Bull Math Biol (2014) 76:2306–2333 DOI 10.1007/s11538-014-0007-y ORIGINAL ARTICLE A Multilayer Grow-or-Go Model for GBM: Effects of Invasive Cells and Anti-Angiogenesis on Growth Olivier Saut · Jean-Baptiste Lagaert · Thierry Colin · Hassan M. Fathallah-Shaykh Received: 16 April 2013 / Accepted: 25 July 2014 / Published online: 23 August 2014 © Society for Mathematical Biology 2014 Abstract The recent use of anti-angiogenesis (AA) drugs for the treatment of glioblas- toma 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 multi- layer 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 funda- mental 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. Keywords Mathematical modeling · Glioblastoma · Partial differential equation · Brain tumors · Angiogenesis · Malignant progression O. Saut · T. Colin IMB, UMR 5251, University of Bordeaux, 33400 Talence, France e-mail: Olivier.Saut@math.u-bordeaux1.fr T. Colin e-mail: Thierry.Colin@math.u-bordeaux1.fr J.-B. Lagaert Grenoble-INP/UJF-Grenoble 1/CNRS, LEGI UMR 5519, 38041 Grenoble, France e-mail: Jean-Baptiste.Lagaert@legi.grenoble-inp.fr H. M. Fathallah-Shaykh (B ) Departments of Neurology and Mathematics, University of Alabama at Birmingham, 1530 Third Ave S, FOT 1020, Birmingham, AL 35294-3410, USA e-mail: hfathall@uab.edu 123