We propose a PDE system intended to describe the earliest stages of the interactions between immune cells and tumor growth. The model is structured in size and space, and it takes into account the migration of the tumor antigen-specific cytotoxic effector cells towards the tumor micro-environment by a chemotactic mechanism. Remarkably, the model exhibits a possible control of the tumor growth by the immune response; nevertheless, the control is not complete in the sense that the asymptotic equilibrium states keep residual tumors and activated immune cells. We will discuss the mathematical modeling, numerical investigation and a few results on the analysis of the system.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

We introduce and study a new model for the progression of Alzheimer's disease incorporating the interactions of A_beta-monomers, oligomers, microglial cells and interleukins with neurons through different mechanisms such as protein polymerization, inflammation processes and neural stress reactions. In order to understand the complete interactions between these elements, we study a spatially-homogeneous simplified model that allows to determine the effect of key parameters such as degradation rates in the asymptotic behavior of the system and the stability of equilibriums. We observe that inflammation appears to be a crucial factor in the initiation and progression of Alzheimer's disease through a phenomenon of hysteresis, which means that there exists a critical threshold of initial concentration of interleukins that determines if the disease persists or not in the long term. These results give perspectives on possible anti-inflammatory treatments that could be applied to mitigate the progression of Alzheimer's disease. We also present numerical simulations that allow to observe the effect of initial inflammation and spatial dependence.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

We investigate the mean-field limits of large-scale networks of interacting biological neurons, represented by the so-called integrate-and-fire models. However, we do not assume any prior structure on the network but consider instead any connection weights that obey certain types of mean-field scaling. When the networks are dense, we are able to achieve a limit that resembles the widely recognized form of mean-field limit, through a graphon limit that tracks the role of individual neurons in the network. When the networks are potentially sparse, mathematically interpreting the role of individual neurons becomes increasingly difficult. Instead, we introduce novel statistical notions that directly describe the large-scale dynamics of networks. This is a joint work with P.-E. Jabin and V. Schmutz.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

Volume exclusion interactions play a key role in many biological systems. In particular, it seems to be the key to explaining spontaneous alignment of anisotropic particles, for example, alignment of myxobacteria or fibers in a network. Most individual-based models impose this type of alignment in their equations. Here we do not wish to impose this type of alignment, but to investigate how it might emerge from volume exclusion interactions. To carry this out, volume exclusion interactions will be modelled via a soft anisotropic repulsion potential (which are vastly used in the literature of liquid crystals). We will present an individual-based model based on this potential and derive the corresponding kinetic and macroscopic equations. This approach allows us to understand how alignment emerges from volume exclusion and how it also affects not only the orientation of the particles, but also their positions.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

Cell migration is a complex biological phenomenon playing an important role in many processes such as embryogenesis, but also in the development of pathologies such as cancer. The main driver of the motility is the actin network, the dynamics of which is regulated by many proteins. Mathematical models have been developped in the last decades to better understand this complexity. One of the difficulty leads on the representation of this moving domain. Several approaches have been proposed: Lagrangian Markers Cells [Edelstein-Kechet et al.(2011)], Level-set methods [Tesson et al.(2020)] or phase fields models [Ziebert-Aronson (2011)]. We propose in this talk to illustrate these methods with two biological issues. In the first one, we will model the impact of the microtubules on the process using the level set method. In the second one, we will use phase field models to explain atypical cases of adhesive haptotaxis [Luo et al (2020)], [Seveau Phd (2022)].

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

The study of collective dynamics is attracting the interest of different research fields, both due to their wide range of applications and to their ability to model self-organization. The emergence of global patterns from local interactions can be easily observed in flock of birds, schools of fish, human crowds, but also cells exhibit collective behaviors in different biological processes characterizing the human body (e.g. in embryogenesis, wound healing, immune response, tumor growth). The main feature of collective cells migration is that the emergent behavior is also driven by chemical stimuli, and not only by mechanical interactions. In this talk I will present a general class of hybrid ODE-PDE models, gathering the advantages of multiscale descriptions. In this context, cells are modeled as discrete entities and their dynamics is given described by a system of second-order ODEs, while the chemical signal influencing the motion is modelled as a continuous signal solving a diffusive equation. The particular coupling of the two scales raises some issues that have been analytically investigated over the last years. Concerning applications, I will present recent advancements on a hybrid mathematical model inspired by Cancer-on-chip experiments, where tumor cells are treated with chemotherapy drug and secrete chemical signals in the environment, thus stimulating immune response.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

The adoption of hybrid models for self-organization dynamics allows to provide an accurate description of cell motion in tissues or organs. From a numerical point of view, the proposed approach may have drawbacks in terms of computational cost, if the number of cells involved significantly increases. In this perspective, the idea is to introduce a fully macroscopic mathematical model, in which cells are treated as a continuous cellular density. Starting from a class of hybrid ODE-PDE models, a new pressureless nonlocal Euler-type model with chemotaxis has been rigorously derived in [1], under the assumption of monokinetic initial data. Outside the monokinetic case, a numerical study has been performed in the one-dimensional case [2]. In this talk I will present some advancements in the numerical approximation of this model in the multidimensional case, to understand the role of different effects in the dynamics. Finally, parameter estimation of the macroscopic model is performed, in order to find the optimal parameters and to provide realistic numerical simulations. I will show different scenarios, comparing the nonlocal Euler-type model with chemotaxis models existing in the literature. This talk is based on an ongoing work with Marta Menci and Roberto Natalini.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

In this talk, I will review some questions that arise around the classical Vicsek model - which is a model for collective dynamics where agents move at a constant speed while trying to adopt the averaged orientation of their neighbours, up to some noise. I will discuss the emergence of bifurcations leading to disordered and ordered motion, depending on the local density of the agents. This is a very interesting phenomenon: it showcases how two completely different observed behaviours can appear simultaneously from agents that interact following the same rules.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

Microbial communities form complex ecosystems that provide beneficial services to humans in a variety of contexts, such as food fermentation, crop protection, bioprocessing or health and well-being. The complexity of microbial interactions makes it difficult to decipher the drivers of community dynamics and functions. Building digital twins of microbial communities could provide insights into their functioning, and strategies for improving the services they provide. In this talk, I will present genome-based models of microbial communities that predict functions and dynamics, and thus represent good candidates for digital twins. However, they induce a high numerical load, especially when coupled with PDE models of microbial populations. A surrogate modeling strategy will be used to provide fast approximations of the genome-based model, in order to overcome this difficulty.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

In this talk I am going to consider a Boltzmann-type equation for the description of a collision dynamic which is not instantaneous. This new class of kinetic equations has been introduced by Kanzler, Schmeiser, and Tora to model ensembles of living agents, where the changes of state are the result of complicated internal processes, and not simple mechanical interactions. We extend their work introducing a first-order approximation to the instantaneous equation, where non-binary collisions are included. This is motivated by the fact that during an extended collision period there is a positive probability that a colliding pair is joined by additional particles. The interaction kernel is of alignment type, where the states of the particles approach each other. For this spatially homogeneous approximation, we check that the formal properties of the system are kept. Furthermore, existence and uniqueness of solutions and instantaneous limit are examined.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

Despite significant advances in the in silico modeling of human physiology, understanding the complex behavior of fluids in the eye and identifying the main factors that influence their dynamics is still a very challenging field. On the one hand, the description requires a multi-scale characterization, since these phenomena encompass a wide range of spatial and temporal scales, from the molecular level to networks of a few meters, between a one-second heartbeat and a lifetime. On the other hand, the fluid dynamics is influenced by the interaction with surrounding tissues and their temperature, which calls for a multi-physics approach. In addition, the geometric representation can be very complex and the availability of real data is scarce. In this challenging context, the aim of this talk is to present our continuous efforts from a modeling and numerical viewpoint to develop a powerful and flexible mathematical and computational framework called the Ocular Mathematical Virtual Simulator. The combined effects of ocular blood flow and different ocular tissues are described by a coupled hemodynamics and biomechanics model. The multi-scale aspect, essential to properly account for systemic effects of the blood circulation coupled with local effects on the tissues of interest, is represented by a coupled partial and ordinary differential equations for fluid flow. The PDE/ODE coupling is handled via (i) operator splitting for the time discretization, which provides modularity of the solution algorithm while preserving the physical energy at the discrete level; and (ii) Hybridizable Discontinuous Galerkin (HDG) method for the PDE discretization, which ensures conservation of fluxes of mass and linear momentum at the discrete level. A special interest is devoted to the issues of verification, validation and treatment of inherent uncertainties. Finally, we discuss some specific applications related to glaucoma, a leading cause of irreversible blindness worldwide, that currently lacks cure and for which existing treatments focus on managing the condition and slowing its progression.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

The pancreatic innervation undergoes dynamic remodeling during the development of pancreatic ductal adenocarcinoma (PDAC). Denervation experiments have shown that different types of axons can exert either pro- or anti-tumor effects, but conflicting results exist in the literature, leaving the overall influence of the nervous system on PDAC incompletely understood. To address this gap, we propose a continuous mathematical model of nerve-tumor interactions that allows in silico simulation of denervation at different phases of tumor development. This model takes into account the pro- or anti-tumor properties of different types of axons (sympathetic or sensory) and their distinct remodeling dynamics during PDAC development. We observe a “shift effect” where an initial pro-tumor effect of sympathetic axon denervation is later outweighed by the anti-tumor effect of sensory axon denervation, leading to a transition from an overall protective to a deleterious role of the nervous system on PDAC tumorigenesis. Our model also highlights the importance of the impact of sympathetic axon remodeling dynamics on tumor progression. These findings may guide strategies targeting the nervous system to improve PDAC treatment.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

In this talk, I’ll be present some recent advancement in the modelling of the size dynamics of adipose cells. Adipose cells or adipocytes are the specialized cells composing the adipose tissue in a variety of species. Their role is the storage of energy in the form of a lipid droplet inside their membrane. Based on the amount of lipid they contain, one can consider the distribution of adipocyte per amount of lipid and observe a peculiar feature : the resulting distribution is bimodal, thus having two local maxima. The aim of this talk is to introduce a model built from the Lifshitz-Slyozov equations that is able to replicate this bimodale feature. I also introduce a microscopic scale model build from the Becker-Döring equations and show a new convergence result toward the Lifshitz-Slyozov-inspired model, which provides a rate of convergence. I will also present some extension to stochastic models, which support some extension of the deterministic model to better approximate data. Regarding the data, I’ll present some parameter estimation on measures from rats.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

Myxococcus xanthus, a social bacterium, exhibits intriguing collective behavior, characterized by coordinated group movement and the ability of each bacterium to change its movement direction by reversing its body axis. This behavior results in the formation of interesting patterns, such as rippling, where cells self-organize into colliding counter-propagating waves, and swarming, where cells align and move together in large groups. The complex nature of this behavior has captured the attention of biologists, physicists, and mathematicians, driving extensive research efforts. This talk has two main goals. First, we present new biological data on Myxococcus xanthus, featuring high-resolution movies of their collective movements. Using advanced algorithms, we segment these movies, track cell trajectories, and analyze reversals. From these observations, we propose a kinetic model explaining the emergence of rippling patterns. Second, we develop a 2D agent-based model where bacterial reversals are closely linked to congestion, a hypothesis confirmed by our data. This model accurately replicates the rippling and swarming dynamics and highlights the crucial role of background anisotropy in the formation and persistence of these patterns. It also shows that the emergence of both rippling and swarming can be explained by the same rules at the individual level. This project is in collaboration with Vincent Calvez (Laboratoire de Mathématiques de Bretagne Atlantique), Tâm Mignot (Laboratoire de Chimie Bactérienne - Marseille) and Jean-Baptiste Saulnier (Laboratoire de Chimie Bactérienne).

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)

The analysis of equations arising in neuroscience raises many challenging questions that always require the development of new tools to answer them. In this presentation we will illustrate two exponential convergence results obtained using the two different techniques of Relative Entropy with Poincaré-type inequality on the one hand and Harris theory on the other. In particular, the presentation will be motivated by the case study of the Integrate and Fire model with random discharge used in mathematical neuroscience to describe the spiking activity of neurons [1, 2]. This study will be an opportunity to highlight some peculiar differences between the two techniques mentioned above [3]. The results we present are an ongoing collaboration with Professor J. A. Canizo and Professor M. J. Caceres from University of Granada, Professor D. Salort from Sorbonne University and Doctor A. Lora-Ramos from University of Granada.

• Thematic program: Mathematics for Biology and Medicine (2024/2025)
• Event: Workshop "Mathematical modeling in life and health sciences" (2024)