In this paper, we develop a simple mathematical model of the vascularization and subsequent growth of a solid spherical tumour. The key elements that are encapsulated in this model are the development of a central necrotic core due to the collapse of blood vessels at the centre of the tumour and a peak of tumour cells advancing towards the main blood vessels together with the regression of newly-formed capillaries. Diffusion alone cannot account for all observed behaviour, and hence, we include 'taxis' in our model, whereby the movement of the tumour cells is directed towards high blood vessel densities. This means that the growth of the tumour is accompanied by the invasion of the surrounding tissue. Invasion is closely linked to metastasis, whereby tumour cells enter the blood or lymph system and hence secondary tumours or metastases may arise. In the second part of the paper, we conduct a travelling wave analysis on a simplified version of the model and obtain bounds on the parameters such that the solutions are nonnegative and hence biologically relevant and also an estimate for the rate of invasion.
Orme, M. E., & Chaplain, M. A. J. (1996). A mathematical model of vascular tumour growth and invasion. Mathematical and Computer Modelling, 23(10), 43–60. https://doi.org/10.1016/0895-7177(96)00053-2