Asymptotic analysis of reaction-diffusion and selection-mutation PDEs
I have focused on the asymptotic analysis of some non-local PDEs which represent individuals that proliferate and die, and can either move randomly (when the structuring variable is the space variable), or undergo (epi-)mutations (when the structuring variable is a trait or a phenotype, such as the size of the cell). There are thus two terms: a reaction term and a diffusion term.
In the phenotype case, and when the mutation term is neglected, Emmanuel Trélat and I have obtained several results thanks to Lyapunov functionals inspired by that developed by Pierre-Emmanuel Jabin and Gaël Raoul. These techniques can be applied to systems and provide a general setting to understand selection of certain phenotypes: the asymptotic limit is a sum of Dirac masses on certain phenotypes, see the following preprint.
The fitness function r(x) is such that the Dirac masses are concentrated on the phenotypes that reach its maximum. When mutations are neglected, however, one does not know which phenotype will take over: it depends on the initial condition. With Tommaso Lorenzi, we are currently investigating how to remove this problem, thanks to a vanishing viscosity method.
The aforementioned Lyapunov functionals also yielded a result in the space case, more particularly for the non-local Fisher-KPP equation. Here is a note on the subject.
Below is a video of a presentation I made at IMPA (Rio de Janeiro, Brazil) on this subject, during the conference Mathematical methods and modeling of biophysical phenomena, in December 2017.
Optimal control, application to cancer therapy
There is no obvious strategy to efficiently use chemotherapy to treat a given cancer. This is due to
1) the unwanted toxicity to the healthy tissue.
2) the emergence of drug resistance: the tumour size might decrease initially in size, but regrowth is often inevitable because the resistant cells have been selected.
With Jean Clairambault, Alexander Lorz and Emmanuel Trélat, we have investigated a 2x2 system (selection PDE, mutations are neglected) for healthy and cancer cells endowed with a resistance phenotype. Based on some Lyapunov functionals (see previous section), we recover that under high constant doses, the tumour cells concentrate on a resistant phenotype.
We thus investigate the optimal control problem consisting in minimising the tumour burden at the end of a given time-window, and the optimal strategy is clear, both numerically and theoretically, when the final time becomes large: the best strategy is to use low doses for as long as possible and then to give the maximum tolerated doses. The cancer cells indeed concentrate on a sensitive phenotype at the end of the first phase and the chemotherapy can be very efficient.
These results have been published in the Journal de Mathématiques Pures et Appliquées, see the pdf file.
The extension of these results to a model accounting for small mutations has been numerically challenging. With Antoine Olivier, we designed a general numerical approach for optimal control problems, based on homotopies, direct methods, and the application of a Pontryagin Maximum Principle (PMP), see the following paper published in Journal of Optimization Theory and Applications.
The underlying principle is to simplify the problem up to a point where it can theoretically be solved by a PMP. The simplified problems and the initial harder one are linked by a parametrised family of optimal control problems. Numerically, these are discretised and their numerical counterpart are large optimisation problems. The homotopy then consists in solving each optimisation problem is solved by taking the solution of the previous problem as an initial guess, starting from the simple one all the way to the hard one.
Patterns in mathematical biology
This project is based on experiments performed in the Laboratoire de Biologie et Thérapeutique des Cancers, by Michèle Sabbah and Nathalie Ferrand. Cancer cells are put in a 3D matrigel mimicking the extracellular-matrix, namely the real conditions encountered by cells in vivo. In the experiments, cells tend to organise as spheroids, and the goal is to explain this, finding the key biological phenomena responsible for it.
Since it is expected that chemotaxis and random movement are involved, we have been trying to find minimal Keller-Segel type models (possibly with nonlinear sensitivity) to recover the patterns, in a work with Luis Almeida, Federica Bubba and Benoît Perthame. These models indeed exhibit Turing instability, see the following preprint.
From the mathematical point of view, there are some challenges for the discretisation of these equations to preserve energy dissipation, positivity and other relevant properties at the discrete level. A paper is in preparation on this subject.
For the application, we are trying to identify the key parameters responsible for different types of patterns, and also to compare quantitatively the results with those of the experiments: number of spheroids, size distribution, etc. See below an example of a 2D simulation.
Vidéos
Control for population dynamics
Together with Emmanuel Trélat and Enrique Zuazua, we are investigating the control of some classical PDEs for population dynamics: the monostable and bistable equations. Is is possible to control towards extinction or dominance of the species? We are trying to tackle these questions with different strategies coming from Control Theory, in particular through the dual observability inequality or by appropriate paths of steady-states.