351 research outputs found

    Pattern Formation And Wound Healing

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    One of the main immediate challenges in the biomedical sciences is the synthesis of the vast amount of data now available at the molecular and cellular levels for development, regulation and repair. This, in turn, requires an understanding of the interaction and coordination of a myriad of complex inter-related processes occurring on very different spatial and temporal scales. Mathematics provides the obvious language in which to develop and interpret these interactions, and a number of mathematical models have already been proposed to account for certain observed biological and medical phenomena. Here, we consider two areas of modelling, namely spatial patterning, and wound healing, both sharing the common underlying processes of cells creating and responding to signalling cues

    Refining self-propelled particle models for collective behaviour

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    Swarming, schooling, flocking and herding are all names given to the wide variety of collective behaviours exhibited by groups of animals, bacteria and even individual cells. More generally, the term swarming describes the behaviour of an aggregate of agents (not necessarily biological) of similar size and shape which exhibit some emergent property such as directed migration or group cohesion. In this paper we review various individual-based models of collective behaviour and discuss their merits and drawbacks. We further analyse some one-dimensional models in the context of locust swarming. In specific models, in both one and two dimensions, we demonstrate how varying the parameters relating to how much attention individuals pay to their neighbours can dramatically change the behaviour of the group. We also introduce leader individuals to these models with the ability to guide the swarm to a greater or lesser degree as we vary the parameters of the model. We consider evolutionary scenarios for models with leaders in which individuals are allowed to evolve the degree of influence neighbouring individuals have on their subsequent motion

    Mesoscopic and continuum modelling of angiogenesis

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    Angiogenesis is the formation of new blood vessels from pre-existing ones in response to chemical signals secreted by, for example, a wound or a tumour. In this paper, we propose a mesoscopic lattice-based model of angiogenesis, in which processes that include proliferation and cell movement are considered as stochastic events. By studying the dependence of the model on the lattice spacing and the number of cells involved, we are able to derive the deterministic continuum limit of our equations and compare it to similar existing models of angiogenesis. We further identify conditions under which the use of continuum models is justified, and others for which stochastic or discrete effects dominate. We also compare different stochastic models for the movement of endothelial tip cells which have the same macroscopic, deterministic behaviour, but lead to markedly different behaviour in terms of production of new vessel cells.Comment: 48 pages, 13 figure
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