[Paper Review] Mechanical models of pattern and form in biological tissues: the role of stress-strain constitutive equations

Chiara Villa, Mark A. J. Chaplain+2 more|arXiv (Cornell University)|Sep 23, 2020

Cellular Mechanics and Interactions77 references20 citations

TL;DR

This paper investigates the impact of different stress-strain constitutive models for the extracellular matrix (ECM) on pattern formation in biological tissues using mechanical models. It demonstrates through linear stability analysis and numerical simulations that fluid-like models (Maxwell and Jeffrey) induce robust spatial patterning, while solid-like models (Kelvin-Voigt and standard linear solid) fail to generate patterns, highlighting the critical role of ECM rheology in morphogenesis.

ABSTRACT

Mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems and describe the physical interactions between cells and their local surroundings. These models generally consist of a balance equation for the cell density, one for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Assuming this 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 type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this work, 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 support the idea that constitutive equations capturing viscous flow and permanent set (Maxwell model, Jeffrey model) have a pattern formation potential much higher than the others (Kelvin-Voigt model, standard linear solid model), further confirmed by the results of our numerical simulations. 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 mechanochemical models of pattern formation with stress-strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.

Motivation & Objective

• To assess how different stress-strain constitutive equations for the extracellular matrix (ECM) influence pattern formation in biological tissues.
• To evaluate the pattern formation potential of various linear viscoelastic models (Kelvin-Voigt, Maxwell, SLS, Jeffrey) under identical conditions.
• To determine whether fluid-like or solid-like mechanical behaviors of the ECM are more conducive to spatial pattern emergence.
• To provide a systematic comparison of constitutive models using linear stability analysis and numerical simulations in 1D and 2D settings.
• To advocate for improved empirical characterization of ECM mechanical properties to enhance the fidelity of mechanochemical models.

Proposed method

• Formulates a mechanical model of pattern formation comprising cell density, ECM density, and force-balance equations for the cell-ECM system.
• Applies linear stability analysis to derive dispersion relations for different constitutive models, assessing their pattern-forming potential.
• Employs the Kelvin-Voigt, Maxwell, standard linear solid (SLS), and Jeffrey models to represent the ECM’s stress-strain behavior.
• Uses nondimensionalized 1D and 2D systems with baseline parameters (e.g., ν = 0.25, E′ = 0.8, η = 1, μ = 0.5) to ensure consistency across models.
• Performs numerical simulations in 2D to validate analytical predictions and observe pattern emergence or absence.
• Applies simplifying assumptions (e.g., μ/η = ν′) to derive tractable forms of the stress-strain relations for series-connected models (Maxwell, SLS, Jeffrey).

Experimental results

Research questions

• RQ1 How do different stress-strain constitutive models of the ECM affect the emergence of spatial patterns in biological tissues?
• RQ2 Which mechanical model—solid-like (e.g., Kelvin-Voigt) or fluid-like (e.g., Maxwell)—is more effective at driving pattern formation?
• RQ3 Does the choice of constitutive model significantly alter the stability properties of the system, as revealed by linear stability analysis?
• RQ4 Can numerical simulations confirm the analytical predictions regarding pattern formation potential across different viscoelastic models?
• RQ5 To what extent do the mechanical properties of the ECM, such as viscosity and elasticity, influence morphogenetic processes in tissue development?

Key findings

• The Maxwell and Jeffrey models, representing fluid-like viscoelastic behavior, exhibit significantly higher pattern formation potential than solid-like models.
• Linear stability analysis shows that the dispersion relations for fluid-like models support unstable modes leading to pattern formation, while solid-like models do not.
• Numerical simulations in 2D confirm that spatial patterns emerge when the Maxwell model is used for the ECM, but no patterns form under the Kelvin-Voigt model.
• The Kelvin-Voigt model fails to generate patterns even under identical parameter settings, indicating its insufficiency for capturing morphogenetic mechanisms.
• The results suggest that the rheological nature of the ECM—particularly its viscous, fluid-like response—is essential for driving pattern formation.
• The study underscores the need for more accurate, tissue-specific experimental data on ECM mechanical properties to refine current mechanochemical models.