Abstract
Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin–Voigt model of linear viscoelasticity to represent the stress–strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this paper, we systematically assess the pattern formation potential of different stress–strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin–Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress–strain relation of the ECM, while no patterns are observed when the Kelvin–Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress–strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.
Highlights
Pattern formation resulting from spatial organisation of cells is at the basis of a broad spectrum of physiological and pathological processes in living tissues (Jernvall et al 2003)
80 Page 2 of 38 stitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin–Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress–strain relation of the extracellular matrix (ECM), while no patterns are observed when the Kelvin–Voigt model is employed
Here we complement and further develop the results presented in Byrne and Chaplain (1996) by systematically assessing the pattern formation potential of different stress–strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues (Byrne and Chaplain 1996; Murray et al 1988; Oster et al 1983)
Summary
Pattern formation resulting from spatial organisation of cells is at the basis of a broad spectrum of physiological and pathological processes in living tissues (Jernvall et al 2003). Mechanochemical and mechanical models of pattern formation in biological tissues have been used to study a variety of biomedical problems, including morphogenesis and embryogenesis (Brinkmann et al 2018; Cruywagen and Murray 1992; Maini and Murray 1988; Murray and Maini 1986; Murray et al 1988; Murray and Oster 1984a, b; Murray et al 1983; Oster et al 1983; Perelson et al 1986), angiogenesis and vasculogenesis (Manoussaki 2003; Scianna et al 2013; Tranqui and Tracqui 2000), cytoskeleton reorganisation (Alonso et al 2017; Lewis and Murray 1991), wound healing and contraction (Javierre et al 2009; Maini et al 2002; Olsen et al 1995; Tranquillo and Murray 1992), and stretch marks (Gilmore et al 2012) These models have been used to estimate the values of cell mechanical parameters, with a particular focus on cell traction forces
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