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Analysis of a mathematical model for the growth of tumors. (English) Zbl 0944.92018

Summary: We study a recently by H.M. Byrne and M.A.J. Chaplain [Math. Biosci. 130, No. 2, 151-181 (1995; Zbl 0836.92011)] 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 \(r=R_0\) (which depends on the various parameteters 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) \(\liminf_{t\to\infty}s(t)>0\), i.e. once engendered, tumors persist in time. Indeed, we further show that
(ii) If \(c\) is sufficiently small then \(s(t)\to R_0\) exponentially fast as \(t\to\infty\), 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 \(\limsup_{t\to\infty}s(t)<\infty\); if however \(c\) is “somewhat” larger than \(\gamma\) then generally \(s(t)\) does not remain bounded and, in fact, \(s(t)\to\infty\) exponentially fast as \(t\to\infty\).

MSC:

92C50 Medical applications (general)
35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35R35 Free boundary problems for PDEs

Citations:

Zbl 0836.92011
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