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Title: Congestion in many-particle systems with volume exclusion constraints : algorithms and applications to modelling in biology
Author: Ferreira, Marina Amado
ISNI:       0000 0004 7427 758X
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2018
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Many-particle systems with congestion are widely found in biology, for example, in cell tissues or herds. Mathematical modelling constitutes an important tool in their study. In contrast to common approaches, we propose two new modelling frameworks that rely on the exact treatment of the contacts between particles: a particle-based and a continuum framework. Both frameworks are based on the same behavioural rules, namely 1) two particles cannot overlap with each other and 2) the particles seek a minimum of a given confining potential at all times. The dynamics is driven by the evolution of the potential and changes in particle characteristics, such as size. In the first part, the static equilibria of the particle-based model are obtained as solutions to a minimization problem. This leads to non-convex optimization under volume exclusion constraints. Classical tools are either not applicable or not efficient. We develop and study a new and efficient minimization algorithm to approximate a solution. The second part concerns the time-evolution of the particle-based framework. We develop new time-stepping schemes involving the resolution of a minimization problem at each time-step, which is tackled with the minimization algorithm developed in the first part. The study of these schemes is performed in the case of a system of hard-spheres undergoing ballistic aggregation on a torus and it succeeds to simulate up to one million particles. These new tools are applied to the study of the mechanics of a cell tissue, which has allowed to validate them in practice. In the third part, we develop a continuum modelling framework describing the evolution of particle density. Our approach differs from previous ones by relying on different modelling assumptions that are more appropriate to biological systems. We show that this novel approach leads to a free-boundary problem and we characterize the dynamics of the boundary.
Supervisor: Degond, Pierre ; Merino-Aceituno, Sara Sponsor: Imperial College London ; Institute of Mathematics and its Applications ; Company of Biologists
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral