately, these models aim to predict tissue-level forces and therefore require accurate tissue models in order to function. Most commonly, ligaments and tendons are represented as non-linear springs, and sometimes dashpots, with an exponential function representing their elastic response, typically tuned to a single cycle or acute response (Lucas et al., 2009, Provenzano et al., 2001). Unfortunately, very little attention is given to the viscoelastic or failure properties of these tissues in these macroscopic biomechanical models. Despite this limitation, these models have shown tremendous success in predicting movement patterns and loads which may predispose an individual to injury.
On the more microscopic level, the study of viscous properties of tissues continues to be an active area of study. Historically, linear rheological models represented the elastic response of tissues with a Hookean spring, and the viscous part with a Newtonian dashpot (Fung, 1967). Various arrangements of these components in series or parallel gave rise to the standard linear viscoelastic solid (SLS) model, which exhibited the three viscous properties of biological tissues: creep, stress-relaxation, and hysteresis. Fung (1967) unveiled a quasilinear viscoelastic (QLV) theory, which was able to combine the complex hyperelastic responses of biological tissues with the familiar viscous properties. This hyper-viscoelastic model has since been successful in modelling tendons, ligaments, and skin (DeFrate and Li, 2007, Lacroix et al., 2013, Nekouzadeh et al., 2007, Toms et al., 2002, Troyer and Puttlitz, 2011). More recently, relaxing the linearity in the viscous components of the model has led to non-linear viscoelastic theories, which have since been applied to ligaments (Abramowitch et al., 2010, Troyer and Puttlitz, 2010, Troyer and Puttlitz, 2011. To our knowledge, these viscoelastic models have only seldom been applied to macroscopic biomechanical models. Conversely, other tissue models have aimed at capturing the failure portion of a ligament or tendon’s stress–strain curve. These range from phenomenological reactive damage mechanics models (Safa et al., 2019a, Safa et al., 2019b) to more mechanistic models that describe the strain distribution in a population of constituent collagen fibres (Barrett and Callaghan, 2017, Bontempi, 2009, Hamedzadeh et al., 2018, Liao and Belkoff, 1999). While they are able to capture the failure portion of the curve, they often omit the viscoelastic properties of the tendon or ligament.
Ultimately, the accurate representation of tissues is critically important for biomechanical models to predict tissue level stresses and strains (Fig. 1). At present, tissue models can accurately characterize either the viscoelastic or the failure properties of tissues, with very few models attempting to bridge the gap between the two (Safa et al., 2019a, Safa et al., 2019b). Therefore, the purpose of this investigation was to present, and subsequently simplify, a one-dimensional hyper-viscoelastic soft tissue model with damage properties. After its derivation, we demonstrate that this final model exhibits failure, creep, stress-relaxation, and hysteresis.
Section snippets
Model Development
We present the model derivation in three stages (Fig. 2). The first is a complete model of collagen fibre recruitment kinematics, represented with a transport equation. This transport equation is able to describe features of damage accumulation through loss of collagen fibres from mechanical load; unfortunately, it is difficult to solve when combined with other rheological models. Therefore, in the second step, we reduce the transport equation to a series of three ordinary differential
Failure simulation
When the model was strained to failure, at a strain rate of 0.01 and 0.02 s−1, it exhibited the characteristic stress–strain curve of ligaments or tendons. These rates are on the lower end of the physiological range, as the breaking function parameters used in this investigation were from quasi-static loading (Mattucci et al., 2013, Barrett and Callaghan, 2017). The response of the model included a toe region, as the fibres uncrimped; a linear region, as the whole population of fibres became
Discussion
In this investigation, we presented a novel rheological model which replicates several important mechanical features of tendons and ligaments. Soft tissues typically adhere to a characteristic stress–strain curve, exhibiting a toe-region, linear-region, and failure region (Fig. 3). Additionally, they portray a number of viscoelastic properties, like creep (Fig. 4), stress-relaxation (Fig. 5) and hysteresis (Fig. 6). The current model can replicate all of these phenomena, as well as failure
Conclusion
In this manuscript we presented a viscoelastic model for ligaments and tendons which included components of damage properties. The model represents a novel method for combining and delineating viscoelasticity and damage in biomechanical models. We explored creep, stress-relaxation, and hysteresis within this modelling framework, and demonstrated that they are all properties achievable by the proposed model. Future improvements to the model may be made in providing a more realistic breaking
CRediT authorship contribution statement
Jeff M. Barrett: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review and editing. Jack P. Callaghan: Conceptualization, Formal Analysis, Funding Acquisition, Investigation, Project Administration, Resources, Supervision, Validation, Writing – review and editing.
References (81)
- et al.
A comparison of the quasi-static mechanical and non-linear viscoelastic properties of the human semitendinosus and gracilis tendons
Clin. Biomech. (Bristol, Avon)
(2010) - et al.
The tensile and stress relaxation responses of human patellar tendon varies with specimen cross-sectional area
J. Biomech.
(1999) Characterization of the passive responses of live skeletal muscle using the quasi-linear theory of viscoelasticity
J. Biomech.
(1994)- et al.
A biomechanical model for fibril recruitment: Evaluation in tendons and arteries
J. Biomech.
(2018) - et al.
Low back three-dimensional joint forces, kinematics, and kinetics during walking
Clin. Biomech.
(1999) - et al.
Biomechanical properties of spinal ligaments and a histological study of the supraspinal ligament in traction
J. Biomech.
(1985) - et al.
EMG assisted optimization: A hybrid approach for estimating muscle forces in an indeterminate biomechanical model
J. Biomech.
(1994) - et al.
A review on the Mullins effect
Eur. Polym. J.
(2009) - et al.
Quantifying Achilles tendon force in vivo from ultrasound images
J. Biomech.
(2016) - et al.
Validation of shear wave elastography in skeletal muscle
J. Biomech.
(2013)
Model-based estimation of muscle forces exerted during movements
Clin. Biomech. (Bristol, Avon)
Rheological analysis of soft collagenous tissue. Part I: theoretical considerations
J. Biomech.
Rheological analysis of soft collagenous tissue Part II: Experimental Evaluations and Verification
J. Biomech.
Partial rupture of the Achilles tendon during a simulated fire ground task: Insights obtained from a case report for the prevention and reporting ofmusculoskeletal injury
Clin. Biomech.
On the constitutive modelling of recruitment and damage of collagen fibres in soft biological tissues
Eur. J. Mech.-A/Solids
Can biomechanical variables predict improvement in crouch gait?
Gait Posture
A theoretical and non-destructive experimental approach for direct inclusion of measured collagen orientation and recruitment into mechanical models of the artery wall
J. Biomech.
Muscle structure and theories of contraction
Prog. Biophys. Biophys. Chem.
Computational biomechanical modeling of scoliotic spine: Challenges and opportunities
Spine Deformity
A failure model for ligaments
J. Biomech.
Biomechanics of tendon injury and repair
J. Biomech.
Simultaneous prediction of muscle and contact forces in the knee during gait
J. Biomech.
Strain rate dependent properties of human craniovertebral ligaments
J. Mech. Behav. Biomed. Mater.
The biomechanics of low back injury: Implications on current practice in industry and the clinic
J. Biomech.
A simplified approach to quasi-linear viscoelastic modeling
J. Biomech.
Cervical spine response in frontal crash
Med. Eng. Phys.
Tendon injury and repair – A perspective on the basic mechanisms of tendon disease and future clinical therapy
Acta Biomater.
Using computed muscle control to generate forward dynamic simulations of human walking from experimental data
J. Biomech.
Human cervical spine ligaments exhibit fully nonlinear viscoelastic behavior
Acta Biomater.
Mechanical properties of the human achilles tendon
Clin. Biomech.
Rheological properties of the human lumbar spine ligaments
J. Biomed. Eng.
The effect of foot strike pattern on achilles tendon load during running
Ann. Biomed. Eng.
Unified viscoelasticity: Applying discrete element models to soft tissues with two characteristic times
J. Biomech.
A mechanistic damage model for ligaments
J. Biomech.
Failure properties of cervical spinal ligaments under fast strain rate deformations
Spine
Probabilistic model of ligaments and tendons: Quasistatic linear stretching
Phys. Rev. E – Stat. Nonlinear Soft Matter Phys.
Low back joint loading and kinematics during standing and unsupported sitting
Ergonomics
Stabilizing function of trunk flexor-extensor muscles around a neutral spine posture
Spine
Effects of sitting and standing on upper extremity physical exposures in materials handling tasks
Ergonomics
Mathematical Head-Neck Models for Acceleration Impacts
Cited by (5)
Lose the stress: Viscoelastic materials for cell engineering
2023, Acta BiomaterialiaCitation Excerpt :For example, one set of experimental stress-relaxation measurements of collagen gels revealed a range of characteristic relaxation times spanning <1 s to >100s, which the authors attributed to fiber/fibril sliding at different length scales [37]. On the macroscopic scale, the viscoelastic behavior of collagen tissues and hydrogels has been widely explored [37,43,46,47,55–69]. The tools of materials engineering have proven valuable for the manipulation of collagen stress-relaxation capabilities.
The rate of tendon failure in a collagen fibre recruitment-based model
2021, Journal of the Mechanical Behavior of Biomedical MaterialsCitation Excerpt :These experimental studies, however, cast doubt on the validity of that assumption. We have found that representing the other constituents with a Voigt-element in series with the current model endows it with both viscoelastic effects and imparts less of the total tissue strain onto the fibrils (Barrett and Callaghan, 2021). However, there was considerable difficulty in arriving at reliable parameter estimates through least-squares.
A discrete shear lag model of the mechanics of hitchhiker plants, and its prospective application to tendon-to-bone repair
2023, Proceedings of the Royal Society A: Mathematical, Physical and Engineering SciencesTheoretical Study on Viscoelastic Property of Biological Soft Tissues with Two Characteristic Time Parameters
2022, Yiyong Shengwu Lixue/Journal of Medical Biomechanics