In this paper, we revisit the linear analysis of the transient evolution of a perturbed tumor interface in two and three dimensions. In Part II, we will study the full nonlinear problem using boundary-integral simulations. The tumor core is nonnecrotic and no inhibitor chemical species are present. A new formulation is developed that demonstrates that tumor evolution is described by a reduced set of two parameters and is qualitatively unaffected by the number of spatial dimensions. One parameter is related to the rate of mitosis. The other describes the balance between vascularization and apoptosis (programmed cell-death). Three regimes of growth are identified with increasing degrees of vascularization: low (diffusion dominated), moderate and high vascularization. We demonstrate that parameter ranges exist for which the tumor evolves self-similarly (i.e., shape invariant) in the first two regimes. In the diffusion-dominated regime, vascularization is weak or absent and self-similar evolution leads to a nontrivial dormant state. In the second regime vascularization becomes significant with respect to apoptosis; self-similar growth is unbounded and is associated with critical conditions of vascularization. Away from these critical conditions, perturbations may either grow with respect to the unperturbed shape, and thus lead to invasive fingering into the external tissues and metastasization, or decay to zero. In the high-vascularization regime, we find that during unbounded growth the tumor shape always tends to the unperturbed shape and neither self-similar nor fingering evolution occur. This last result is in agreement with recent experimental observations of in vivo tumor growth and angiogenesis, and suggests that the metastatic growth of highly-vascularized tumors is associated to vascular and elastic anisotropies, which are not included in our model.
Institute for Mathematics and Its Applications>IMA Preprints Series
Cristini, Vittorio; Lowengrub, John; Nie, Qing.
Nonnecrotic tumor growth and the effect of vascularization. I. Linear analysis and self-similar evolution.
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