Research PaperA micromechanical model of tendon and ligament with crimped fibers
Introduction
Tendons and ligaments serve as fundamental load-bearing tissues in the musculoskeletal systems of vertebrates by connecting bones to bones and bones to muscles, respectively (Reese et al., 2010; Shearer, 2015a, 2015b). Understanding the mechanical behavior of tendons and ligaments, especially its relation to their microstructures, is essential for the study of injury, reconstruction, surgery operations, and tissue engineering (Hansen et al., 2002; Lin et al., 2016; Li et al., 2020). Both tendons and ligaments are characterized by a hierarchical structure composed of numerous fibrous subunits at different length scales (Reese et al., 2010; Shearer, 2015a, 2015b; Sherman et al., 2015). The most basic microstructural subunit is collagen fibrils, which are arranged in a crimped pattern to form collagen fibers that further constitute tendons and ligaments (Evans and Barbenel, 1975; Reese et al., 2010; Yahia and Drouin, 1988, 1989; Zhu et al., 2012). Such microstructural organization deeply affects macroscopic elastic response of tendons and ligaments, including their uniaxial stress–strain behavior and volumetric behavior (e.g. Poisson's ratio) (Ahmadzadeh et al., 2015; de Botton and Oren, 2013; Maceri et al., 2010, 2012; Reese et al., 2010).
Specifically, the stress–strain curves of tendon and ligament feature the J-shape due to the crimped morphology of fibers (Maceri et al., 2010). The effect of fiber waviness on the mechanical properties of tendon and ligament is a key question in understanding their biomechanical properties and guiding diagnosis and treatment of related diseases. In addition, an intriguing volumetric property of tendon and ligament is that their Poisson's ratios may exceed 0.5, which is the upper limit for homogenous isotropic solids. For instance, the reported tensile Poisson's ratios in the axial direction along the fiber are about 0.80.3, 2.01.9, and 2.982.59 in rat tail tendon (Cheng and Screen, 2007), human hip joint capsule ligament (Hewitt et al., 2001), and bovine flexor tendon (Lynch et al., 2003), respectively. Such large Poisson's ratios have been considered to indicate significant transverse compressive forces and volume change during deformation, which are of fundamental importance to signal mechanotransduction for adjacent cells and nutrients transportation within these tissues (Cheng and Screen, 2007; Hewitt et al., 2001; Lavagnino et al., 2015; Reese et al., 2010). Despite such important roles of the large Poisson's ratio in controlling physiological functions of tendons and ligaments, its microstructural origins are poorly understood. Specifically, both planar and helical arrangements of collagen fibers have been found in tendons and ligaments (Evans and Barbenel, 1975; Yahia and Drouin, 1988, 1989; Zhu et al., 2012), yet it remains unclear how these microstructures contribute to the large Poisson's ratios (Garnich and Karami, 2004; Reese et al., 2010).
Therefore, in-depth mechanistic study is needed to elucidate the structure–function relation of tendons and ligaments, which is a crucial question to understand their biomechanics as well as to guide the design and manufacturing of tissue-engineered tendons and ligaments. From a mechanical point of view, tendon and ligament can be theorized as fiber-reinforced biocomposites, to which a considerable amount of mechanical researches have been dedicated. However, there is still a lack of mechanical models that can elucidate their structure–function mechanisms underlying the effect of crimped fiber morphology and the origin of the large Poisson's ratio.
Molecular dynamics (MD) simulations offer a powerful approach to understand the structure–function relationship of biomaterials from the bottom up. For example, this method has been used to investigate the molecular deformation mechanisms during stretching of collagen microfibrils (Gautieri et al., 2011), the origin of the longitudinal heterogeneity in collagen fibrils (Tang et al., 2017), and the interaction between an inorganic biomaterial and actin (Li et al., 2014). However, MD simulations are still hard to predict the macroscopic mechanical properties of biocomposites with complicated, hierarchical microstructures.
To date, continuum mechanics that regards biological tissues as homogenized or functional gradient media, is still the most frequently used in clinical applications to describe the mechanical behaviors by an overall constitutive relation of specific form. The constitutive parameters are then measured via uniaxial tension (Holzapfel et al., 2000; Natali et al., 2010; Pavan et al., 2015), indentation, or other experimental techniques. However, the obtained constitutive parameters have not been well correlated to the microstructure or mechanical properties of the constituents (Ault and Hoffman, 1992a). In addition, in previous mechanical analysis of soft tissues, the volume incompressibility was often assumed, and thus the effects of Poisson's ratio have not been well included (de Botton and Oren, 2013; Gasser et al., 2006; Holzapfel et al., 2000, 2015, 2019).
In contrast, microstructural models normally assume an approximate shape of crimped collagen fibrils, such as in-plane zig-zag (Diamant et al., 1972), sinusoidal (Comninou and Yannas, 1976) and helical shape (Beskos and Jenkins, 1976), and thus can predict the mechanical properties of these fibrous tissues from microstructural features. Recently, by considering the crimped morphology of fibers in tendons and ligaments, Reese et al. (2010) constructed a micromechanics-based finite element model to predict the J-shape of the experimental stress–strain curves under uniaxial tension. Importantly, they found that the helical superstructure rather than the planar crimp of biocomposites is capable of predicting large Poisson's ratios, which is in agreement with Garnich and Karami's statement (Garnich and Karami, 2004) that planar crimp alone is insufficient to generate such a distinctive property. However, a critical question that needs to be elucidated further is how the planar crimp of biofibers contributes to the large Poisson's ratio of tendons and ligaments.
Micromechanics predicts the macroscopic mechanical properties of a composite from the microstructure and mechanical parameters of its constituents. In the past decades, micromechanical methods have been extensively applied to predict the elastic properties of composites such as polycrystalline materials (Hershey, 1954) and carbon nanotube-reinforced composites (Shao et al., 2009; Shi et al., 2004). As the constitutive relation of biological tissues attracts extensive attention across the scientific community, micromechanical methods have been extended to analyze the structure–function relationship of biological tissues. Lin et al. (2016) showed that the micromechanical method is feasible in predicting the elastic properties of soft composites from the microstructures and constitutive relations of their constituent phases. Fritsch and Hellmich (2007) related the hierarchical structural organization of bone to its effective elastic properties by utilizing the micromechanical model. Importantly, theoretical results were shown to be in good agreement with experimental measurements, therefore validating the suitability of micromechanical model for predicting mechanical properties of biocomposites. Given such success of micromechanical methods in other contexts, herein we extend its application to analyze the structure–function relationship of tendons and ligaments.
This study is aimed to investigate the structure–function mechanism in tendons and ligaments so as to understand how material parameters and microstructural characteristics might regulate the macroscopic mechanical properties of the material. Specifically, we adopt the Mori–Tanaka method, which has been demonstrated as a simplified yet powerful micromechanical method for estimating the constitutive relations of composites (Mori and Tanaka, 1973). Thereby, the interaction among the constituents in a biocomposite with planar crimped fibers is accounted for. For simplicity, both the planar crimped fibers and the matrix are assumed to be isotropic and elastic. Based on a unit cell model, we examine the effect of fiber waviness on the elastic properties of tendons and ligaments. Finite element simulations will also be performed to validate the theoretical results. In the following sections, Sections 2 and 3 present the micromechanical model and the finite element model, respectively, to predict the constitutive relation of a biocomposite with ligament/tendon-like microstructure. Section 4 exhibits the theoretical and numerical results, followed by discussion over the implications of the micromechanical model in Section 5. Finally, Section 6 summarizes the main conclusions drawn from this study.
Section snippets
Micromechanical model
In this study, a tendon or ligament is considered as a composite with periodic structures consisting of aligned and curved fibers. The Mori–Tanaka method is used to predict the constitutive relation of such a fiber-reinforced composite. With a concise unit cell model, the relation between the overall elastic properties and microstructural features of the composite will be established.
Finite element model
To test the accuracy of the theoretical solutions above, a finite element model is developed to numerically solve the mechanical properties of tendon/ligament-like composites. In this finite element model, we define a unit cell which contains all necessary microstructural information and has periodic boundary conditions.
Results
In this section, the effect of crimp angle on the elastic properties of tendon/ligament-like composites is quantified via both theoretical and numerical analysis. Parametric analysis on the Poisson's ratio is also presented, followed by an examination of local stresses during axial stretching.
Discussion
Predicting constitutive relations of biological tissues are of central importance in understanding their biomechanics and guiding the tissue engineering. Micromechanics-based approaches that consider biomimetic microstructures and the mechanical behavior of individual constituents are accurate and efficient for characterizing macroscopic mechanical properties of biocomposites together with analyzing their structure–function relationship. Previous micromechanical models, which usually treated
Conclusions
In this paper, we have presented a micromechanical model to investigate the mechanical properties of tendons and ligaments, which are treated as planar crimped fiber-reinforced composites. By using the Mori–Tanaka method, the mechanical interaction between the fibers and the matrix is reasonably reflected. Explicit analytical expressions of the macroscopic elastic modulus and the Poisson's ratio of the composites under the loading along the axial direction are obtained, correlating to the crimp
Credit author contribution statement
Shengsheng Xiao: conducted theoretical analysis, performed simulations; all authors wrote the paper. Yue Shao: conducted theoretical analysis, and. Bo Li: conducted theoretical analysis. Xi-Qiao Feng: designed research and conceived the project.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
Support from the National Natural Science Foundation of China (Grant No. 11921002) is acknowledged.
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