Mathematics

In recent years, the application of continuum mechanics to the study of biological systems has revealed new perspectives in understanding complex biological processes. Furthermore, the interplay of technological and scientific advancements has offered novel tools for developing mathematical models with potential clinical applications. In this talk, we will explore recent advancements in the mathematical modelling of brain mechanics. Initially, we will examine the active mechanics of axons and their relevance to various neurological disorders, spanning from neurodegenerative diseases to viral infections. Our focus will be on elucidating the auto-regulation of the actin cortex's active contraction under mechanical deformation, critical for maintaining axonal equilibrium. Disruptions in this equilibrium, often induced by pathological conditions, result in the deformation of the axon's natural cylindrical shape and compromise its electrochemical signal transmission. We will establish that this morphological degeneration is primarily attributed to microtubule depolymerization, leading to subsequent mechanical instability. In the latter part of the presentation, we will explore the application of continuum mechanics in modeling the brain as a whole. Specifically, we will spotlight the role of mechanical instability in the formation of cerebral sulci, emphasizing the influence of spatially inhomogeneous brain growth. This growth induces mechanical stress, culminating in the development of the characteristic furrows of the brain. Additionally, we will share preliminary insights into the mathematical modeling of the growth of glioblastoma, an aggressive brain tumor with highly individualized progression patterns. Employing a minimal mathematical model based on mixture theory, we expedite the numerical simulations using model order reduction techniques and neural networks. Machine learning techniques are also used for patient-specific parameter identification through the analysis of tumor distribution at two distinct time points.