Abstract. In this paper we present several ODE systems encoding the most essential observations and assumptions about the complex hierarchical interactive processes of tumor neo-vascularization (angiogenesis). From experimental results we infer that a significant marker of tumor aggressiveness is the oscillatory behavior of tumor size, which is driven by its vascularization dynamics. To study the forces underlying these oscillations we perform a Hopf point analysis of the angiogenesis models. In the analyzed models Hopf points appear if and only if a nontrivial set of time-delays is introduced into the tumor proliferation or the neo-vascularization process. We suggest to examine in laboratory experiments how to employ these results for containing cancer growth.1. Introduction. Growth of malignant tumors beyond the diameter of 1 − 2mm critically depends on their neo-vascularization, which provides vital nutrients and growth factors, and also clears toxic waste products of cellular metabolism [12]. Indeed, the role of angiogenesis -the formation of new blood vessels by budding from existing ones -as a target for cancer therapy, is currently a focus of intensive research [12], [8], [19].In order to establish successful anti-angiogenic treatment rationale, the dynamics of angiogenesis must be better understood. These dynamics are very complex, involving many interacting oscillatory processes, which operate on several scales of time and space. Their essential constituents are briefly described below.Having reached a certain size and, therefore, a certain critical volume/surface ratio, a shortage of oxygen (denoted hypoxia) and nutrients is created within the tumor. Under hypoxia the tumor produces proteins, notably Vascular Endothelial Growth Factor (V EGF ). Increasing V EGF levels lead to increased proliferation and mobility of endothelial cells, and, as a result, to increased formation of immature vessels by these cells. Consequently the blood supply of the tumor is augmented, encouraging tumor proliferation [22]
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.
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