A complementary energy approach accommodates scale differences in soft tissues

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Abstract

The mechanics of biological entities, from single molecules to the whole organ, has been extensively analyzed during the last decades. At the smaller scales, statistical mechanics has fostered successful physical models of proteins and molecules, which have been later incorporated within constitutive models of rubber-like materials and biological tissues. At the macroscopic scale, the additive decomposition of energy functions i.e., a parallel arrangement of the tissue constituent, has been recurrently used to account for the internal heterogeneity of soft biological materials. However, it has not yet been possible to unite the mechanics at the tissue level with the actual response of the tissue components. Here, we exemplify our approach using cardiovascular tissue where the mechanical response at the tissue scale is in the range of kPa whereas the elastic modulus of collagen, the main component of the vascular tissue, is in the range of MPa GPa. In this work we develop a novel theoretical framework based on a complementary strain energy function that builds-up on a full network model. The complementary strain energy function introduces naturally an additive decomposition of the deformation gradient for the tissue constituents, i.e an arrangement in series of the constituents. We demonstrate that the macroscopic response of the tissue can be reproduced by just introducing the underlying mechanical and structural features of the micro-constituents, improving in a fundamental manner previous attempts in the mechanical characterization of soft biological tissues. The proposed theoretical framework unveils a new direction in the mechanical modeling of soft tissues and biological networks.

Introduction

The influence of mechanics in biology spans from single proteins through cells and tissues to entire organs (Hunter, Borg, 2003, Iskratsch, Wolfenson, Sheetz, 2014, Parsons, Horwitz, Schwartz, 2010), so physicists, mathematicians, and engineers have focused on its characterization for several decades. For example, the mechanics of adhesion molecules (Yao et al., 2014) determine the dynamics of cell adhesion (Elosegui-Artola, Oria, Chen, Kosmalska, Perez-Gonzalez, Castro, Zhu, Trepat, Roca-Cusachs, 2016, Erdmann, Schwarz, 2004, Geiger, Spatz, Bershadsky, 2009) and, therefore, of cell motility. The mechanics of actin have fundamental implications in a large number processes in mechanobiology such as cytokinesis and ring contraction (Biron, Alvarez-Lacalle, Tlusty, Moses, 2005, Salbreux, Prost, Joanny, 2009) and cancer progression (Butcher et al., 2009). Organization of the extracellular-matrix (ECM) at the tissue and organ level dictates the mechanics of the tissue in health and diseases such as aneurysm formation (Sakalihasan et al., 2005). Understanding how all these constituents arrange in biological networks is key to both understanding their mechanical response and also for the design of biomedical materials (Hench, Polak, 2002, Lutolf, Hubbell, 2005) and tissue engineering (Griffith, 2002, Langer, Tirrell, 2004).

Today, most of the mechanical models for soft tissues come from the early developments in rubber elasticity Flory (1961). The theory of non-linear hyperelasticity has been the starting point from which phenomenological continuum models for anisotropic soft biological tissues have been developed at the macroscale. On the contrary, the concepts of Gaussian and non-Gaussian statistical mechanics have been used at the microscale. The Freely Joined Chain or the Worm-like chain have described successfully the mechanics of DNA molecules (Bustamante et al., 2003) and collagen molecules (Buehler and Wong, 2007). The introduction of suitable averaging methods allowed to obtain the macroscopic constitutive law of networks, such as actin filaments (Bozec, Horton, 2005, Kuhl, Garikipati, Arruda, Grosh, 2005, MacKintosh, Kas, Janmey, 1995). In this regard, different continuum models based on single filament statistical mechanics have been also proposed (Alastrué, Martinez, Doblare, Menzel, 2009, Arruda, Boyce, 1993, Miehe, Göktepe, Lulei, 2004, Ogden, Saccomandi, Sgura, 2006) to describe the mechanical behavior of tissues with outstanding accuracy. However, despite using an accurate description of the microstructure, a significant mismatch in the mechanical properties of the constituents at the microscale is observed.

A clear example of multi-scale modeling in biological networks appears in cardiovascular tissue which is among the most studied biological entities in mechanics. Arterial wall, the specific system we focus on in this work to exemplify our theoretical derivation, is a highly heterogeneous material with a complex architecture made up of collagen and elastin (see Fig. 1). In this regard, most models for arterial mechanics rely on the mechanical contribution of these two components. The stiffness of the cardiovascular tissue (Fig. 1a) has been reported in the range of kPa (see Dobrin, 1978, Fung, Fronek, Patitucci, 1979, Holzapfel, Sommer, Gasser, Regitnig, 2005, Peña, Martínez, Peña, 2015 among many others) whereas the mechanical stiffness of the micro-constituents at the fiber and fibril scale e.g., collagen, (Fig. 1b) are found to be orders of magnitude higher, in the range of MPa GPa (Eppell, Smith, Kahn, Ballarini, 2006, Shen, Dodge, Kahn, Ballarini, Eppell, 2008, Tang, Ballarini, Buehler, Eppell, 2010). To date, no mechanical model has been able to reconcile the differences between the mechanical properties measured at the microscale with those obtained at the tissue level.

Section snippets

Continuum modeling of soft tissues

To motivate our work, we start by describing the basic kinematics and the energy function, or Strain Energy Density Functions (SEDF) Ψ, required to derive a macroscopic stress-strain relation as P=FΨ, with P the first Piola-Kirchhoff stress tensor, and F the deformation gradient (Marsden and Hughes, 1994). The deformation gradient F=Xφ(X,t):TΩ0TΩt, represents the linear tangent map from the tangent space TΩ0 to the time–dependent tangent space TΩt, and P is the thermodynamic force conjugate

Results

The presented framework has been applied to describe the mechanical behavior of arterial tissue. We first characterize the behavior of isolated collagen fibrils using Eq. 3. We then proceed to describe the mechanical behavior of arterial tissue (carotid) using the classical SEDF approach and the proposed complementary SEDF approach.

Conclusions

In short, we have introduced a new reasoning in the mechanical modeling of soft biological tissue, in which the complex architecture of the network components works concomitantly in series and not in parallel, as usually assumed. Here, we evaluate the behavior of the complementary SEDF against a standard SEDF within the same full-network model. Further comparison between different chain models, full-network models and full-network versus invariant and stretch-based models can be found

Declaration of Competing Interests

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.

CRediT authorship contribution statement

Pablo Saez: Conceptualization, Methodology, Investigation, Validation, Writing - original draft. Steven J. Eppell: Validation, Writing - review & editing. Roberto Ballarini: Validation, Writing - review & editing. Jose F. Rodriguez Matas: Conceptualization, Methodology, Investigation, Validation, Writing - original draft.

Acknowledgements

P.S acknowledges the support of the Generalitat de Catalunya 2017-SGR-1278.

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