We put forward an algorithm describing the three principal interconnected sub-processes that influence tumor and vasculature dynamics: (i) tumor cell proliferation (ii) angiogenesis, that is, the formation and regression of immature vessels (IV), and (iii) maturation, i.e., the formation and destabilization of mature vessels (MV). This algorithm takes account of the crucial quantitative interactions of these sub-processes, occurring across the molecular, cellular and organ levels. Implementing this complex algorithm in a computer model, one can evaluate the correlations between various factors influencing angiogenesis and their influence on tumor progression at any given moment. Moreover, the computer simulations enable analysis of the versatile effects of drugs on the growth and decay of both the tumor and the immature and mature blood vessels, as well as on the induction of an array of relevant growth factors such as angiopoietin-1 (Ang1), angiopoietin-2 (Ang2), vascular endothelial growth factor (VEGF) and platelet-derived growth factor (PDGF). Simulation results suggest that vessel maturation and destabilization of MV drive the otherwise non-linearly growing system into a very dynamic region, having irregular, scale-invariant, fluctuations, around certain asymptotic values of all the involved quantities. Destabilization itself adequately explains the experimentally observed eventual decrease of tumor growth, with no need to implicate additional assumptions, such as a new tumor growth inhibitory, or anti-angiogenic, factors. Our results further suggest that mono-therapy alone can slow tumor growth, but is not capable of eliminating it altogether. In contrast, the combined treatment of anti-angiogenic and anti-maturation drugs causes prolonged suppression of tumor growth and a significant linear decrease in average tumor size. Laboratory experiments are warranted for validating our predictions and for providing in vivo evaluated parameters.