The mechanics of biological growth: A study through the vertex model

This doctoral thesis investigates the mechanics of growth and remodeling in biological tissues through a discrete framework known as the vertex model. While this model has been extensively used for numerical simulations in biological contexts, the present work focuses on a fundamental aspect: the generation of elastic stresses in vertex-based systems. This study introduces the concept of incompatibility in such tissues, a well-established source of residual stresses in continuous mechanics in the absence of external loads. A key contribution of this thesis is the identification and characterization of two distinct types of incompatibility in the vertex model. The first, termed internal incompatibility, arises when the target area and perimeter of individual cells violate the isometric inequality. Internal incompatibility is recognized as a regulator of the critical transition between fluid-like and solid-like cell behavior, which plays a crucial role in processes such as cancer cell migration. The second, termed external incompatibility, pertains to the manner in which cells are interconnected to form specific tissue morphologies. Both types of incompatibility act as sources of residual stresses in tissues described through the vertex model. The second part of this work explores the consequences of elastic stress accumulation on the possibility of inelastic tissue evolution. Specifically, the study examines phase transitions, including T1, T2, and T3 transitions, as well as cell division, analyzing how such non-elastic processes enable the system to evolve toward a minimal energy configuration. This evolution represents a potential pathway for growth in biological tissues. The analysis, implemented using a MATLAB code, further investigates the influence of parameters such as area stiffness, perimeter stiffness, and line tension on growth progression, to single out the conditions leading to the instabilization of initially flat or regular surfaces. Preliminary results suggest that incompatibilities and stiffness parameters significantly contribute to the onset of corrugations at the interface between domains with differing control parameters. These findings open the door to explore connections with tumor growth, particularly in the context of metastasis spreading into healthy tissues.

Publisher

University of Galway

Rights

CC BY-NC-ND

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University of Galway Theses (PhD Theses)