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Our research interests include:
Please clink on the links above to find out more.
Cancer modelling
We are interested in modelling the dynamics of cancer progression and treatment from a number of different view points and on various
spatial and temporal scales. Below some areas of interest are outlined in more detail.
Please contact Professor Philip K. Maini or Dr Eamonn Gaffney
for more details.
Chemotherapeutic and Anti-angiogenesis Treatments
Most of the existing mathematical models for tumour growth and tumour-induced angiogenesis neglect blood flow, yet this is an important
factor on which both nutrient and metabolite supply depend. Under funding from the EU 5th and 6th Frameworks we have developed mathematical
models which show how blood flow and red blood cell heterogeneity influence the growth of tissue composed of normal and cancer cells. We
first determine the distribution of oxygen in a vascular network, incorporating into our model features of blood flow and vascular dynamics
such as structural adaptation, complex rheology and red blood cell circulation. We then study the dynamics of the tissue using a cellular
automaton formulation. Our results show that blood flow and red blood cell heterogeneity play a major role in the dynamics of cancer cell
growth. In collaboration with colleagues from Israel (Prof Zvia Agur, Institute for Medical Biomathematics, Bene
Ataroth, Israel) we have investigated drug delivery protocols for Doxorubicin treatment for non-Hodgkin's lymphomas (NHL) by including, in
this modelling framework, vessel maturity, NHL cell-cycle kinetics, and Doxorubicin pharmacokinetics and pharmacodynamics.
At the same time, we have been investigating the effects of chemotherapeutic interventions on tumour cells. We have shown that the standard
model for this in the presence of multiple drugs and acquired drug resistance breaks down when the modelling assumptions are made more
biologically realistic. In particular, the inclusion of the effects of the cell cycle in the presence of cell cycle phase specific drugs
combined with limitations on the frequency with which drugs can be delivered results in a total breakdown of previous scheduling hypotheses.
In a new collaboration with Professor David Kerr CBE (Department of Clinical Pharmacology, University of Oxford) we are now aiming to combine
these two models to study the effects of therapies in which both anti-angiogenesis and chemotherapeutic drugs are applied.
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Therapeutic Implications of the Acid-Mediated Invasion Hypothesis
In a collaboration involving Professors Robert A. Gatenby and Robert J. Gillies (Department of Radiobiology, University of Arizona), David J.
Gavaghan (Computing Laboratory, University of Oxford), Sir Michael Brady (Department of Engineering Sciences, University of Oxford) we have
developed a mathematical model of reaction-diffusion type to predict the excess extracellular hydrogen ion concentration in a growing tumour.
This model predicts that avascular tumours will always be benign, but that vascular tumours may become malignant (that is, grow uncontrollably)
if a certain combination of model parameters breaches a threshold. This actually suggests the counter-intuitive treatment of adding excess acid
to the tissue as a possible method of controlling tumour growth (as it will poison the tumour). Professor Gatenby is presently pursuing this
experimentally. We are currently extending our model to incorporate further aspects of the glycolytic pathway and the relationship between
extracellular oxygen concentration and cellular acid production.
As the tumour microenvironment becomes more toxic, adaptation becomes essential, and we have developed a hybrid cellular automaton model to
investigate the cell-microenvironmental interactions that mediate somatic evolution of cancer cells during the development of ductal carcinoma in
situ. We are in the process of experimentally verifying this model and preliminary results show that it does indeed correctly
predict key aspects of the spatio-temporal evolution of invasive breast cancer. The model suggests further that the transition to aggressively
invasive phenotypes may be delayed through novel strategies directed towards interrupting the hypoxia-glycolysis-acidosis cycle and future work will
explore the therapeutic potential of this finding.
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Applications to Barrett's Esophagus
One of the strengths of a mathematical approach is that sometimes we find that a modelling framework developed to tackle one problem can be
easily adapted and used to investigate another problem. Recently we were approached by Dr Janusz Jankowski (Gastroenterological Oncology,
Department of Clinical Pharmacology, University of Oxford) to help investigate the problem of oesophageal adenocarcinoma
which is increasing dramatically in the western world and has a poor prognosis mainly because individuals present at a late stage.
It is thought that environmental interactions may play a crucial role for the clonal expansion and propagation of metaplastic premalignant lesions.
Diet could well play an important role here and we are in the process of modifying our hybrid cellular automaton model (described above) to take
account of particular aspects contributing to this disease, such as acid reflux. The aim here is to use models to generate a variety of hypotheses
and compute the outcome of their interactions. Experiments in the Jankowski laboratory will then help us chose which hypotheses and interactions
are consistent with observations and thus refine the model. By continuing this iteration back and forth between experiment and theory we will gain
a better understanding of this disease and what causes and aggravates it. In turn, this will suggest ways to improve diet to lower incidence of this
cancer as well therapeutic strategies to control the disease.
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Colorectal Cancer
One of the main areas of research targeted by the E-Science pilot project on Integrative Biology is the development of models at a range of spatial
scales to investigate colorectal cancer. In collaboration with Sir Walter Bodmer (Wetherall Institute of Molecular Medicine, University of Oxford),
and Professor S. Jonathan Chapman (OCIAM) we have been developing mathematical models of cell population dynamics in the colonic crypt and in colorectal
cancer. We have highlighted shortcomings in previous models which have been used by experimentalists and have developed a more
biologically-realistic model which allows us to study the effects of asynchronous cell division and feedback controls. In particular we have shown how
the latter (ignored in previous models) is vital for homeostasis. Moreover, we have shown that abnormal disruption of the feedback mechanism can lead
eventually to colonal expansion and neoplasia. An application of this research is to determine how control can be restored to the correct level.
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Geometric properties of tumour growth
While a huge amount of data on this process is available, it is not easy to find fundamental unifying principles that help us to understand the
underlying growth mechanisms. One exception is the fractal properties that have been observed in the growth of a broad class of solid tumours, suggesting
a degree of universality. These findings suggest a particular fractal structure of the tumour surface, that evolves in time in a self-similar fashion,
sharing some of the properties of other complex growth phenomena in physics. The application of models developed for physics problems is, however, not
straightforward, as the geometry of the tumour is radically different. New equations thus have to be developed in order to describe the tumour
self-organization, keeping at the same time both the fractal properties of the tumour surface and the geometrical symmetries of the problem. The analysis
of these equations is also important in order to obtain information that can be contrasted with the experiments, and it is also suggestive of new
empirical studies that can be accomplished in this area.
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Spatial and spatio-temporal pattern formation
Specifically, we are interested in partial differential equation modelling of chemical and mechanical aspects of the generation of pattern and form in
embryology and development. Applications include skeletal patterning in the vertebrate limb, primitive streak formation, somitogenesis, skin organ formation
(feather bud formation, tooth initiation), tissue movement during invagination processes, tissue-tissue interactions in, for example, determining
lung morphology, cell aggregation in Dictyostelium, patterning generation in Hydra.
Please contact Professor Philip K. Maini or Dr Ruth E. Baker for more details.
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Somitogenesis
Somitogenesis, the sequential production of a periodic pattern along the head-tail (HT) axis of vertebrate embryos, is one of the most obvious examples
of the patterning processes that take place during embryogenesis. Segmentation is visualized by the sequential formation of bilateral blocks of cells,
the somites, from the rostral extremity of the presomitic mesoderm (PSM). Genetic or environmental factors can disturb somitogenesis, and there are many
recognized clinical conditions that can occur as a result. In humans, disturbance of the somitogenic processes can cause abnormal segmentation of the
vertebral column, leading to defects such as rib fusions and wedge and butterfly vertebrae. Studying the developmental mechanisms in vertebral patterning
will aid in the identification of protective or potentially disruptive factors for normal somitogenesis, and could lead towards treatments for the
prevention of vertebral patterning disorders.
There is very strong evidence to support the fact that somite formation is controlled by two factors: a segmentation clock which controls when the
boundaries of somites form, and an FGF8 wavefront which controls where the boundaries of somites form . The PSM is not a homogeneous tissue, but in fact
consists of two distinct parts, one in which somite formation has begun; cells have become committed to form part of a somite and are fixed with respect
to their AP polarity, and the other in which the cells are not committed or immature. These two regions are divided by their level of FGF8 signalling;
low in the cranial region where segmentation has begun and high in the caudal region where it has not. The boundary between these two regions is known
as the determination front. An interaction between the segmentation clock and FGF8 determination front gates the cells into somites.
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Wound Healing
Matrix orientation plays a crucial role in determining the severity of scar tissue after dermal wounding. We have developed a multiscale modelling
framework which allows us to examine the interaction of many of the factors involved in orientation and alignment. Briefly, the model considers a fibrin
clot into which cells (modelled as discrete objects) move, degrading the clot and laying down collagen. The fibrin and the collagen matrix are modelled
as continuous vector fields whose direction and length represent, respectively, the predominant orientation of fibres and their density (Dallon et al., 1999).
We have shown that this model predicts patterns of alignment on a macroscopic length scale that are lost in a continuum model of cell population
and have used the model to investigate several factors which influence the alignment of collagen. Specifically, we related the model
to current anti-scarring therapies using Transforming Growth Factor β (TGF-β). Using our model, we were able to tease out which of the many behavioural
changes induced by TGF-β were the crucial factors influencing alignment, and hence scarring. This work was carried out in
collaboration with mathematical colleagues Professor Jonathan A. Sherratt (Heriot-Watt University, Edinburgh) and Dr John Dallon (Brigham Young University, USA)
and experimental colleague Professor Mark J. Ferguson, CBE (Faculty of Life Sciences, University of Manchester, Renovo Ltd). Intriguingly, many of the
model predictions appear to hold true and we are presently investigating, in collaboration with the Ferguson laboratory, ways to verify the other model
predictions before using the model to potentially aid in drug design for better healing.
Please contact Professor Philip K. Maini for more details.
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Tear Film Dynamics
The pre-ocular tear film is a stratified incompressible fluid which is roughly ten micrometers thick. Pathologies of this film or its production are
typically responsible for dry eye, which is a common, age-related affliction responsible for considerable disability and days lost through sick leave.
It is important to understand how tear film osmolarity varies across the whole ocular surface using empirical indicators of dry eye syndromes, since dry
eye-induced tear hyperosmolarity is recognised as a major cause of ocular damage during dry eye. However, relationships between dry eye indicators and
corneal surface osmolarity are experimentally inaccessible, highlighting the need for quantitative modelling in this area.
There has been significant work modelling the fluid dynamics of the tear. One study used lubrication theory to study the deposition of the tear film in
the up-blink, while another used similar techniques for investigating tear film break-up in the inter-blink and emphasised the importance of evaporation
and gravity. These studies, however, have not considered the solute balance within such flows. In a collaboration with Professor Anthony J. Bron and Dr
John M. Tiffany (The Nuffield Laboratory of Ophthalmology, University of Oxford), we are pursuing a coupled solute-balance-fluid flow model of the tear
film to address such questions. This will encompass currently known data and ideas with a generalization of current compartmental models to specifically
investigate the hyperosmolarity associated with dry eye. This will allow us to approximate and delimit the overall behaviour of solute concentrations
within each compartment and to investigate numerous questions of relevance to dry eye, such as whether lacrimal gland secretions of hypotonic fluid can
offset the effects of high tear film evaporation rate (due to a deficiency in tear film lipid production).
Please contact Dr Eamonn Gaffney for more details.
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Ciliary Dynamics in the Lung
The design of treatments for lung surface pathologies such as chronic obstructive pulmonary disease (COPD), asthma and cystic fibrosis is based on the
quantitative details of lung muco-ciliary flows, and transport within these, despite the fact that we have a very poor understanding of these phenomena.
Our research in this area, in collaboration with colleagues from the University of Birmingham, has provided a biophysical basis for why the (rare)
electrolyte metabolism disorder, pseudo-hypoaldosteron, does not induce lung failure despite excess airway surface liquid production. It has also
predicted that mucolytic drugs are relatively ineffective in increasing muco-ciliary clearance in the lung, especially compared to enhancing the cilia
beat cycle time with β-andrenergic agents such as salbutamol, this is in general agreement with clinical observation.
Future work will investigate predictions on how one may attempt to enhance transport for lung epithelia with patches lacking cilia in the presence of
mucus hypersecretion, a phenotype existing in COPD. This phenotype appears to be present after fatal asthma attacks. It would thus be instructive to
understand if it has a severe deleterious effect on muco-ciliary clearance. This, in turn, would indicate whether such tethering could be a contributing
factor to the build-up of airway blocking mucus plugs observed after fatal asthma attacks and thus whether it should be targeted therapeutically. This
question is especially pertinent given the debate in the literature on the role of mucus plug in asthma deaths.
Please contact Dr Eamonn Gaffney for more details.
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Sperm Dynamics
We have been collaborating with Dr Jackson Kirkman-Brown (The Assisted Conception Unit, Birmingham Women's Hospital), Professor John R. Blake
(Department of Mathematics, University of Birmingham) and Dr David J. Smith (Departments of Mathematics and Medicine, University of Birmingham, and
Honorary Research Fellow, The Assisted Conception Unit, Birmingham Women's Hospital) on a problem concerning sperm dynamics.
Falling male sperm counts and subfertility have been regularly reported and has become of national and international concern. However, traditional
semen analysis provides only limited information for a couple seeking fertility treatment, motivating the investigation of further tests for sperm
fitness and function. Mathematical modelling has a clear potential to play a vital role in providing the necessary understanding for the design of
novel tests in this area.
Traditionally, the modelling of sperm dynamics has assumed spherical heads and helical tail beats which are oversimplifications. For example, under
the microscope, sperm are observed to swim along boundaries with their heads undergoing oscillatory rotations in synchrony with the period of the
flagellum beat. Drag is orientation dependent for a non-spherical particle translating and rotating close to a boundary in Stokes flow. In addition,
oblate spheroid rigid bodies are predicted to exhibit complex dynamics in external flows once they are close to a boundary. Consequently, the
oblateness of the sperm head geometry is likely to have significant influence on the forces and torques experienced by sperm, and therefore on
their overall motility.
In this recently established collaboration, we are developing techniques to analyse sperm movies and to subsequently predict the forces on sperm as
they swim, without resort to simplified geometries. The helical flagellum beat model has already been shown to be inappropriate for human sperm and
motility signatures have been developed which can differentiate between different types of sperm beating.
Possible future applications are numerous; in the short term we will investigate how the forces on and motility of a sperm flagellum alter on
perturbation of the surrounding viscosity and rheology, which would offer insight into the mechanism of action of the progesterone-only pill. In
the long term, our clinical aim will be to scale up these techniques to correlate different flagellum beat patterns, motility characteristics and
force profiles between normal versus infertile sperm for improving the screening of sperm samples for male fertility treatments.
Please contact Dr Eamonn Gaffney for more details.
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Stochastic modelling
Reaction-diffusion processes
Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species, for example, problems in
developmental biology, genes and enzymes. Such reaction-diffusion processes can be mathematically modelled using either deterministic
partial-differential equations or stochastic simulation algorithms. Stochastic models provide a more detailed understanding of the reaction-diffusion
processes. Such a description is often necessary for the modelling of biological systems where small molecular abundances of some chemical species
make deterministic models inaccurate or even inapplicable. Stochastic models are also necessary when biologically observed phenomena depend on stochastic
fluctuations, for example, switching between two favourable states of the system. There are several stochastic (molecular-based or mesoscopic) approaches
for modelling chemical reactions and molecular diffusion. Coupling models of these two fundamental processes together offers several challenging
mathematical problems. The goal of this research area is to design reliable, correct and efficient methods for the stochastic
simulation of reaction-diffusion processes in biology.
Please contact Dr Radek Erban for more details. Lecture notes on stochastic modelling of reaction-diffusion
processes can be found here.
The cell cycle
Another important effect that can arise in a small chemical system is the coupling between the inherent stochasticity, due to the low numbers of reagents,
and an external signal. The interplay between these effects yields in general a nontrivial output, and might be the cause of several resonances. As
these resonances enhance the periodic behaviour of the system, its study could be of relevance for the understanding of some periodic biochemical
reactions, such as the cell cycle. The cell cycle is known to be influenced by environmental conditions, and, although some models are able to reproduce
the periodic behaviour without internal fluctuations nor external perturbations, we do not know to what extent these affect the dynamics.
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Chemotaxis
Motile organisms sense their environment and can respond to it by either directed movement toward or away from a signal, or by changing their speed
of movement and/or frequency of turning. Such responses to extracellular chemical signals are often called chemotaxis. The first strategy is used by
cells that move by crawling through their environment, provided they have receptors in the cell membrane and are large enough to detect typical
differences in the signal over their body length. Examples include leukocytes (part of our immune system) or the model organism Dictyostelium
discoideum. Small cells such as bacteria cannot effectively make a "two-point in space", measurement over their body length,
and therefore they adopt the second strategy; measuring temporal variation in the signal as they move through the field.
We investigate the mechanisms underlying chemotaxis on different space scales and time scales from biochemical intracellular signal transduction
pathways to the behaviour of a single organism and the collective properties of cellular populations. Due to the coupling of many nonlinear
processes, mathematics can offer unique insights into the chemotaxis processes and provide connections between different levels of description.
We also investigate fundamental questions about mathematical models of chemotaxis, for example, the global existence and behaviour of solutions of
the complex chemotaxis partial-differential equations. One of the most impressive effects of chemotaxis is the so-called
"chemotactic collapse". This collapse happens when the cells aggregate into small regions of space, forming high density configurations. Classical
chemotaxis equations model the balance between collective chemotactic drift and individual cell diffusion. Depending on the particular situation,
the drift might "win", and an aggregate is formed, or diffusion wins and cells disperse. This problem can be generalised to the context in which
the cells perform some sort of anomalous diffusion and this balance between dispersal and aggregation can be studied too.
Bacterial chemotaxis
Bacteria, too small to sense a change in their extracellular environment using their length, use a robust intracellular phosphotransfer cascade
to signal between membrane receptors at one end of the cell and flagella motors at the opposing end. We are currently investigating the role that
the spatial localisation of proteins within the cell plays in affecting the overall response of a single bacterium to attractant gradients. In
collaboration with Prof. Judy Armitage (Department of Biochemistry, University of Oxford) reaction diffusion models of the intracellular signalling
cascade within E. coli are being explored. This work is currently being extended to the more complex signalling cascade found in R.
spaheroides to understand the role that multiple signalling pathways play in affecting the bacterial reponse. We have recently conducted two
extensive reviews on the mathematical modelling work undertaken in this area on both the single cell and population scales.
Please contact Professor Philip K. Maini,
Dr Marcus Tindall for more details.
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Gene regulatory networks
Various ways to model gene-regulatory networks exist, ranging from logical (boolean), deterministic (ordinary differential equations) to stochastic
models. The stochastic approach takes into account fluctuations due to the inherently random nature of biochemical reactions. This intrinsic noise
gives rise to significant effects when either the molecular abundances of protein or mRNA molecules are small or the kinetics of the transitions
between chemical states are slow. The disadvantage of detailed stochastic modelling is its computational intensity; this is due to the relatively
large number of chemical species (tens or hundreds) in currently modelled biochemical pathways.
Since biochemical reactions occur over different time scales, the temporal evolution of the network can often be described by a smaller number of
variables. We investigate the application of computational data mining techniques (in particular, spectral graph theory, diffusion maps and the
resulting low-dimensional description of high-dimensional data) to gene regulatory network models to find suitable
low-dimensional descriptions. Knowledge of good observables is vital to the creation of effective reduced models of complex gene regulatory
networks. Having found such observables (either from experimental evidence or by computational data mining), we develop methods for analysis of
gene regulatory network models.
We also study the reverse engineering of gene regulatory networks. Having the experimental gene expression data, we look for the uknown topology
(wiring) of the gene regulatory network (for example using Shannon's entropy, theory of finite fields, or Bayesian networks). In particular, we
investigate the connections between Boolean, algebraic, deterministic and stochastic approaches to modelling of gene regulation.
Please contact Dr Radek Erban for more details.
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Multiscale modelling in biology
Whilst modelling biological systems, we are often presented with information at a fine level of description (stochastic, individual-based),
while we want to study the behaviour at a macroscopic coarse-grained (continuum, population) level. Computational methods for obtaining an
approximation to the macroscopic evolution of the system without explicitly obtaining macroscopic equations have been developed in collaboration
with Professor Ioannis Kevrekidis (Princeton University) - the so-called equation-free methods. Equation-free methods can be viewed as a
computational superstructure wrapped around a microscopic model of a biological system. They have been applied to problems where macroscopic
dynamics is either deterministic, or stochastic. The efficiency of equation-free methods can be further
improved if one knows some extra information about the problem. For example, in some morphogenesis applications, we are able to easily derive
approximate mean-field partial differential equation which can be used to design the so called equation-assisted methods.
Equation-free methods for problems with continuous symmetries (travelling or self-similar behaviour) can be designed more efficiently and accurately
if one takes the corresponding symmetry into account.
Please contact Dr Radek Erban for more details.
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Gene therapy
One of the strategies for introducing therapeutic genes into the body is to modify common viruses. These professional pathogens (controlled as
therapeutic viral vectors) offer great promise for the treatment of a wide variety of clinical conditions such as cancer, genetic disorders
and infectious diseases through techniques like vaccination and gene therapy. Although highly effective in animal models, these techniques
often fail to show efficacy in humans. One of the major hurdles to the clinical success of viral vectors is the presence or generation of antibodies
to the virus capsoid proteins, resulting in the neutralisation of the administered dose. Polymer modification of virus particles has become established
as one of the most effective ways to prevent antibodies and other molecules, such as complement proteins, from accessing the virus surface.
Mathematical modelling of polymer modification of viruses includes chemisorption, the irreversible attachment of polymer molecules to the viral
surface. Models of various degrees of complexity have been developed for this process. This work is being conducted in collaboration with
Professor Len Seymour and Dr. Kerry Fisher (Department of Clinical Pharmacology, University of Oxford).
Please contact Dr Radek Erban for more details.
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From Individual to Collective Behaviour in Ecology
We are using various mathematical methods to investigate the relationship between the behaviour of individual animals and the dynamics of their
population. A focus of this research work has been on social insects and honey bees in particular. We are interested in how insects use simple rules
and local information to generate complex and functional patterns. More recent work has concentrated on applying these techniques to the dynamics of
populations in ecological systems. We are working in collaboration with a number of research groups (Steve Simpson (Sydney), Iain Couzin (Princeton),
David Sumpter (Sweden)).
Please contact Professor Philip K. Maini, Dr Radek Erban
for more details.
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Webpage last updated 14th May 2009 by S. Jolliffe
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