Research PaperComparison of structural anisotropic soft tissue models for simulating Achilles tendon tensile behaviour
Graphical abstract
Introduction
The Achilles tendon is the largest tendon in the body and the most frequently injured tendon in humans. It connects the gastrocnemius muscle (calf muscle) to the calcaneus bone (heal bone) and serves a mechanical function by transmitting muscle loads to the bone and enables efficient locomotion. However, repetitive loading of tendons can lead to chronic tendon injuries (tendinopathy) (Kader et al., 2002) that are signified by inflammation, pain and very slow recovery (Scott et al., 2013). Treatments of tendinopathy involves the application of controlled mechanical loading (Scott et al., 2013), but there is no consensus on the timings and magnitudes of loading to enhance the tissue restoration process (Sussmilch-Leitch et al., 2012). Also, our understanding of the magnitudes and type of re-occurring loading that initially causes tendinopathy is limited to hypotheses about load-induced microtrauma (Maffulli et al., 2004, Riley, 2004) and damage accumulation in the tissue (Maffulli et al., 2004, Neviaser et al., 2012). Constitutive mechanical models for tendons can be helpful to better understand tendinopathy incidence and prevention, as they can phenomenologically simulate the biomechanical behaviour of the tissue and its constituents.
Many constitutive soft tissue models have been specifically developed for tendons and ligaments, describing the tissues as for example viscoelastic (Kahn et al., 2010, Pioletti and Rakotomanana, 2000, Pioletti et al., 1998), poroelastic (Yin and Elliott, 2004) and hyperelastic (Ciarletta et al., 2008, Tang et al., 2011) using both isotropic and anisotropic formulations. Existing models are often general and categorise tendon and ligaments under the common nomenclature of soft connective tissues. But experimental studies show tissue and site-specific function and properties (Birch et al., 2013, Bojsen-Moller et al., 2004, Kutsumi et al., 2005). Thus, site- and tendon-specific models are necessary to unravel the complex synergies between mechanical loading and the tissue׳s structure, composition and function.
Tendons have a hierarchical tissue structure where the collagen fibres are the main structural components that confer the mechanical strength in tension, which results in an anisotropic behaviour of the tissue (Voleti et al., 2012). These fibres are recruited and align to the direction of loading. Experimental studies report that excessive loading can lead to disruption of collagen fibres and their orientation, which in turn affects the biomechanical behaviour of the whole tendon (Sereysky et al., 2012, Sereysky et al., 2010, Wren et al., 2003). Researchers have developed computational models on mesoscopic and microscopic tendon level, modelling the collagen fibres, their crimping pattern and cross-links (Depalle et al., 2014, Gautieri et al., 2013, Grytz and Meschke, 2009, Kastelic et al., 1980, Reese et al., 2013). These models can describe localised phenomena but lack the ability to consider interaction-effects for example between collagen and the interstitial fluid, as well as how that is scaled up to the tissue level behaviour of tendons.
One way to overcome this limitation is to use structural constitutive models, where the tissue response is assumed to be the additive effect of the biomechanical behaviour of its tissue components. This idea was first introduced by Lanir (1980) who developed a structural model for soft tissues. Later, this concept was adopted and applied for many different tissue types such as cardiovascular tissues (Fan and Sacks, 2013, Fan and Sacks, 2014, Gasser et al., 2006, Holzapfel et al., 2002), ligaments (De Vita and Slaughter, 2006, Limbert et al., 2003, Rodriguez et al., 2006), cartilage (Julkunen et al., 2007, Wilson et al., 2005), dermal tissue (Ni Annaidh et al., 2012, Sacks, 2003) and tendons (Ciarletta et al., 2006, Kahn et al., 2013, Puxkandl et al., 2002). We have previously developed a structural model for the Achilles tendon by modifying an existing material model for articular cartilage (Khayyeri et al., 2015). This model can capture the tensile behaviour of intact rat tendons during tensile loading. The structural Achilles tendon model consists of three main tendon components – water, collagen fibres and non-collagenous matrix – and could depict the load-bearing role of each constituent. However, it simulates fluid inflow to the tissue during tension, which according to experiments is reported to be the reverse (Hannafin and Arnoczky, 1994, Lanir et al., 1988, Wellen et al., 2004). Furthermore, a recent report has shown that the hyperelastic structural model of Gasser et al. (2006) (GOH-model) often used for soft tissues like cardiovascular tissues (Holzapfel et al., 2002, Holzapfel and Ogden, 2009, Holzapfel and Ogden, 2010) can capture the biomechanical behaviour of Achilles tendons undergoing healing after total rupture (Bajuri et al., 2016). The GOH-model is a well-accepted hyperelastic anisotropic model that has been included into the commercial finite element modelling software Abaqus.
This study compares three structural constitutive models. Two were originally developed for other tissues, namely cartilage (Wilson et al., 2004) and cardiovascular tissues (Gasser et al., 2006), but they have both recently been implemented to describe the tensile behaviour of the Achilles tendon (Bajuri et al., 2016, Khayyeri et al., 2015). The study further introduces a third new model that was modified from Khayyeri et al. (2015), which models the non-fibrillar matrix of the tendon as transversely isotropic (instead of isotropic as in Khayyeri et al. (2015)) in order to capture the exudation of interstitial fluid from the tendon during tensile loading. This addition is motivated by the fact that fluid makes out approximately 70% of the tendon weight and plays a significant role for tendon biomechanical function. Studies have for example suggested that the stress-relaxation response of tendons is partly due to fluid exudation (Hannafin and Arnoczky, 1994, Lanir et al., 1988), and others have shown that fluid interacts with the collagen molecules creating water-generated tensile stresses (Masic et al., 2015). Accurate representation of the water in tendons is essential as the role of fluid in tissue function, health and regeneration is still not fully understood. The aim of this paper is to investigate and compare three structural anisotropic models for capturing rat Achilles tendon biomechanical behaviour during tensile loading.
The constitutive models that will be investigate are the following:
- 1.
An isotropic hyperelastic model with anisotropic fibre description (GOH-model short for Gasser–Ogden–Holzapfel model) (Bajuri et al., 2016, Gasser et al., 2006).
- 2.
A visco-hyper-poro-elastic model (earlier presented as Fibre Reinforced Poro-Viscoelastic (FRPVE)-model) with isotropic hyperelastic matrix and anisotropic viscoelastic fibre description (FRPVEiso-model) (Khayyeri et al., 2015, Wilson et al., 2004).
- 3.
A visco-hyper-poro-elastic model with transversely isotropic hyperelastic matrix and anisotropic viscoelastic fibre description (FRPVEtrans-model).
The models will be optimised to average experimental data retrieved from a mechanical tensile experiment on rat Achilles tendons (Eliasson et al., 2007). The Achilles tendons from Sprague Dawley rats (n=9, female, 16 weeks old) were harvested with the calcaneus bone-end and the gastronomous muscle. The bone-end was attached to clamps in a mechanical testing machine with 30° dorsal flexion and the muscle-end was fixed with sandpaper to the machine clamps. After reaching a pre-load of 1 N, cyclic tensile loading was performed between 1 and 20 N, for 20 consecutive cycles (only the first load cycle was used in this study). Data from the control group of this study was used (please refer to Eliasson et al. (2007) for more details on the experiment).
Section snippets
The anisotropic hyperelastic model
The GOH-model is a structural hyperelastic model that has been widely implemented for describing the biomechanical behaviour of collagenous tissues (Gasser et al., 2006). It assumes that the energy potential is a sum of energy in the non-fibrillar matrix and the structural collagen fibres. The strain energy function is formulated aswhere and are the potentials for the non-fibrillar matrix and collagen fibres respectively, , where C is the right
Results
Unloading of tendons is a time-dependent behaviour that includes energy loss in the system, visible as viscoelastic hysteresis. Therefore, simulations of a complete load cycle does not permit the hyperelastic GOH-model to capture the viscoelastic behaviour with high accuracy (Fig. 3A, RMS=3.228, Simulation 1 in Table 1). The GOH-model was also optimised against data from only one load step, i.e. tensile load only without unloading. In these simulations, the model performed well and could
Discussion
This study compared the ability of three structural constitutive models to simulate the biomechanical behaviour of rat Achilles tendon under tensile loading. It investigates two constitutive models previously developed for cardiovascular tissue (Gasser et al., 2006) and articular cartilage (Wilson et al., 2004) (also for Achilles tendon (Khayyeri et al., 2015)) and presents an entirely novel structural model. The results show that all models can be optimised to capture tensile mechanical
Funding
This research project is supported by a Marie Curie Intra-European Fellowship for Career Development (PIEF-GA- 2012-626941).
Acknowledgements
We would like to thank Prof Per Aspenberg, Linköping University, Sweden, for kindly providing us the experimental data on rat Achilles tendon used in this study.
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