Amira Meddah,
"Stochastic hybrid dynamical systems for simulating low-grade Glioma evolution"
, 10-2024
Original Titel:
Stochastic hybrid dynamical systems for simulating low-grade Glioma evolution
Sprache des Titels:
Englisch
Original Kurzfassung:
Gliomas stand out as the most common and aggressive type of brain tumours, characterised by their rapid cell growth and significant ability to penetrate surrounding brain tissue. Despite advancements in medical treatments, these tumours often show a resilience to therapies, leading to a poor prognosis with high rates of tumour recurrence being a principal contributor to mortality. The mechanisms behind glioma progression remain poorly understood, particularly in terms of tumor migration and invasion.
In this thesis, we present a comprehensive study of glioma progression through several mathematical models that capture both microscopic and macroscopic scales. A significant part of this work is dedicated to the introduction of piecewise diffusion Markov processes (PDifMPs), a type of generalised stochastic hybrid system (GSHS), which provides a valuable resource for future research and establishes the basic understanding of PDifMPs, facilitating their application in subsequent studies.
As a first step in these studies, we propose a hierarchical stochastic model using PDifMPs for a detailed representation of glioma evolution. In particular, this model combines a stochastic model of cell motility with a deterministic model of migration in response to environmental cues. We then used the properties of PDifMP to derive a macroscopic equation for overall tumour mass evolution that captures the impact of micro-environmental features on tumour progression. In particular, the model illustrates how changes in microscopic parameters such as the jump rate function and diffusion coefficient are related to glioma spreading behaviour and the critical transition from low-grade to high-grade glioma.
Another significant part of this work focuses on the challenges of numerical approximation of PDifMPs, particularly in scenarios where explicit flow maps are not available. This work investigates both mean square convergence and weak convergence for the proposed approximation scheme, establishing weak convergence by means of a Martingale problem formulation. As an application, we employ these results to simulate the migration patterns exhibited by moving glioma cells at the microscopic level.
This thesis highlights the important role of a mathematically rigorous approach in the analysis of glioma progression. Through detailed modelling of the stochastic and deterministic behaviour of glioma cells, it provides in-depth insights into tumour growth and progression to malignancy. It paves the way for future research aimed at refining therapeutic strategies and improving our understanding of glioma progression using the hierarchical model developed. In addition, it lays the foundation for further development of numerical methods for PDifMP.
Forschungs-