Slow and fast invasion waves in a model of acid-mediated tumour growth. (English) Zbl 1178.35120

The authors study a front-type travelling waves (TWs) in the context of tumour invasion. The latter phenomenon is a characteristic feature of tumoural processes which largely accounts for the threat they represent to patients. The early growth stages of a primary solid tumour are merely a consequence of the successive divisions of their initial constituents. A reaction-diffusion system is then proposed as a model to describe acid-mediated cancer invasion, and the properties of travelling waves that can be supported by such a system are considered, a rich variety of wave propagation dynamics both fast and slow which is compatible with the model are shown. In particular, asymptotic formulae for admissible wave profiles and bounds on their wave speeds are provided. It is noted that from a modelling point of view, the use of TWs to describe tumour invasion suggests possible scenarios in which the same mechanism could be used to inhibit such a process. For instance, this could be done by means of an immune response represented by a wave propagating in the opposite direction, which annihilates the invading front upon contact. Mathematically, this amounts to considering reaction-diffusion systems (with additional components to those considered in this study) with TW solutions propagating in opposite directions and possessing the aforementioned annihilation property.


35C07 Traveling wave solutions
35K57 Reaction-diffusion equations
92B05 General biology and biomathematics
92C15 Developmental biology, pattern formation
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI


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