APPLICATIONES MATHEMATICAE 36,3 (2009), pp. 313–331 Jan Poleszczuk and Urszula Foryś (Warszawa) ANGIOGENESIS PROCESS WITH VESSEL IMPAIRMENT FOR GOMPERTZIAN AND LOGISTIC TYPE OF TUMOUR GROWTH Abstract. We propose two models of vessel impairment in the process of tumour angiogenesis and we consider three types of treatment: standard chemotherapy, antiangiogenic treatment and a combined treatment. The models are based on the idea of Hahnfeldt et al. that the carrying capacity for any solid tumour depends on its vessel density. In the models proposed the carrying capacity also depends on the process of vessel impairment. In the ﬁrst model a logistic type equation is used to describe the neoplastic cell dynamics, while in the second one we use the Gompertz type of growth. Simulations of solutions show that a vascular dormant state of the tumour can be reached in two diﬀerent ways. In addition in each case eﬃciency of treatments is diﬀerent. 1. Introduction. In adults, the normal physiological role of angio- genesis—the process of new vessel formation—is restricted to wound healing, menstrual cycle and pregnancy. In addition, angiogenesis is critical during fetal development. Unfortunately, it is also essential to the growth and de- velopment of solid tumours. After reaching an avascular dormant state, a tumour can grow further only by inducing vessels in host tissue to sprout capillary tubes which migrate towards and ultimately penetrate the tumour, providing it with a circulating blood supply and, therefore, an additional source of nutrients [20, 21]. Despite the essential role of angiogenesis in tumour growth, it has been discovered that tumour angiogenesis is highly pathological. Incorrect struc- 2000 Mathematics Subject Classiﬁcation : 34A12, 34A34, 92C50. Key words and phrases : tumour, angiogenesis, Hahnfeldt et al. model, vessel impairment, chemotherapy, antiangiogenic treatment, ordinary diﬀerential equations, steady state, in- variant set. DOI: 10.4064/am36-3-5 [313] c  Instytut Matematyczny PAN, 2009