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Influence of delays on the models of angiogenesis process and immunotherapy of cancer (Angiodel)
Start date: 01 Nov 2008, End date: 31 Oct 2011 PROJECT  FINISHED 

Tumour growth, dynamics and interactions with an immune system are very important processes from medical point of view and can be also considered as a topic of mathematical modelling and analysis. We would like to focus on the models of tumour angiogenesis process and immunotherapy of cancer known from the literature and study the influence of time delays on the dynamics of these models solutions. It is well-known that the signaling pathways can be modelled using time delays. Therefore, the models described by systems of ordinary differential equations can be modified to the systems of delay differential equations. Such modifications have been considered for example in the case of Hahnfeldt et al. model of angiogenesis process which is the most accepted model of this process. Immune processes are also considered as delayed with respect to the stimuli. The first model which reflects this effect was proposed by G.I. Marchuk (1980). At present there arises many models describing different aspects of anti-tumour immunity and immunotherapy of cancer. Within the project we will study the influence of time delays on the dynamics of solutions to mathematical systems of differential equations describing tumour angiogenesis and immunotherapy of cancer focusing on the models with at least two delays. From the mathematical point of view the dynamics of systems with several delays is not as well-known as in the case of one discrete delay and such analysis is a challenging problem and can bring also some insight to the medical knowledge and treatment of cancer.
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