Current research
My research is in the area of mathematical modelling, particularly in mathematical biology and ecology. The motivation for this type of research comes very much from real-world problems and the emphasis is on qualitative mathematical models that capture the essential behaviour of a particular phenomenon. I currently have research interests in a variety of areas, including:
- Cardiovascular disease.
- Kidney function and regulation.
- Tumour-induced angiogenesis (growth of new blood vessels).
- Invasive species (weeds).
- Scale insect ecology.
- Epidemiology.
- Size-structured populations.
- Biological random walks.
Some potential projects for Honours, Masters or PhD students.
One of the challenges in research of this nature is in identifying the model and mathematical techniques appropriate to a particular problem. Different problems require different tools and the models borrow analytical and numerical techniques from a range of areas, for example: dynamical systems; partial differential equations; perturbation theory; probability and stochastic processes; data analysis. A common theme in my research is looking at the effects of stochasticity, i.e. what happens when not everything follows the same predictable behaviour, but there is some uncertainty in the system.
Here are some of the people I have worked with. And here is a list of publications.
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Cardiovascular disease
Cardiovascular disease in the form of atherosclerotic plaque is highly localised to regions of the artery wall that are subject to disturbed flow characteristics, such as flow recirculation and, in particular, low wall shear stress. The early stages of plaque formation involve a complex interplay between arterial geometry, blood flow characteristics, lipoprotein mass transport and cellular processes. The aim of this research is to model the combined effect of these mechanical, biochemical and cellular factors and to understand their relative contributions to the initiation of cardiovascular disease. The key focus is the interaction of spatial variations in wall shear stress and advective-diffusive transport of cell signalling molecules with intracellular processes (in particular calcium signalling) that affect the pathophysiology of the artery wall.
Kidney function and regulation
The human kidney exhibits a remarkable degree of control over renal blood flow and plasma filtration rate over a wide range of renal arterial blood pressures. This control is known as autoregulation and is effected primarily via altered smooth muscle tone in afferent arteriolar vessels. The signal to the smooth muscle cells is a combination of a local myogenic response, a mechanism known as tubuloglomerular feeback (TGF) and possibly other unknown mechanisms. TGF is goverened by concentrations of electrolytes in the filtered fluid. Thus the autoregulation process occurs as an interaction between transmural pressures, vascular resistances, blood flow rates, filtration rates and electrolyte concentrations. Impaired autoregulation can lead to chronic renal failure. These processes can be modelled using a variety of differential equation-based approaches. A particularly interesting question concerns the effects of stochastic (i.e. random) changes in renal function and interaction between individual nephrons.
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Tumour-induced angiogenesis
The growth of a solid tumour is constrained by oxygen supply to a maximum diameter of 1-2 mm. Further growth is only possible if the tumour induces the growth of new blood vessels from the existing host vasculature (a process termed angiogenesis). This is typically achieved via the secretion, by the oxygen-starved cancer cells, of a range of angiogenic growth factors, eventually resulting in the formation of a new capillary network and improved blood supply to the tumour. Angiogenesis occurs as a result of cell migration and proliferation, processes that are regulated by the interaction of a range of growth factors and growth inhibitors with cell receptors. These processes may be modelled either at a macroscopic level (continuum models) or at a microscopic level (individual cell-based models). The focus of this research is on using cell-based models of angiogenesis, and investigating their relationship to the cotinuum limit. The are important questions about the mechanisms by which cells aggregate and differentiate to form stable vessels, the effects of blood flow in the newly formed capillaries, and the roles of a number of growth factor and inhibitors. The effect of growth inhibitors is a particularly important area because of their potential as anti-cancer treatments.
Scale insect ecology
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Hondeydew-producing scale insects are fun creatures to model! They are tiny insects that live in the bark of beech trees, and are common in large areas of New Zealand. They insert their mouthparts into the tree, allowing them to eat the tree's sap. Excess sap is excreted through a long waxy tube, and droplets of honeydew (which is an important food source for other species in the forest ecosystem) frequently form on the end of these tubes. Little is known about what regulates the rate of honeydew 'production' – is it simply the net pressure gradient between the interior of the tree and the atmoshphere, or are the bugs playing an active role in the process? The rate of droplet formation is also affected by environmental conditions, especially the relative humidity of the air. The processes of droplet formation and evaporation can be modelled using a dynamical system based on simple fluid dynamics, mass transport and evaporation. The model results can be compared to field measurements of the volume and sugar content of droplets.
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Modelling tree growth
Look at these pictures of a tree! They were created using the fast marching method, which is a really nice method for numerically calculating the motion of a surface in 3D. There isn't any tree physiology in the model, just really simple rules for the growth of the tree's main stem and branches. The surfaces are all bumpy because the numerics are a bit shonky, but you get the general idea!Epidemiology
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Epidemiology is the study of infectious diseases. An emerging phenomenon in epidemiology is superspreading, which is when relatively few infected individuals are responsible for a disproportionately large number of secondary cases, whereas the majority of infected individuals cause no secondary cases. Superspreading has been implicated in the SARS outbreaks of 2003 and can have a major effect on the course of an epidemic. For example, repeated field measurements of a low value of R0 (the basic reproduction number) may lead to serious underestimation of the epidemic potential of a particular disease. The graph shows data for a number if disease outbreaks: the point where the curve crosses the vertical line shows the percentage of secondary cases caused by the most infections 20% of all infected individuals. The outbreaks passing above the (20,80) point comply with the '20/80' rule (the most infectious 20% of individuals cause at least 80% of all cases) and are likely to be dominated by superspreading.
Superspreading can be modelled as a stochastic process of superspreading events (SSEs), which occur relatively infrequently, but have the potential to cause a very large number of secondary infections. An SSE might occur when an infected individual interacts closely with a large number of susceptibles over a short period, for example in a hospital or busy marketplace. Comparing the SSE model with an ordinary stochastic process with the same value of R0 allows the effects of superspreading to be quantified.
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Size-structured populations
The traditional framework for studying populations of different species interacting via predation and competition is food webs. These represent the different species as nodes in a network, with links between nodes indicating interactions between species. However, the strengths of these interactions are often difficult to measure, and are constantly changing due to variations between individuals within a given node in the food web. An alternative to the food web approach is to use body size, rather than species, as an indication of trophic position. As an individual grows, the size of its prey items also changes. In this way, the ecosystem may be represented as a continuous spectrum of body sizes. This system may be modelled using a combination of stochastic processes, corresponding to events like predation, growth, birth and death, which occur at the level of an individual organism. This individual-based model can be approximated by a continuous PDE which, under the right conditions, may be used to describe properties of the system as a whole. For example, will the size spectrum settle down to a steady state, such as the well known power law relationship of density with size? Or will it exhibit more complex behaviour, such as sustained oscillations?
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Biological random walks
Random walks have been used to model a wide variety of biological phenomena, from cell migration inside the human body to animal foraging movements. In its simplest form, a random walk is just a set of probabilities of moving in different directions, although this basic idea can be generalised in many ways. From this simple picture at the level of an individual, complex behaviour and patterns can emerge at the level of the population as a whole. In recent years, a particular class of random walk known as the Lévy walk has gained widespread popularity in ecological models. Their super-diffusive and fractal (scale-free) properties can, in certain circumstances, be advantageous for exploiting patchily distributed resources and avoiding the backtracking associated with simple Brownian motion. However, important questions remain as to what ecological mechanisms and environmental factors are underyling these observed patterns.