Shimi Chettiparambil Mohanan

• This thesis explores the mathematical modelling of tissue dynamics, focusing on processes such as tissue regeneration and degradation, influenced by cell migration and proliferation. The first chapter examines meniscus cartilage regeneration. Utilising kinetic transport equations, we apply an upscaling technique to the lower-scale dynamics of stem cells and chondrocytes, deriving a macroscopic system of reaction-diffusion-(taxis) equations (RD(T)Es) coupled with an ordinary differential equation (ODE) for hyaluron. Analysis of pattern formation in a simplified version of this system highlights the critical role of the taxis coefficient. Numerical simulations complement this analysis, demonstrating pattern formation and system stability for varying taxis coefficient values. The second chapter investigates wound formation and spread in Buruli ulcer, caused by bacteria named Mycobacterium ulcerans. Starting from kinetic transport equations (KTEs) and using parabolic scaling, we derive a macroscopic reaction-diffusion-taxis equation for bacteria, incorporating multiple taxis. An RDTE-RDE-ODE system is developed to model interactions among bacteria, the toxin mycolactone, normal tissue, and necrotic matter. We provide an analysis proving the existence of classical solution. Numerical simulations explore five scenarios with different taxis sensitivity parameters and initial conditions, revealing distinct dynamics for each solution component. The final chapter incorporates microscopic-level bacterial dynamics through a random jump approach, resulting in a RDTE for bacteria. We analyse an RDTE-RDE-ODE system modelling bacteria, toxin, normal tissue, and necrotic matter, demonstrating solution existence in classical sense. Simulations offer deeper insights into the underlying biological processes. We conclude with a numerical simulation that highlights the differences in dynamics between the system derived from the random jump approach and the model based on kinetic transport equations. In sum, this thesis contributes to developing and analysing multiscale mathematical models that enhance our understanding of tissue dynamics, with possible applications to tissue regeneration and wound formation.