The growth of a tumour in a rigid walled cylindrical duct is examined in order to model the initial stages of tumour cell expansion in ductal carcinoma in situ (DCIS) of the breast. A nutrient-limited growth model is formulated, in which cell movement is described by a Stokes flow constitutive relation. The effects on the shape of the tumour boundary of the material properties (i.e. the viscosity) and the extent to which the cells adhere to the duct wall are studied using numerical and asymptotic methods. It is shown how stable, non-planar, interface configurations result and that, during these initial stages, before the duct wall has been breached, few cells die and a nutrient-rich model is usually sufficient to capture the behaviour. Finally, we discuss the relevance of this approach to DCIS and suggest possible avenues for further work.
The stability of a planar tumour growing into neighbouring tissue is examined and, in particular, its dependence on the properties of the tumour and of the surrounding material studied. An abundant supply of nutrient is assumed, so the proliferation of cells is uninhibited (resulting in exponential growth). We consider two possible constitutive relations. Darcy's law and Stokes flow, in describing the deformation of the tissue and the resulting model takes the form of a coupled system comprising a nonlinear reaction-diffusion-convection equation for the tumour cell concentration and an elliptic system for the deformation and stress fields. Using a combination of linear-stability analysis, numerical methods and thin-film approximations. the evolution of the advancing tumour boundary is determined. It is shown that when the tumour and surrounding material properties are the same, a planar interface is always linearly unstable, with the Stokes flow problem being reducible to the Darcy one. We treat the subsequent (nonlinear) evolution and suggest possible extensions to this work.
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