Reduced models of chemical reaction in chaotic flows

We describe and evaluate two reduced models for nonlinear chemical reactions in a chaotic laminar flow. Each model involves two separate steps to compute the chemical composition at a given location and time. The “manifold tracking model” first tracks backwards in time a segment of the stable manifold of the requisite point. This then provides a sample of the initial conditions appropriate for the second step, which requires solving one-dimensional problems for the reaction in Lagrangian coordinates. By contrast, the first step of the “branching trajectories model” simulates both the advection and diffusion of fluid particles that terminate at the appropriate point; the chemical reaction equations are then solved along each of the branched trajectories in a second step. Results from each model are compared with full numerical simulations of the reaction processes in a chaotic laminar flow.

A two-fluid model for tissue growth within a dynamic flow environment

We study the growth of a tissue construct in a perfusion bioreactor, focussing on its response to the mechanical environment. The bioreactor system is modelled as a two-dimensional channel containing a tissue construct through which a flow of culture medium is driven. We employ a multiphase formulation of the type presented by G. Lemon, J. King, H. Byrne, O. Jensen and K. Shakesheff in their study (Multiphase modelling of tissue growth using the theory of mixtures. J. Math. Biol. 52(2), 2006, 571–594) restricted to two interacting fluid phases, representing a cell population (and attendant extracellular matrix) and a culture medium, and employ the simplifying limit of large interphase viscous drag after S. Franks in her study (Mathematical Modelling of Tumour Growth and Stability. Ph.D. Thesis, University of Nottingham, UK, 2002) and S. Franks and J. King in their study Interactions between a uniformly proliferating tumour and its surrounding: Uniform material properties. Math. Med. Biol. 20, 2003, 47–89). The novel aspects of this study are: (i) the investigation of the effect of an imposed flow on the growth of the tissue construct, and (ii) the inclusion of a chanotransduction mechanism regulating the response of the cells to the local mechanical environment. Specifically, we consider the response of the cells to their local density and the culture medium pressure. As such, this study forms the first step towards a general multiphase formulation that incorporates the effect of mechanotransduction on the growth and morphology of a tissue construct. The model is analysed using analytic and numerical techniques, the results of which illustrate the potential use of the model to predict the dominant regulatory stimuli in a cell population.