A bi-parametric model for the tumour angiogenesis and antiangiogenesis therapy

In this work we study a family of phenomenological super-macroscopic models describing the tumour angiogenesis and antiangiogenesis therapy, introduced by Folkman and coworkers [P. Hahnfeldt, D. Panigrahy, J. Folkman, L. Hlatky, Tumour development under angiogenic signaling: A dynamical theory of tumour growth, treatment response, and postvascular dormancy, Cancer Res. 59 (1999) 4770–4775], and depending on two positive parameters α∈[0,1]α∈[0,1] and γ∈[0,5/3]γ∈[0,5/3]. Here we extend the model by considering not only vessel-disrupting antiangiogenic therapies, but also drugs purely inhibiting the proliferation of endothelial cells, as well as drugs exerting both these actions. By means of cooperativity theory, we show that in order for the family to describe the biological phenomena under study one has to add a further constraint on the two parameters: γ≤αγ≤α. We find conditions for the therapy-induced tumour eradication, and for the local and global stability of the equilibria in the case of low-level therapy or in the absence of it. We also consider the case γ>αγ>α. Finally, in the concluding remarks, we stress some limitations of our super-macroscopic approach, and outline some steps of improvement that might lead to more comprehensive detailed multiscale modelling of tumour angiogenesis.