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We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe(More)
A system of nonlinear partial differential equations is proposed as a model for the growth of an avascular-tumour spheroid. The model assumes a continuum of cells in two states, living or dead, and, depending on the concentration of a generic nutrient, the live cells may reproduce (expanding the tumour) or die (causing contraction). These volume changes(More)
We consider non-negative solutions on the half-line of the thin film equation h t + (h n h xxx) x = 0, which arises in lubrication models for thin viscous films, spreading droplets and Hele–Shaw cells. We present a discussion of the boundary conditions at x = 0 on the basis of formal and modelling arguments when x = 0 is an edge over which fluid can drain.(More)
Gene expression at the single-cell level incorporates reaction mechanisms which are intrinsically stochastic as they involve molecular species present at low copy numbers. The dynamics of these mechanisms can be described quantitatively using stochastic master-equation modelling; in this paper we study a generic gene-expression model of this kind which(More)
We consider a new model for groundwater ow. The model diiers from previous models in that the saturation-pressure relation is extended with a dynamic term, namely the time derivative of the saturation. The resulting model equation is of nonlinear degenerate pseudo-parabolic type. We give a rigorous analysis of travelling wave solutions and the local(More)
Many solid tumour growth models are formulated as systems of parabolic and/or hyperbolic equations. Here an alternative, two-phase theory is developed to describe solid tumour growth. Versions of earlier models are recovered when suitable limits of the new model are taken. We contend that the multi-phase approach represents a more general, and natural,(More)
OBJECTIVES The luminal surface of the gut is lined with a monolayer of epithelial cells that acts as a nutrient absorptive engine and protective barrier. To maintain its integrity and functionality, the epithelium is renewed every few days. Theoretical models are powerful tools that can be used to test hypotheses concerning the regulation of this renewal(More)
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(More)
In this paper we adapt an avascular tumour growth model to compare the effects of drug application on multicell spheroids and on monolayer cultures. The model for the tumour is based on nutrient driven growth of a continuum of live cells, whose birth and death generates volume changes described by a velocity field. The drug is modelled as an externally(More)
The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We present an analysis of the asymptotic behaviour of singularities arising in this flow for a special class of solutions which generalises a known (radially symmetric) reduction. Specifically , the rate at which blowup occurs is investigated in settings with(More)