Preface

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Preface

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On the foundations of cancer modelling: selected topics, speculations, & perspectives

This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution

Speed of reaction diffusion in embryogenesis

Reaction diffusion systems have been proposed as mechanisms for patterning during many stages of embryonic development. While much attention has been focused on the study of the steady state patterns formed and the robustness of pattern selection, much less is known about the time scales required for pattern formation. Studies of gradient formation by the diffusion of a single morphogen from a localized source have shown that patterning can occur on realistic time scales over distances of a millimeter or less. Reaction diffusion has the potential to give rise to patterns on a faster time scale, since all points in the domain can act as sources of morphogen. However, the speed at which patterning can occur has hitherto not been explored in depth. In this paper, we investigate this issue in specific reaction diffusion models and address the question of whether patterning via reaction diffusion is fast enough to be applicable to morphogenesis

Complex pattern formation in reaction diffusion systems with spatially-varying parameters

Spontaneous pattern formation in reaction–diffusion systems on a spatially homogeneous domain has been well studied. However, in embryonic development and elsewhere, pattern formation often takes place on a spatially heterogeneous background. We explore the effects of spatially varying parameters on pattern formation in one and two dimensions using the Gierer–Meinhardt reaction–diffusion model. We investigate the effect of the wavelength of a pre-pattern and demonstrate a novel form of moving pattern. We find that spatially heterogeneous parameters can both increase the range and complexity of possible patterns and enhance the robustness of pattern selection

Weak Interactions in Binary Mixtures of Chlorobenzene & Bromobenzene with o-, m- & p-Xylenes

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Achieving coexistence - Comment on "Modelling rain forest diversity:The role of competition by Bampfylde et al. (2005)

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Leaky vessels as a potential source of stromal acidification in tumours

Malignant tumours are characterised by higher rates of acid production and a lower extracellular pH than normal tissues. Previous mathematical modelling has indicated that the tumour-derived production of acid leads to a gradient of low pH in the interior of the tumour extending to a normal pH in the peritumoural tissue. This paper uses mathematical modelling to examine the potential of leaky vessels as an additional source of stromal acidification in tumours. We explore whether and to what extent increasing vascular permeability in vessels can lead to the breakdown of the acid gradient from the core of the tumour to the normal tissue, and a progressive acidification of the peritumoural stroma. We compare our mathematical simulations to experimental results found in vivo with a tumour implanted in the mammary fat pad of a mouse in a window chamber construct. We find that leaky vasculature can cause a net acidification of the normal tissue away from the tumour boundary, though not a progressive acidification over time as seen in the experiments. Only through progressively increasing the leakiness can the model qualitatively reproduce the experimental results. Furthermore, the extent of the acidification predicted by the mathematical model is less than as seen in the window chamber, indicating that although vessel leakiness might be acting as a source of acid, it is not the only factor contributing to this phenomenon. Nevertheless, tumour destruction of vasculature could result in enhanced stromal acidification and invasion, hence current therapies aimed at buffering tumour pH should also examine the possibility of preventing vessel disruption