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 Prof. Philip Maini or Dr Eamonn Gaffney for further information.


Colorectal cancer

The intestinal epithelium is folded into flask-shaped invaginations called crypts, which increase the absorptive surface area of the mucosa.The stem cells housed at the base of crypts are the most likely originators of colorectal cancers (CRCs), the third most common cancer diagnosed in the UK. This is because the stem cells are long-lived and actively dividing, hence able to accrue the multiple mutations required to initiate tumour growth. Such mutations can alter cell behaviour, by altering the orientation of cell division, increasing the size of the proliferative compartment, altering cell adhesion or directly regulating competition between stem cells. A key step to improving our understanding of CRC is to establish those cell behaviours that are selectively advantageous within a crypt and predict the transient behaviour ofsporadic mutants. Alongside histological and genetic studies, modelling is crucial for gaining mechanistic insights into these dynamics. Using mouse models of colorectal cancer for parameterization and validation, we have developed an off-lattice individual cell-based model of a colonic crypt, which we have used to investigate the mechanisms underlying homeostasis and carcinogenesis. For example, we have used the model to study the process of monoclonal conversion, where the progeny of onecell eventually dominates the entire crypt, facilitating the spread of mutations. Varying the mutant 'phenotype' revealed a strong dependence of probability of mutant monoclonal conversion on dysregulation of cell-cell adhesion, and provided strong evidence for the 'bottom-up' theory of crypt invasion, where successful mutants originate at the crypt base and propagate further through crypt fission. 

Left: hematoxylin and eosin stained section through normal healthy human colonic mucosae, showing surface epithelium (S), crypts (C) and the lamina propria separating them (L). Right: schematic showing the proliferative compartments within the crypt. Signalling cues, such as a Wnt gradient along the crypt axis, are thought to regulate proliferative behaviour. Taken from Fletcher et al (2012).

WCMB members working in this area

Key references 

  • I. M. M. van Leeuwen, G. R. Mirams, A. Walter, A. Fletcher, P. Murray, J. Osborne, S. Varma, S. J. Young, J. Cooper, B. Doyle, J. Pitt-Francis, L. Momtahan, P. Pathmanathan, J. P. Whiteley, S. J. Chapman, D. J. Gavaghan, O. E. Jensen, J. R. King, P. K. Maini, S. L. Waters, H. M. Byrne (2009). An integrative computational model for intestinal tissue renewal. Cell Prolif. 42:617-636. (eprints)
  • P. J. Murray, A. Walter, A. G. Fletcher, C. M. Edwards, M. J. Tindall and P. K. Maini (2011). Comparing a discrete and continuum model of the intestinal crypt. Phys. Biol. 8:026011. (eprints)
  • S.-J. Dunn, A. G. Fletcher, S. J. Chapman, D.J. Gavaghan and J.M. Osborne (2012). Modelling the role of the basement membrane in the colonic epithelium. J Theor Biol, 298:82-91.
  • A. G. Fletcher, C. J. W. Breward and S. J. Chapman (2012). Mathematical modelling of monoclonal conversion in the colonic cryptJ. Theor. Biol.300:118-133. (eprints)
  • G. R. Mirams, A. G. Fletcher, P. K. Maini and H.M. Byrne (2012). A theoretical investigation of the effect of proliferation and adhesion on monoclonal conversion in the colonic crypt. J. Theor. Biol. 74(9): 2204-2231. (eprints)


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. 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. 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. 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.

WCMB members working in this area
Key references
  • E. A. Boston and E. A. Gaffney (2011). The influence of toxicity constraints in models of chemotherapeutic protocol escalation. Math. Med. Biol. 28:357-384. (eprints)
  • H. Perfahl, H. M. Byrne, T. Chen, V. Estrella, T. Alarcon, A. Lapin, R. A. Gatenby, R. J. Gillies, M. C. Lloyd, P. K. Maini, M. Reuss and M. R. Owen (2011). Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions. PLoS One 6. (eprints)
  • H. C. Monro and E. A. Gaffney (2009). Modelling chemotherapy resistance in palliation and failed cure. J. Theor. Biol. 257:292-302. (eprints)
  • M. R. Owen, T. Alarcon, P. K. Maini, H. M. Byrne (2009). Angiogenesis and vascular remodelling in normal and cancerous tissues. J. Math. Biol. 58:689-721. (eprints)


Therapeutic implications of the acid-mediated invasion hypothesis

In a collaboration involving Professors Robert A. Gatenby and Robert J. Gillies (Moffitt Cancer Center, Florida) and David J. Gavaghan (Computing Laboratory, 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. 

Vascular tumour
Examples of vascular tumours.
Images kindly supplied by Professor Robert Gatenby (Moffitt Cancer Center, Florida).
WCMB members working in this area
Key references
  • N. K. Martin, E. A. Gaffney, R. A. Gatenby, R. J. Gillies, I. F. Robey and P. K. Maini (2011). A mathematical model of tumour and blood pHe regulation: The HCO-3/CO2 buffering system. Math. Biosci. 230:1-11. (eprints)
  • N. K. Martin, E. A. Gaffney, R. A. Gatenby and P. K. Maini (2010). Tumour-stromal interactions in acid-mediated invasion: a mathematical model. J. Theor. Biol. 267:461-470. (eprints)
  • R. A. Gatenby, K. Smallbone, P. K. Maini, F. Rose, J. Averill, R. B. Nagle, L. Worrall, R. J. Gillies (2007). Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer. Brit. J. Cancer 97:646-653. (eprints)