Journal of Mathematical Biology

, Volume 38, Issue 3, pp 262-284

Analysis of a mathematical model for the growth of tumors

  • Avner FriedmanAffiliated withSchool of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN 55455, USA. e-mail: friedman@math.umn.edu;reitich@math.umn.edu
  • , Fernando ReitichAffiliated withSchool of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN 55455, USA. e-mail: friedman@math.umn.edu;reitich@math.umn.edu

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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=R 0 (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 t→∞ 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)→R 0 exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further,

(iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞.

Key words: Tumors Parabolic equations Free boundary problems