Phase-Field Based Simulations of Embryonic Branching Morphogenesis

Abstract

While the study of the embryonic organ development is an active research field, many of the fundamental questions regarding the emergence of shape and growth control remain unanswered. Mathematical modelling stands as a powerful tool to help investigate morphogenesis dynamics during embryonic development. Out of all the proposed mechanisms, receptor-ligand Turing Pattern frameworks are the only models that can accurately match branching events. Turing Patterns are spatio-temporal stable formulations that emerge from a system of diffusing morphogens such as proteins, which interact in a non-linear manner and give rise to patterns. Our group has previously used the Arbitrary Lagrangian-Eulerian (ALE) framework for solving the system of PDEs that describes our receptor-ligand Turing model. However, complex mesh deformations during lung growth limits the number of branch generations that can be simulated. To circumvent this, we previously proposed a Phase-Field approach that avoids mesh deformations by labelling the surface of the modelling domains as interfaces between phases, and by coupling the reaction-diffusion framework to said surfaces. In this thesis, we report a rigorous comparison between the Phase-Field approach and the ALE-based simulation. The Phase- Field simulation can reproduce the Turing Patterns predicted by the ALE approach with a relative error below 2% on the implicit surface representation. Furthermore, we propose an extended model that includes a second independent phase-field to represent the outer mesenchymal layer and restrict the production of the ligand to this outer surface. The resulting 3D growth simulations are numerically much more stable than the ALE-based simulations and handle a range of deformations with ease. In spite of this, Phase-Field simulations require a high resolution of the narrow surface, which leads to a considerable larger linear system to solve for. Consequently, to reduce the computational burden of our memory-bounded time-dependent problem, the reported scaling results on different mesh sizes required the tuning of our cluster environment and the use of the iterative solver GMRES with an algebraic multigrid-preconditioner.