Linear stability analysis for a free boundary problem modeling tumor growth in the presence of high-molecular mass drugs

Abstract

In this paper, we consider a free boundary problem modeling tumor growth in the presence of drugs with high molecular mass. This model is an improvement of a similar model proposed by Byrne and Chaplain ((1995) [1]). The novelty in our model is to better account for the fast diffusion coefficient of nutrients and the slow diffusion coefficient of drugs with high molecular mass, which is more realistic. The introduction of these terms, however, makes the analysis of linear stability more challenging in the spectral analysis of the linearized operator. A significant challenge is the lack of explicit expressions for their locations. We shall give a complete classification of parameters λσ, λβ, τ, σ‾, β‾ and σ˜ under which either (i) there exist exact one radially symmetric stationary solution (σS,βS,pS) with the boundary r=R2, or (ii) exact two solutions (σS,βS,pS) with the boundary r=R1 or r=R2 (R1<R2), or (iii) no solutions. We further show that for all γμ>0, (σS,βS,pS,R1) is linearly unstable under non-radially symmetric perturbations, while there exists a threshold value c⁎⩾0 such that (σS,βS,pS,R2) is linearly stable for γμ>c⁎ and linearly unstable for γμ∈(0,c⁎). As a biological implication, we discuss the positive and negative effects of drugs with high molecular mass on tumor growth and get a high risk region for drugs. Our results provide a strategy for tumor treatment.