A microstructurally motivated constitutive description of collagenous soft biological tissue towards the description of their non-linear and time-dependent properties

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Abstract

A versatile constitutive model for load-carrying soft biological tissue should incorporate salient microstructural deformation mechanisms and be able to reliably predict complex non-linear viscoelastic behavior. The advancement of treatment and rehabilitation strategies for soft tissue injuries is inextricably linked to our understanding of the underlying tissue microstructure and how this defines its macroscopic material properties. Towards this long-term objective, we present a generalized multiscale constitutive framework based on a novel description of collagen, the most mechanically significant extracellular matrix protein. The description accounts for the gradual recruitment of undulated collagen fibrils and introduces proteoglycan mediated time-dependent fibrillar sliding. Crucially, the proteoglycan deformation allows for the reduction of overstressed fibrils towards a preferential homeostatic stress. An implicit Finite Element implementation of the model uses an interpolation strategy towards collagen fiber stress determination and results in a memory-efficient representation of the model. A number of test cases, including patient-specific geometries, establish the efficiency of the description and demonstrate its ability to explain qualitative properties reported from macroscopic experimental studies of tendon and vascular tissue.

Introduction

The constitutive modeling of load-carrying soft biological tissues, such as musculoskeletal (tendon, ligament, muscle) and cardiovascular (arterial, myocardium) tissue, presents as a valuable tool towards enhancing the understanding of injury and healing mechanisms, as well as degenerative diseases. While cells play a minor role, the tissue’s mechanical properties are principally determined by extracellular matrix (ECM) constituents, such as collagen, elastin and proteoglycans (PG’s). The complex arrangement and interaction of these different components defines the macroscopic structural properties of the tissue. The microstructural architecture is specific to every tissue type and each is mechanically optimized to fulfill their individual biomechanical function.

Fibrous tissues are characteristically anisotropic and non-linear, showing stiffening at higher strains, Fung (1967). These features have historically been attributed to the organization and successive recruitment of undulated (or crimped) fibrous collagen across different length scales, Rigby et al. (1959) and Roach and Burton (1957). The collagen in fibrous tissues is organized hierarchically, where tropocollagen molecules form collagen fibrils, which in turn assemble into collagen fibers, and then further integrate into higher order structures such as fascicles, Fratzl (2008).

Fibrous tissues are also described by load-history dependent and time-dependent mechanical behavior; exhibiting hysteresis during cyclic loading, the Mullins effect in repeated cycles, stress relaxation at constant strain and creep at constant stress, Fung (1993). Furthermore, this viscoelasticity is highly non-linear, whereby hysteresis is strain rate insensitive and the rates of creep and relaxation are a function of the applied stress or strain level, respectively, Lanir and Fung (1974). The extent of this behavior varies to different degrees depending upon the tissue in question and its anatomical location. The mechanisms underpinning the temporal adjustment of the microstructure are not well understood, yet individual components of the ECM are known to exhibit independent time-dependent behavior, a feature that is often suppressed in the characterization of soft biological tissues.

The cross-linking of collagen, in particular, is thought to greatly influence the various deformation mechanisms that act throughout its hierarchical architecture. Specifically, collagen fibrils are mutually connected by fibril-associated PG’s such as decorin; anti-parallel anionic glycosaminoglycan (GAG) duplexes that are bound non-covalently to collagen fibrils through a protein core at intervals of approximately 60 nm, Cribb and Scott (1995). The reversible deformability of PG’s is critical to their function as shape-maintaining modules, Scott (2003), with both fast and slow modes of deformation having been suggested, Scott (2008).

Whilst a plethora of evidence exists in the literature, the mechanical influence of PG’s is still not fully understood, with no clear role having been established. Some results suggest that they promote the relative sliding of adjacent fibrils as opposed to interfibrillar load transitioning, Rigozzi et al., 2009, Rigozzi et al., 2010 and Screen et al. (2005). The selective removal of GAG’s in vascular tissue has been shown to cause a significant reduction in stress relaxation, and result in an earlier transition point of the non-linear stress–strain curve, with multiphoton microscopy displaying reduced undulation amongst fibrous collagen, Mattson et al., 2017, Williams et al., 2009 and Lin et al. (2016). Similarly, altered stress relaxation, Elliott et al. (2003) and Legerlotz et al. (2013), greater strain-rate sensitivity, Robinson et al. (2004), higher fibril strains, Rigozzi et al. (2013), and reduced mechanical integrity, Gordon et al. (2015), have been reported in GAG-depleted tendon tissue. Relaxation times for isolated collagen fibrils were also found to be smaller than those reported for tissue-level relaxation, Shen et al. (2011). GAG’s have additionally been found to contribute to the elasticity and viscoelasticity of other soft tissues types such as cartilage, eye sclera, and heart valve tissue, Halper, 2014, Murienne et al., 2015 and Ross et al. (2019). Together this indicates that proteoglycans are involved in tissue viscoelasticity and the recruitment behavior of collagen. However, others have contrarily observed the impact of PG’s to be negligible; with PG knockout leading to no significant change in tendon, Fessel and Snedeker, 2011, Svensson et al., 2011, and ligament, Lujan et al. (2009), mechanical properties.

The conflicting nature of the aforementioned observations has prompted the conception of several competing theories regarding the structure-function relationship between collagen fibrils and PG’s in biological soft tissues. PG’s (decorin specifically) are thought to control the alignment and distance between adjacent fibrils, Connizzo et al. (2013) and Chen et al. (2020), as such it has been suggested that a reduction in PG’s may lead to increased friction amongst fibrils and the consequent reduction of interfibrillar sliding. Another theory counters this assertion and postulates that PG’s instead behave more in a preventative manner whereby they act as a viscous adhesive between fibrils. There still remains much controversy and a lack of common agreement concerning the mechanical role of PG’s in relation to soft tissue viscoelasticity.

A constitutive description that encapsulates the diverse time dependent material properties of load carrying tissues, demands a sufficiently detailed approach. Several models have been proposed that approximate soft tissue as a quasi-linear viscoelastic (QLV) material, in both one and three dimensions, Puso and Weiss, 1998, DeFrate and Li, 2007 and Johnson et al. (1996). Whilst informative, these models present only a macroscopic (single scale) view of the tissue, prohibiting an explicit description of any localized microstructural reorganization. Furthermore, Despite using a large number of parameters, they fail to adequately capture the dependency of creep and relaxation rates upon stress and strain levels. Others have used non-linear viscoelastic descriptions to overcome this issue, Pioletti and Rakotomanana, 2000, Provenzano et al., 2002, Troyer et al., 2012, Kahn et al., 2010 and Peña et al. (2011), yet while such models may calibrate better to experimental data, they are still highly phenomenological and associated material parameters lack any real histological foundation.

Conversely, microstructurally based models permit the introduction of information concerning soft tissue composition and structure. The macroscopic stress state is considered to arise from the contribution and interaction of mechanically significant ECM constituents, where for many applications collagen fibers and PG’s are of greatest importance, Gasser et al. (2006) and Federico and Gasser (2010). Specifically, such models allow for the inclusion of constituent specific stress-free configurations, which allows not just for the modeling of individual constituent time-dependent properties, Raz and Lanir, 2009, Sverdlik and Lanir, 2002, Shearer et al., 2020 and Liu et al. (2019), but also an explicit description of collagen recruitment to characterize soft tissue nonlinearity, Martufi and Gasser, 2011, Martufi and Gasser, 2012, Polzer et al., 2015, Shearer, 2015, Weisbecker et al., 2015 and Hill et al. (2012). Others have also included the mechanical contribution of GAG’s towards collagen engagement within their constitutive models, Mattson et al. (2019) and Linka et al. (2016).

The present work proposes a constitutive framework that includes salient microstructural deformation mechanisms at multiple length scales, to capture the macroscopic time dependent response of fibrous soft tissue. To this end, continuum level tissue properties are derived through the integration of a 1D collagen fiber model over the unit sphere, where said model incorporates collagen fibril recruitment and PG mediated interfibrillar sliding. A multiscale microfiber approach of this kind allows for the direct integration of histological features at the microscale, towards the effective characterization of the discussed viscoelastic phenomena. We apply the model to the uniaxial deformation of the Calcaneal (Achilles) tendon. A tissue whose time-dependent properties have been investigated extensively, due in part to its complex transient loading history and clinical relevance. Furthermore, the model is applied to the pressure inflation of the abdominal aorta, both a cylindrical and a patient-specific aortic bifurcation geometry. Given these applications, our description predicts non-linear viscoelastic properties and other notable phenomena known from the experimental characterization of fibrous tissues.

Section snippets

Histomechanical modeling assumptions

We consider a fibrous tissue comprised of collagen fibers that reinforce an otherwise isotropic matrix material, which encapsulates the combined mechanical properties of all non-collagenous ECM constituents. Collagen fibers are assembled by numerous collagen fibril proteoglycan-complexes (CFPG-complexes), i.e., nanoscale mechanical sub-units consisting of undulated collagen fibrils that are interconnected by PG cross-linking bridges, see Fig. 1. Upon deformation, fibrils are sequentially

Results

In order to explore the ability of the proposed model to capture the time-dependent mechanical properties of fibrous tissues, and to demonstrate its finite element implementation, four numerical examples were analyzed: (i) the properties of a single CFPG-complex under cyclic loading, (ii) the uniaxial deformation of the human Achilles tendon for a variety of testing protocols, (iii) the pressure inflation of a cylindrical segment of the healthy and aneurysmatic abdominal aorta, and (iv) the

Discussion and conclusions

Collagen is the most abundant structural protein in biological tissues, and its ability to carry load is fundamental to the proper physiological function of many organs. The mechanical functionality of the collagenous microstructure should therefore be properly reflected, or characterized, in the constitutive description of fibrous tissues. We have developed a generalized multiscale constitutive framework for soft biological tissue that is based on the hierarchical organization of collagen. The

CRediT authorship contribution statement

Christopher Miller: Concept, Design, Analysis, Writing, Revision of the manuscript. T. Christian Gasser: Concept, Design, Analysis, Writing, Revision of the manuscript.

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.

Acknowledgements

This research has been supported by the project grant 2015-04476 from the Swedish Research Council (VR) .

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