Emma Leschiera, Mathematical modelling of the CD8+T cell immune response to heterogeneous tumours
The number of sub-populations generating a tumour, as well as the immunogenicity of tumour cells are two major components of intra-tumour heterogeneity (ITH) and play a key role in the immune response against solid tumours. Mathematical models allow to evaluate the influence of these two effects on anti-tumour immunity in a controlled manner. Separating these two components, we can investigate their effects on tumour aggressiveness independently or together.
In this talk, we present a spatially explicit stochastic individual-based model capturing the interactions between tumour cells and CD8+ T cells. Considering different initial compositions of the tumour, we investigate how ITH affects the anti-tumour immune response. Tumour cells are characterized by different antigen profiles and a level of antigen presentation. In our model, ITH can vary with the number of tumour antigens (i.e. the number of sub-populations of tumour cells) and with the level of antigen presentation (i.e. the immunogenicity of tumour cells).
Computational simulations show that both components play a role in the anti-tumour immune response. In addition, our results reproduce qualitatively biological experiments.
Jules Guilberteau, Long-time behavior of a phenotype-structured PDE with advection and non-local growth
Cell differentiation processes, such as epithelial-mesenchymal transition (EMT), hematopoietic stem cells or embryonic stem cells can be modeled by ODE systems which describe the behavior of several molecules in interaction which determine the phenotype adopted by a cell. A natural way of modeling a cell colony undergoing cell differentiation is thus to consider a PDE model with an advection term related to the associated ODE, and then to add other terms which describe phenomena related to cell-cell interactions. In this talk, we focus on such advection PDEs to which we add a non-local growth term which models cell proliferation. We prove that the long-time behavior of this PDE leads to concentration phenomena closely dependent on the equilibria of the associated ODE and to the values of the growth function. These results will be compared to those of the PDEs with advection only and with growth only, in order to understand the role of each of these terms.
Mathieu Mezache, A bi-monomeric polymerisation/depolymerisation model to capture the oscillatory kinetics in prion dynamics
Prions are associated with neurodegenerative disorders such as Alzheimer, Parkinson and Huntington diseases. During the evolution of prion pathology, the monomeric prion protein (PrPC) is converted into misfolded aggregating conformers (PrPSc). PrPSc assemblies have the ability to self-replicate and self-organise in the brain through a still unresolved molecular mechanism.
We introduce a new polymerisation/depolymerisation model capable of explaining oscillations, which have been observed experimentally in the time-course of prion protein polymerisation experiments. First, we study the asymptotic behavior of the model in a discrete size setting. For a reduced system, we prove the exponential convergence toward a pathological equilibrium despite its oscillatory behavior. As for the complete system, we prove its wellposedness and study the linear stability of disease-free equilibria.
Furthermore, we are also interested in the bi-monomeric model in a continuous size setting. It gives rise to a Lifschitz-Slyosov-type equation coupled to two ODE. Depending on conditions on the polymerization and depolymerisation rates, we prove that the solution is either periodic or converges exponentially fast to a Dirac mass.