A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport

Using basic thermodynamical principles we derive a Cahn--Hilliard--Darcy model for tumour growth including nutrient diffusion, chemotaxis, active transport, adhesion, apoptosis and proliferation. The model generalise earlier models and in particular include active transport mechanisms which ensures thermodynamical consistency. We perform a formally matched asymptotic expansion and develop several sharp interface models. Some of them are classical and some new ones which for example include a jump in the nutrient density at the interface. A linear stability analysis for a growing nucleus is performed and in particular the role of the new active transport term is analysed. Numerical computations are performed to study the influence of the active transport term for specific growth scenarios.