Tumor Gompertzian growth by cellular energetic balance

A microscopic model of tumor growth in vivo is still an open problem. It requires a detailed description of cellular interactions and a control on the large variety of in situ conditions related to the distributions of nutrient, of oxygen, of growth inhibitors, of blood vessels and capillarity and to the mechanical effects due to tissue elasticity and heterogeneity. On the other hand, in spite of the previous large set of potential parameters, tumors have a peculiar growth pattern that is generally described by a Gompertzian curve [1], often considered as a pure phenomenological fit to the data. More precisely there is an initial exponential growth (until 1-3 mm in diameter) followed by the vascular Gompertzian phase[2]. Then it seems reasonable to think that cancer growth follows a general pattern that one can hope to describe by macroscopic variables, constrained by a set of environmental conditions crucial to understand the fundamental features of the growth as, for example, the shape and the maximum possible size of the tumor and the onset of the tissue invasion and metastasis. Following this line of research, for example, the universal model proposed in [3] has been recently applied to cancer[4]. In this paper we consider a macroscopic model of tumor growth that: i) gives an energetic basis to the Gompertzian law; ii) clearly distinguishes among the general evolution patterns, which include feedback effects and external constraints; iii) can give indications on the different tumor phases during its evolution. The proposed macroscopic approach is not in competition with microscopic models [5], but it is a complementary instrument for the description of the tumor growth. The Gompertzian curve is solution of the equation