Analysis of a mathematical model for the growth of tumors

Abstract

In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r = s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r = R0 (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf s(t) > 0, i.e. once engendered, tumors persist in time. t-->infinity Indeed, we further show that (ii) If c is sufficiently small then s(t)-->R0 exponentially fast as t-->infinity, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not "sufficiently small" but is smaller than some constant gamma determined explicitly by the parameters of the problem, then t-->infinity lim sup s(t) < infinity; if however c is "somewhat" larger than gamma then generally s(t) does not remain bounded and, in fact, s(t)-->infinity exponentially fast as t-->infinity.

Nonlinear Modeling and Simulation of Tumor Growth

“…In [212] we use a reformulation of several well-known continuum-scale models of growth and angiogenesis ( [90,38,39,46,11,80,25,16]. We build upon the biological complexity of previous models that used boundary integral methods [59,127] (described in Section 1.0), where tissue is represented as a constant density fluid whose elasticity is neglected.…”

“…The growth component of the model is inspired by the work of [90,38,39,46,80,59]. The angiogenesis model is essentially that of [11] with some additions gleaned from [52,159,157].…”

Radially symmetric growth of nonnecrotic tumors

“…During the last four decades an increasing number of mathematical models have been proposed to describe the growth of solid tumors (see [2,[6][7][8] and the literature therein). There is a three level approach in modeling the complex phenomena influencing and describing the processes inside a tumor.…”

“…Using algebraic manipulations Cristini et al obtain in [5] a new mathematical formulation of an existing model (see [4,8,11]) which describes the evolution of nonnecrotic tumors in different regimes of vascularisation (see the cases (i)-(iii) presented in Sect. 2).…”

“…2). The simpler model, as presented in [4,8,11], has been studied extensively by different authors, see, e.g. [3,6,7,9] and the references therein.…”