Research Interests

My research addresses several Interconnected Challenges in Mathematical Biology, focusing on Cancer Growth and Metastasis, Cell Migration, Evolutionary Dynamics, and their rigorous study through problem-specific Numerical Methods and Analytical Techniques. The sections below outline my research interests in each one of this directions:

Cancer Growth and Metastasis

This part of my research research focuses on the mathematical modelling, analysis, and numerical simulations of cancer growth, invasion, and metastatic spread across multiple organs.

We develop genuinely hybrid and multiscale models that integrate stochastic and deterministic frameworks to describe the invasion and dissemination of cancer cells across multiple organs. These models capture both individual and collective cell migration, accounting for biomechanical and biochemical interactions within the tumour microenvironment. They incorporate key biological processes such as tissue-specific tropism and heterotypic cell-cell interactions. Additionally, these models describe the full metastatic cascade, including intravasation, circulation, extravasation, and organ-specific metastasis. Our approach aims to bridge mathematical modelling with clinically relevant tumour progression dynamics and uncover potential personalised treatment strategies. 

From a numerical perspective, we have developed and implemented high-resolution Implicit-Explicit Finite Volume methods on both Uniform and Adaptively Refined Meshes (h-refinement) to enhance computational efficiency. These numerical schemes enable large-scale simulations of metastatic invasion and the formation of secondary foci across different tissue environments. From an analytical standpoint, we have established results on the existence, uniqueness, and boundedness of classical solutions, ensuring the mathematical well-posedness of these models.

Cytoskeleton Dynamics & Live Cell Motility 

This research explores the mathematical modelling, analysis, and numerical simulation of actin-based motility in living cells, with a particular focus on cancer cell migration. The actin cytoskeleton is a key driver of cell motility, enabling cells to navigate complex microenvironments during invasion and metastasis.

At the modelling level, we develop Filament-Based Lamellipodium Models (FBLM) derived from energy minimisation principles and variational techniques. These models extend to hundreds of interacting cells, capturing large-scale collective motility dynamics and emergent behaviours observed in multicellular migration. These models describe the mechanical properties of actin filaments, treating the lamellipodium as a two-phase, fourth-order parabolic delay system. They account for interactions with the extracellular matrix and the impact of chemotactic and haptotactic cues on cell migration.

From a numerical perspective, we have designed and implemented problem-specific Finite Element Methods (FEM) tailored for FBLM simulations. These numerical schemes enable the reconstruction of realistic experimental scenarios, allowing for the simulation of various cell morphologies in different environmental conditions, including non-uniform surfaces and multi-signal chemical gradients. From an analytical perspective, we investigate stability, convergence, and sensitivity properties of these models, ensuring their robustness for predicting cell motility dynamics.

We are able to reproduce biologically realistic experimental scenarios of various shapes of moving cells over non-uniform surfaces and under the influence of multiple chemical signals. As such, this work provides a mathematical foundation for understanding cancer cell migration and informs computational approaches for testing mechanistic hypotheses in cellular motility research.

Adaptive Mesh Reconstruction/Refinement Methods 

This work includes the development of r-, h-, and hr-refinement techniques for a wide range of problems, spanning from Conservation Laws and Hamilton-Jacobi equations to Euler systems and Advection-Reaction-Diffusion models arising in Mathematical Biology. In particular, we have devised, developed, and implemented an hr-refinement for which all the necessary stages of the h-, r-, and hr-refinements are integrated.

In addition to developing these methods, we place special emphasis on the numerical analysis of the r-refinement method, particularly its stabilisation properties. We have derived a tool for the analysis of r-refinement, incorporating both the reconstruction of the mesh and the time evolution of the numerical solution.

Adaptive Dynamics and Evolution of Dinosaur birds

We conduct multi-scale modelling and numerical simulations to explore the atomistic description of per capita energy intake and investment versus the evolution of body mass. Additionally, we characterise population dynamics in relation to the evolution of communities over endothermicity and size axes. The corresponding macroscopic models, assuming small mutations, lead to parabolic Lotka-Volterra equations, which we numerically simulate using high-order, robust FV methods with AMR techniques.