Frank Ernesto Alvarez

Diffusion dans des modèles d’agrégation et de cancer: Cette thèse est consacrée à l'étude de plusieurs problèmes issus de la modélisation mathématique des tumeurs. Plus spécifiquement, l’intérêt principal est orienté vers les interactions ayant lieu au sein de la tumeur et avec son environnement. Néanmoins, certains des modèles et méthodes présentés au cœur de la thèse ont une portée bien plus générale que l’étude du cancer. Les principaux résultats sont divisés en cinq chapitres. Dans le premier chapitre, par une nouvelle analyse mathématique comparant la taille des tumeurs entre traitements non pas en fonction du temps, mais en fonction de la taille de la population résistante, nous établissons une comparaison entre les résultats de différentes stratégies de traitement appliquées à une tumeur composée de deux sous-populations, une de cellules sensibles et une autre de cellules résistantes. Dans le deuxième chapitre, nous dérivons l'expression asymptotique d'un cycle limite apparaissant dans un modèle d'interaction tumeur-système immunitaire. Le troisième chapitre est consacré à la modélisation du bet-hedging, une stratégie évolutive d'intérêt pour la théorie atavique du cancer. L'existence et le caractère unique de la solution du modèle sont prouvés et deux phénomènes d'intérêt biologique sont mis en évidence par des simulations. Le chapitre quatre est un complément au troisième chapitre. On y développe une discussion philosophique sur la théorie atavique du cancer et on esquisse deux modèles différents pour l'émergence de la coopération. Le chapitre cinq concerne l'étude d'une méthode particulaire pour un modèle d'advection-réaction-diffusion non local d'une grande importance dans le domaine de les dynamiques adaptatives. La conservation du comportement asymptotique est analysée pour le schéma numérique proposé. Les chapitres six et sept sont consacrés à l'étude du système de Keller-Segel parabolique-parabolique où nous donnons respectivement quelques estimations de la solution et déterminons le comportement asymptotique pour le cas non radial.

We consider the parabolic-parabolic Keller–Segel equation in the plane and prove the nonlinear exponential stability of the self-similar profile in a quasi-parabolic-elliptic regime. We first perform a perturbation argument in order to obtain exponential stability for the semigroup associated to part of the first component of the linearized operator, by exploiting the exponential stability of the linearized operator for the parabolic-elliptic Keller–Segel equation. We finally employ a purely semigroup analysis to prove linear, and then nonlinear, exponential stability of the system in appropriated functional spaces.

We discuss the mathematical modelling of two of the main mechanisms that pushed forward the emergence of multicellularity: phenotype divergence in cell differentiation and between-cell cooperation. In line with the atavistic theory of cancer, this disease being specific of multicellular animals, we set special emphasis on how both mechanisms appear to be reversed, however not totally impaired, rather hijacked, in tumour cell populations. Two settings are considered: the completely innovating, tinkering, situation of the emergence of multicellularity in the evolution of species, which we assume to be constrained by external pressure on the cell populations, and the completely planned—in the body plan—situation of the physiological construction of a developing multicellular animal from the zygote, or of bet hedging in tumours, assumed to be of clonal formation, although the body plan is largely—but not completely—lost in its constituting cells. We show how cancer impacts these two settings and we sketch mathematical models for them. We present here our contribution to the question at stake with a background from biology, from mathematics and from philosophy of science.

Clinical and pre-clinical data suggest that treating some tumors at a mild, patient-specific dose might delay resistance to treatment and increase survival time. A recent mathematical model with sensitive and resistant tumor cells identified conditions under which a treatment aiming at tumor containment rather than eradication is indeed optimal. This model however neglected mutations from sensitive to resistant cells, and assumed that the growth-rate of sensitive cells is non-increasing in the size of the resistant population. The latter is not true in standard models of chemotherapy. This article shows how to dispense with this assumption and allow for mutations from sensitive to resistant cells. This is achieved by a novel mathematical analysis comparing tumor sizes across treatments not as a function of time, but as a function of the resistant population size.

The well-posedness of a non-local advection-selection-mutation problem deriving from adaptive dynamics models is shown for a wide family of initial data. A particle method is then developed, in order to approximate the solution of such problem by a regularised sum of weighted Dirac masses whose characteristics solve a suitably defined ODE system. The convergence of the particle method over any finite interval is shown and an explicit rate of convergence is given. Furthermore, we investigate the asymptotic-preserving properties of the method in large times, providing sufficient conditions for it to hold true as well as examples and counter-examples. Finally, we illustrate the method in two cases taken from the literature.

Confronted with the biological problem of managing plasticity in cell populations, which is in particular responsible for transient and reversible drug resistance in cancer, we propose a rationale consisting of an integro-differential and a reaction-advection-diffusion equation, the properties of which are studied theoretically and numerically. By using a constructive finite volume method, we show the existence and uniqueness of a weak solution and illustrate by numerical approximations and their simulations the capacity of the model to exhibit divergence of traits. This feature may be theoretically interpreted as describing a physiological step towards multicellularity in animal evolution and, closer to present-day clinical challenges in oncology, as a possible representation of bet hedging in cancer cell populations.