The recent use of anti-angiogenesis (AA) drugs for the treatment of glioblastoma multiforme (GBM) has uncovered unusual tumor responses. Here, we derive a new mathematical model that takes into account the ability of proliferative cells to become invasive under hypoxic conditions; model simulations generate the multilayer structure of GBM, namely proliferation, brain invasion, and necrosis. The model is able to replicate and justify the clinical observation of rebound growth when AA therapy is discontinued in some patients. The model is interrogated to derive fundamental insights int cancer biology and on the clinical and biological effects of AA drugs. Invasive cells promote tumor growth, which in the long run exceeds the effects of angiogenesis alone. Furthermore, AA drugs increase the fraction of invasive cells in the tumor, which explain progression by fluid-attenuated inversion recovery (FLAIR) signal and the rebound tumor growth when AA is discontinued.

Abstract. In this article we propose a model to describe the inflammatory process which occurs during ischemic stroke. First, an introduction to some basic concepts about the biological phenomenon is given. Then, a detailed derivation of the model and the numerical scheme used are presented. Finally, the studies of the model robustness and sensitivity are showed and some numerical results on the time and space evolution of the process are presented and discussed.Résumé. Dans cet article, nous nous attachons à modéliser le processus inflammatoire lors d'un accident vasculaire cérébral ischémique. Tout d'abord, nous expliquons brièvement les bases biologiques de ce phénomène. Puis, nous décrivons les équations et les schémas numériques retenus pour le modéliser et l'implémenter. Nous présentons ensuite des études de robustesse et de sensibilité et montrons et discutons les premiers résultats de simulation obtenus.