Mathematical Modelling of Solid Tumour Growth: Applications of Pre-pattern Formation

The year 2002 saw both the 50th anniversary of Turing’s seminal paper on morphogenesis [33], and the 30th anniversary of Gierer and Meinhardt’s equally important paper concerning activator-inhibitor theory [9]. These two papers have had a huge influence on the application of reaction-diffusion pre-pattern theory as a mechanism to describe spatio-temporal pattern formation in many biological systems. Specific applications of the theory (to name but a few) can be found in processes in developmental biology, population biology, ecology and interacting chemical systems. It is not our intention in this chapter to discuss the range of applications — for a comprehensive account of the theory and references to the many other applications, the interested reader is referred to the books [17, 22]. Instead, here we apply reaction-diffusion pre-pattern theory to a specific problem on a spherical domain, that of a growing avascular solid tumour We also suggest actual chemicals known to be produced by tumours (autocrine growth factors) which could give rise to the pre-patterns and examine their relevance in the light of clinical and experimental observations.