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During my PhD thesis, I mainly worked on three projects involving mathematical and bioinformatics methods in immuno-oncology. 1) Project 1: development of the SmartFACS deconvolution tool. I developed a deconvolution tool called SmartFACS in collaboration with the Mechanics and Applied Mathematics Department (ULB) and the Immunobiology Laboratory (ULB). The role of this computer program is to estimate the cellular composition of a sample from RNA sequencing data. This tool has been developed in the context of immunology, where it is often of interest to identify immune cells present in the blood or in tumors. At the time this project began, no deconvolution tool for mouse data was yet available, despite the fact that much biomedical research is carried out in mouse models. During my thesis, I therefore decided to develop a tool adapted to mouse data, while trying to improve existing tools for human data by detecting a wider range of immune populations. 2) Project 2: research of clinical biomarkers for anti-PD1 immunotherapy treatment. In collaboration with the Immunobiology group (ULB), I worked on anti-PD1 therapy, which is already used clinically for certain cancers. Unfortunately, the success rate of this treatment is still relatively moderate. This is why the aim of my project was to determine predictive markers of response to anti-PD1 in order to better understand the mechanisms of resistance and find avenues of improvement for this treatment, as well as to better identify responder patients. To achieve these objectives, a mouse model was developed in the Immunobiology Laboratory. This model enabled us to reproduce the dichotomous response observed in the clinic. My work focused on analyzing pre-treatment tumor sequencing data obtained using the mouse model, in order to compare the immune infiltrate of responder and non-responder mice. My data analyses, together with experiments conducted by Jelena Gabrilo, revealed two immune populations important for treatmen, Doctorat en Sciences, info:eu-repo/semantics/nonPublished

We consider an infinite fluid queue modulated by a Markovian environment which varies slowly between two different modes of behaviour. We derive the series expansion of the stationary density of its buffer content as a function of the aggregated stationary densities of the buffer contents of the two environmental fluid queues., SCOPUS: cp.p, info:eu-repo/semantics/published

We consider the Poisson equation (I -P)u = g, where P is the transition matrix of a quasi-birth-and-death process with infinitely many levels, g is a given infinite dimensional vector, and u is the unknown. Our main result is to provide the general solution of this equation. To this purpose we use the block tridiagonal and block Toeplitz structure of the matrix P to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples., SCOPUS: ar.j, info:eu-repo/semantics/published

info:eu-repo/semantics/published

The Erlangian approximation of Markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block-triangular, block-Toeplitz structure. To this end, we propose some algorithms which exploit the Toeplitz structure and the properties of generators. Such algorithms allow us to compute the exponential of very large matrices, which would otherwise be untreatable with standard methods. We also prove interesting decay properties of the exponential of a generator having a block-triangular, block-Toeplitz structure., info:eu-repo/semantics/published

A Markov process is defined by its transition matrix. A skip-free Markov process is a stochastic system defined by a level that can only change by one unit either upwards or downwards. A regular perturbation is defined as a modification of one or more parameters that is small enough not to change qualitatively the model.This thesis focuses on a category of methods, called matrix analytic methods, that has gained much interest because of good computational properties for the analysis of a large family of stochastic processes. Those methods are used in this work in order i) to analyze the effect of regular perturbations of the transition matrix on the stationary distribution of skip-free Markov processes; ii) to determine transient distributions of skip-free Markov processes by performing regular perturbations.In the class of skip-free Markov processes, we focus in particular on quasi-birth-and-death (QBD) processes and Markov modulated fluid models.We first determine the first order derivative of the stationary distribution - a key vector in Markov models - of a QBD for which we slightly perturb the transition matrix. This leads us to the study of Poisson equations that we analyze for finite and infinite QBDs. The infinite case has to be treated with more caution therefore, we first analyze it using probabilistic arguments based on a decomposition through first passage times to lower levels. Then, we use general algebraic arguments and use the repetitive block structure of the transition matrix to obtain all the solutions of the equation. The solutions of the Poisson equation need a generalized inverse called the deviation matrix. We develop a recursive formula for the computation of this matrix for the finite case and we derive an explicit expression for the elements of this matrix for the infinite case.Then, we analyze the first order derivative of the stationary distribution of a Markov modulated fluid model. This leads to the analysis of the mat, Doctorat en Sciences, info:eu-repo/semantics/nonPublished