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Mathematical modelling of the effects of mitotic inhibitors on avascular tumour growth. (English) Zbl 0981.92012
Summary: We build on our previous mathematical model [see IMA J. Math. Appl. Med. Biol. 16, No. 2, 171-211 (1999; Zbl 0943.92019); ibid. 14, No. 1, 39-69 (1997; Zbl 0866.92011)] to study the effects of high molecular mass mitotic inhibitors released at cell death. The model assumes a continuum of living cells which, depending on the concentration of a generic nutrient, generate movement (described by a velocity field) due to the changes in volumes caused by cell birth and death. The necrotic material is assumed to consist of two diffusible materials: 1) basic cellular material which is used by living cells as raw material for mitosis; 2) a generic non-utilizable material which may inhibit mitosis.
Numerical solutions of the resulting system of partial differential equations show all the main features of tumour growth and heterogeneity. Material 2) is found to act in an inhibitive fashion in two ways: i) directly, by reducing the mitotic rate and ii) indirectly, by occupying space, thereby reducing the availability of the basic cellular material. For large time the solutions to the model tend either to a steady-state, reflecting growth saturation, or to a traveling wave, indicating continual linear growth. The steady-state and traveling wave limits of the model are derived and studied, the regions of existence of these two types of long-time solution being explored in parameter space using numerical methods.

MSC:
92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
65N99 Numerical methods for partial differential equations, boundary value problems
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