Mathematical modelling of avascular-tumour growth. II: Modelling growth saturation.

We build on our earlier mathematical model (Ward & King, 1997, IMA J. Appl. Math Appl. Med. Biol., 14, 39-69) by incorporating two necrotic depletion mechanisms, which results in a model that can predict all the main phases of avascular-tumour growth and heterogeneity. The model assumes a continuum of live cells which, depending on the concentration of a generic nutrient, may reproduce or die, generating local volume changes and thus producing movement described by a velocity field. The necrotic material is viewed as basic cellular material (i.e. as a generic mix of proteins, DNA, etc.) which is able to diffuse and is utilized by living cells as raw material to construct new cells during mitosis. Numerical solution of the resulting system of partial differential equations shows that growth ultimately tends either to a steady-state (growth saturation) or becomes linear. Both the travelling-wave and steady-state limits of the model are therefore derived and studied. The analysis demonstrates that, except in a very special case, passage of cellular material across the tumour surface is necessary for growth saturation to occur. Using numerical techniques, the domains of existence of the large-time solutions are explored in parameter space. For a particular limit, asymptotic analysis makes explicit the main phases of growth and gives the location of the bifurcation between the long-time outcomes.