|My principal research interests are in applying
mathematics and fluid mechanics to biology, physiology and medical engineering.
In particular, I have modelled the biology and behaviour of individual
micro-organisms and suspensions of micro-organisms. Recently, I have been
extending my research into modelling arterial disease in collaboration
with vascular surgeons at Leeds and Birmingham. I also study plankton population
dynamics and travelling waves in moving media such as the oceans. See below
for more details.
Modelling of Arterial Disease
I am modelling the early stages of atherosclerosis, specifically intimal hyperplasia, in which there is a feedback mechanism between the permeability of the endothelium lining the artery, the absorption of cholesterol, swelling of the intima in the wall of the artery which pushes the endothelium into the lumen of the artery, and the shear stress exterted on the wall by the flow of blood. This work is being carried out in consultation with vascular surgeons at St. James's University Hospital and physiologists at Imperial College. The ultimate aim is to predict the effects of proposed treatments on the progression of the disease. Two PhD students are working on this project, funded by the EPSRC and the MRC. I am also studying the progression and rupture of abdominal aortic aneurysms together with colleagues from vascular surgery both at Leeds and at Good Hope Hospital in Birmingham. One MRC-funded PhD student is working on this and the work was awarded the Prize for the Best Poster in the Cardio-Vascular Section at the ASME 2001 Bioengineering Conference. Both projects will lead to a better understanding of the function and disease processes of the arterial wall and there is potential for further funding from companies that make vascular grafts and stents.
Modelling of Control Mechanisms in Micro-Organisms and Animals
Many micro-organisms are motile and actively seek optimum environments, for example bacteria swim up and down chemical gradients (chemotaxis) and the ciliate Tetrahymena swims vertically upwards (negative gravitaxis). I have worked on simple models for the important control mechanism of phototaxis (responses towards light) which enables swimming algal cells to find the best habitat for photosynthesis. A model based on the shading of the cell's photoreceptor which periodically scans its environment as the whole cell rotates when swimming along was developed to explain experiments on responses to multiple sources of light (Hill & Vincent 1993). Recently, I have extended this work to include on responses to polarised light (Hill & Plumpton 2000). I have also carried out experimental and theoretical work on the measurement and analysis of trajectories of swimming cells which has lead to new techniques for assessing cell motility (Hill & Häder 1997). I am investigating the use of stochastic differential equations and random walks to better describe the average motion and dispersion of populations of swimming cells. In collaboration with Prof. Alan Roberts (Biology, Bristol), I have explained how immature tadpoles without a fully-developed inner ear control complicated escape manoeuvres using passive mechanical torques. These are all fundamental problems in biology and important in their own right, but they may find future application in the control of microscopic robots made possible by developments in nanotechnology.
Pattern Formation by Oceanic Plankton
The modelling of plankton blooms in the Atlantic and Pacific Oceans as an excitable system has been pioneered by the research group in biomathematics at Leeds. I have collaborated with colleagues in Applied Mathematics and Physiology at Leeds to study the effects of the relative motion of the excitable medium on wave propagation as a model for the effects of ocean currents on the blooms (Biktashev et al. 1998). A PhD student and I are now modelling the effects of microzooplankton in a three-component model for oceanic blooms. The microzooplankton are thought by ecologists to play an important role in controlling blooms because they reproduce at a similar rate to the phytoplankton on which they forage. Understanding the blooms and the factors that control them is of great environmental and economic importance because the plankton are at the bottom of the oceanic food chain.
Suspensions of swimming micro-organisms spontaneously form patterns associated with up- and down-welling of the fluid. I have helped to elucidate the physical and biological mechanisms in this paradigm for biological complexity by developing mathematical models and analysing them using a variety of linear, nonlinear and computational techniques. A current interest is the development of statistical methods for the quantitative description of the patterns which promise to have a wide range of applications.
Fluid Mechanics of Suspensions
Previously I have worked on dispersion in sedimenting colloidal suspensions (Davis & Hill 1992) and a research project is available for a PhD student to work on pairwise hydrodynamic interactions between small particles in a chamber rotating about a horizontal axis. This is expected to lead to an interesting dynamical system and it has application in chemical engineering and to experiments on the reorientation of swimming micro-organisms (Kessler et al. 1998).