A free boundary problem for a singular system of differential equations: An application to a model of tumor growth

Cui, Shangbin; Friedman, Avner

doi:10.1090/S0002-9947-03-03137-4

A free boundary problem for a singular system of differential equations: An application to a model of tumor growth
HTML articles powered by AMS MathViewer

by Shangbin Cui and Avner Friedman PDF

Trans. Amer. Math. Soc. 355 (2003), 3537-3590 Request permission

Abstract:

In this paper we consider a free boundary problem for a nonlinear system of two ordinary differential equations, one of which is singular at some points, including the initial point $r=0$. Because of the singularity at $r=0$, the initial value problem has a one-parameter family of solutions. We prove that there exists a unique solution to the free boundary problem. The proof of existence employs two “shooting” parameters. Analysis of the profiles of solutions of the initial value problem and tools such as comparison theorems and weak limits of solutions play an important role in the proof. The system considered here is motivated by a model in tumor growth, but the methods developed should be applicable to more general systems.

References

J. Adam, A simplified mathematical model of tumor growth, Math. Biosci. 81(1986), 224–229.
N. Britton and M. Chaplain, A qualitative analysis of some models of tissue growth, Math. Biosci. 113(1993), 77–89.
H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci. 135(1996), 187–216.
Shangbin Cui and Avner Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math. Biosci. 164 (2000), no. 2, 103–137. MR 1751267, DOI 10.1016/S0025-5564(99)00063-2
Shangbin Cui and Avner Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl. 255 (2001), no. 2, 636–677. MR 1815805, DOI 10.1006/jmaa.2000.7306
M. Dorie, R. Kallman and M. Coyne, Effect of cytochalasin b, nocodazole and irradiation on migration and internalization of cells and microspheres in tumor cell spheroids, Exp. Cell Res. 166(1986), 370–378.
M. Dorie, R. Kallman, D. Rapacchietta and et al, Migration and internalization of cells and polystyrene microspheres in tumor cell spheroids, Exp. Cell Res. 141(1982), 201–209.
Avner Friedman and Fernando Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999), no. 3, 262–284. MR 1684873, DOI 10.1007/s002850050149
H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math. 51(1972), 317–340.
H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol. 56 (1976), no. 1, 229–242. MR 429164, DOI 10.1016/S0022-5193(76)80054-9
F. Hughes and C. McCulloch, Quantification of chemotactic response of quiescent and proliferating fibroblasts in Boyden chambers by computer-assisted image analysis, J. Histochem. Cytochem. 39(1991), 243–246.
D. McElwain and G. Pettet, Cell migration in multicell spheroids: swimming against the tide, Bull. Math. Biol. 55(1993), 655–674.
G. Pettet, C. P. Please, M. J. Tindall and et al, The migration of cells in multicell tumor spheroids, Bull. Math. Biol. 63(2001), 231–257.
J. Sherrat and M. Chaplain, A new mathematical model for avascular tumor growth, J. Math. Biol. 43(2001), 291–312.
K. Thompson and H. Byrne, Modelling the internalisation of labelled cells in tumor spheroids, Bull. Math. Biol. 61(1999), 601–623.
J. Ward and J. King, Mathematical modelling of avascular-tumor growth II: Modelling growth saturation, IMA J. Math. Appl. Med. Biol. 15(1998), 1–42.