Stochastic individual-based models of adaptive dynamics and applications to cancer immunotherapy

Abstract

In this thesis stochastic individual-based models describing Darwinian evolution of asexual, competitive populations are studied. A specialization of these models is developed to describe tumor development under immunotherapy and an arising extended model is analyzed mathematically. In the first part (Chapter II) we consider a population with a large but non-constant population size characterized by a natural birth rate, a logistic death rate modeling competition, and a probability of mutation at each birth event. In this individual-based model the population state at a fixed time is given as a measure on the space of phenotypes and the evolution of the population is described by a continuous time, measure-valued Markov process. We investigate the long-term behavior of the system in the limits of large population size (K → ∞), rare mutations (u → 0), and small mutational effects (σ → 0), proving convergence to the canonical equation of adaptive dynamics. This limit equation is an ODE that describes the evolution in time of the phenotypic value in a population consisting essentially of one single phenotype. The main difficulty is that we take the three limits simultaneously, i.e. u = u_K and σ = σ_K, tend to zero with K, subject to conditions that ensure that the time scale of birth and death events remains separated from that of successful mutational events. This slows down the dynamics of the microscopic system and leads to serious technical difficulties that require the use of completely different methods than in comparable works where the limits are taken separately. More precisely, the time until a mutant phenotype fixates is diverging (in K) and thus, we cannot use the law of large numbers to approximate the stochastic system. In the second part (Chapter III) we propose an extension of the individual-based model, which broadens the range of biological applications. The primary motivation was to model cancer immunotherapy in order to simulate and describe qualitative the experiments reported in Landsberg et al. [92], where tumors resist immunotherapy through inflammation-induced reversible dedifferentiation. The main expansions are that we have three different actors in this context (T-cells, cytokines, and cancer cells), that we distinguish cancer cells by phenotype and genotype, that we include environment-dependent phenotypic plasticity, and that we take into account the therapy effects. With this new setup we are able to model various phenomena arising in immunotherapy. We argue why stochastic models may help to understand the resistance of tumors to therapeutic approaches and may have non-trivial consequences on tumor treatment protocols. Furthermore, we show that the interplay of genetic mutations and phenotypic switches on different time scales as well as the occurrence of metastability phenomena raise new mathematical challenges. The present thesis focuses more on these aspects. More precisely, we study the behavior of the individual-based model which includes phenotypic plasticity on a large (evolutionary) time scale and in the simultaneous limits of large populations (K → ∞) and rare mutations (u_K → 0), proving convergence to a Markov jump process, which is a generalization of the usual polymorphic evolution sequence. This can be seen as an extension of the results by Champagnat and Méléard (cf. [25, 30]).