Mathematical analysis of some models of tumour growth and simulations

Our main interest is to characterize mathematical models of tumour growth by clarifying the equivalence and difference between them mathematically. We first discuss the solvability and the asymptotic profile of the solution to some parabolic ODE systems described tumour angiogenesis([12]-[15], [23], [24]). These models are proposed independently, one arises from reinforced random walk, proposed by Othmer and Stevens and another from a number of researches in biology and biomedicine, proposed by Anderson and Chaplain. We show a rigorous relationship between them and give a mathematical framework of the solvability and asymptotic profiles of the solutions of them. Finally we study a mathematical model on generic solid tumour growth at the avascular stage, proposed by Anderson and Chaplain. The focus of their model is on an aspect of tissue invasion. Although it is the different phenomenon from angiogenensis, we can find a consistency in their mathematical structures. Then we will apply the approach used in mathematical models of tumour angiogenesis to it and show the solvability and the asymptotic profile of the solution of it. On the other hand, we show some results of computer simulations of these models with the help of our mathematical analysis.