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Acidic pH is a common characteristic of human tumours. It has a significant impact on tumour progression and response to therapies. In this paper, we develop a simple model of three-dimensional tumour growth to examine the role of acidosis in the interaction between normal and tumour cell populations. Both vascular and avascular tumour dynamics are investigated, and a number of different behaviours are observed. Whilst an avascular tumour always proceeds to a benign steady state, a vascular tumour may display either benign or invasive

A common experimental technique for viewing in vivo angiogenesis utilises tumours implanted into a test animal cornea. The cornea is avascular but the tumour promotes vascularisation from the limbus and the new blood vessels can be readily observed through the transparent cornea. Many of the early mathematical models for tumour angiogenesis used this scenario as their experimental template and as such assumed that there is a large gap, of the order of 2 mm, between the tumour and neighbouring vasculature at the onset of angiogenesis. In this work we consider whether the assumption that there is a significant gap between the tumour and neighbouring vasculature is unique to intra-cornea tumour implants, or whether this characterises avascular tumour growth more generally. To do this we utilise a simple scaling argument, derive a multi-compartment model for tumour growth, and consider in vivo images. This analysis demonstrates that the corneal implant experiments and the corresponding mathematical models cannot generally be applied to a clinical setting.

Systems biology takes an interdisciplinary approach to the systematic study of complex interactions in biological systems. This approach seeks to decipher the emergent behaviors of complex systems rather than focusing only on their constituent properties. As an increasing number of examples illustrate the value of systems biology approaches to understand the initiation, progression, and treatment of cancer, systems biologists from across Europe and the United States hope for changes in the way their field is currently perceived among cancer researchers. In a recent EU-US workshop, supported by the European Commission, the German Federal Ministry for Education and Research, and the National Cancer Institute of the NIH, the participants discussed the strengths, weaknesses, hurdles, and opportunities in cancer systems biology.

Malignant tumours are characterised by a low, acidic extracellular pH (pHe) which facilitates invasion and metastasis. Previous research has proposed the potential benefits of manipulating systemic pHe, and recent experiments have highlighted the potential for buffer therapy to raise tumour pHe, prevent metastases, and prolong survival in laboratory mice. To examine the physiological regulation of tumour buffering and investigate how perturbations of the buffering system (via metabolic/respiratory disorders or changes in parameters) can alter tumour and blood pHe, we develop a simple compartmentalised ordinary differential equation model of pHe regulation by the View the MathML source buffering system. An approximate analytical solution is constructed and used to carry out a sensitivity analysis, where we identify key parameters that regulate tumour pHe in both humans and mice. From this analysis, we suggest promising alternative and combination therapies, and identify specific patient groups which may show an enhanced response to buffer therapy. In addition, numerical simulations are performed, validating the model against well-known metabolic/respiratory disorders and predicting how these disorders could change tumour pHe.

Background: Clinical positron emission tomography imaging has demonstrated the vast majority of human cancers exhibit significantly increased glucose metabolism when compared with adjacent normal tissue, resulting in an acidic tumour microenvironment. Recent studies demonstrated reducing this acidity through systemic buffers significantly inhibits development and growth of metastases in mouse xenografts. Methods: We apply and extend a previously developed mathematical model of blood and tumour buffering to examine the impact of oral administration of bicarbonate buffer in mice, and the potential impact in humans. We recapitulate the experimentally observed tumour pHe effect of buffer therapy, testing a model prediction in vivo in mice. We parameterise the model to humans to determine the translational safety and efficacy, and predict patient subgroups who could have enhanced treatment response, and the most promising combination or alternative buffer therapies. Results: The model predicts a previously unseen potentially dangerous elevation in blood pHe resulting from bicarbonate therapy in mice, which is confirmed by our in vivo experiments. Simulations predict limited efficacy of bicarbonate, especially in humans with more aggressive cancers. We predict buffer therapy would be most effectual: in elderly patients or individuals with renal impairments; in combination with proton production inhibitors (such as dichloroacetate), renal glomular filtration rate inhibitors (such as non-steroidal anti-inflammatory drugs and angiotensin-converting enzyme inhibitors), or with an alternative buffer reagent possessing an optimal pK of 7.1–7.2. Conclusion: Our mathematical model confirms bicarbonate acts as an effective agent to raise tumour pHe, but potentially induces metabolic alkalosis at the high doses necessary for tumour pHe normalisation. We predict use in elderly patients or in combination with proton production inhibitors or buffers with a pK of 7.1–7.2 is most promising.

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour.

Malignant tumours are characterised by a low, acidic extracellular pH (pHe) which facilitates invasion and metastasis. Previous research has proposed the potential benefits of manipulating systemic pHe, and recent experiments have highlighted the potential for buffer therapy to raise tumour pHe, prevent metastases, and prolong survival in laboratory mice. To examine the physiological regulation of tumour buffering and investigate how perturbations of the buffering system (via metabolic/respiratory disorders or changes in parameters) can alter tumour and blood pHe, we develop a simple compartmentalised ordinary differential equation model of pHe regulation by the HCO3-/CO2 buffering system. An approximate analytical solution is constructed and used to carry out a sensitivity analysis, where we identify key parameters that regulate tumour pHe in both humans and mice. From this analysis, we suggest promising alternative and combination therapies, and identify specific patient groups which may show an enhanced response to buffer therapy. In addition, numerical simulations are performed, validating the model against well-known metabolic/respiratory disorders and predicting how these disorders could change tumour pHe.

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour.