Stochastic resonance in a suspension of magnetic dipoles under shear flow

We show that a magnetic dipole in a shear flow under the action of an oscillating magnetic field displays stochastic resonance in the linear response regime. To this end, we compute the classical quantifiers of stochastic resonance, i.e. the signal to noise ratio, the escape time distribution, and the mean first passage time. We also discuss limitations and role of the linear response theory in its applications to the theory of stochastic resonance.Comment: 17 pages, 5 figures, approved for publication in PR

A design principle for vascular beds: the effects of complex blood rheology

We propose a design principle that extends Murray's original optimization principle for vascular architecture to account for complex blood rheology. Minimization of an energy dissipation function enables us to determine how rheology affects the morphology of simple branching networks. The behavior of various physical quantities associated with the networks, such as the wall shear stress and the flow velocity, is also determined. Our results are shown to be qualitatively and quantitatively compatible with independent experimental observations and simulations

A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells

The evolution of the cell-cycle is known to be influenced by environmental conditions, including lack of extracellular oxygen (hypoxia). Notably, hypoxia appears to have different effects on normal and cancer cells. Whereas both experience hypoxia-induced arrest of the G1 phase of the cell-cycle (i.e. delay in the transition through the restriction point), experimental evidence suggests that only cancer cells undergo hypoxia-induced quiescence (i.e. the transition of the cell to a latent state in which most of the cell functions, including proliferation, are suspended). Here, we extend a model for the cell-cycle due to Tyson and Novak (J. Theor. Biol. 210 (2001) 249) to account for the action of the protein p27. This protein, whose expression is upregulated under hypoxia, inhibits the activation of the cyclin dependent kinases (CDKs), thus preventing DNA synthesis and delaying the normal progression through the cell-cycle. We use a combination of numerical and analytic techniques to study our model. We show that it reproduces many features of the response to hypoxia of normal and cancer cells, as well as generating experimentally testable predictions. For example our model predicts that cancer cells can undergo quiescence by increasing their levels of p27, whereas for normal cells p27 expression decreases when the cellular growth rate increases

Towards whole-organ modelling of tumour growth

Multiscale approaches to modelling biological phenomena are growing rapidly. We present here some recent results on the formulation of a theoretical framework which can be developed into a fully integrative model for cancer growth. The model takes account of vascular adaptation and cell-cycle dynamics. We explore the effects of spatial inhomogeneity induced by the blood flow through the vascular network and of the possible effects of p27 on the cell cycle. We show how the model may be used to investigate the efficiency of drug-delivery protocols

Periodic Modulation Induced Increase of Reaction Rates in Autocatalytic Systems

We propose a new mechanism to increase the reactions ratesin multistable autocatalytic systems. The mechanism is based upon the possibility for the enhancement of the response of the system due to the cooperative behavior between the noise and an external periodic modulation. In order to illustrate this feature we compute the reaction velocities for the particular case of the Sel'Kov model, showing that they increase significantly when the periodic modulation is introduced. This behavior originates from the existence of a minimum in the mean first passage time, one of the signatures of stochastic resonance.Comment: Submitted to J. Chem. Phy

Cancer modelling: Getting to the heart of the problem

Paradoxically, improvements in healthcare that have enhanced the life expectancy of humans in the Western world have, indirectly, increased the prevalence of certain types of cancer such as prostate and breast. It remains unclear whether this phenomenon should be attributed to the ageing process itself or the cumulative effect of prolonged exposure to harmful environmental stimuli such as ultraviolet light, radiation and carcinogens (Franks and Teich, 1988). Equally, there is also compelling evidence that certain genetic abnormalities can predispose individuals to specific cancers (Ilyas et al., 1999). The variety of factors that have been implicated in the development of solid tumours stems, to a large extent, from the fact that ‘cancer’ is a generic term, often used to characterize a series of disorders that share common features. At this generic level of description, cancer may be viewed as a cellular disease in which controls that usually regulate growth and maintain homeostasis are disrupted. Cancer is typically initiated by genetic mutations that lead to enhanced mitosis of a cell lineage and the formation of an avascular tumour. Since it receives nutrients by diffusion from the surrounding tissue, the size of an avascular tumour is limited to several millimeters in diameter. Further growth relies on the tumour acquiring the ability to stimulate the ingrowth of a new, circulating blood supply from the host vasculature via a process termed angiogenesis (Folkman, 1974). Once vascularised, the tumour has access to a vast nutrient source and rapid growth ensues. Further, tumour fragments that break away from the primary tumour, on entering the vasculature, may be transported to other organs in which they may establish secondary tumours or metastases that further compromise the host. Invasion is another key feature of solid tumours whereby contact with the tissue stimulates the production of enzymes that digest the tissue, liberating space into which the tumour cells migrate. Thus, cancer is a complex, multiscale process. The spatial scales of interest range from the subcellular level, to the cellular and macroscopic (or tissue) levels while the timescales may vary from seconds (or less) for signal transduction pathways to months for tumour doubling times The variety of phenomena involved, the range of spatial and temporal scales over which they act and the complex way in which they are inter-related mean that the development of realistic theoretical models of solid tumour growth is extremely challenging. While there is now a large literature focused on modelling solid tumour growth (for a review, see, for example, Preziosi, 2003), existing models typically focus on a single spatial scale and, as a result, are unable to address the fundamental problem of how phenomena at different scales are coupled or to combine, in a systematic manner, data from the various scales. In this article, a theoretical framework will be presented that is capable of integrating a hierarchy of processes occurring at different scales into a detailed model of solid tumour growth (Alarcon et al., 2004). The model is formulated as a hybrid cellular automaton and contains interlinked elements that describe processes at each spatial scale: progress through the cell cycle and the production of proteins that stimulate angiogenesis are accounted for at the subcellular level; cell-cell interactions are treated at the cellular level; and, at the tissue scale, attention focuses on the vascular network whose structure adapts in response to blood flow and angiogenic factors produced at the subcellular level. Further coupling between the different spatial scales arises from the transport of blood-borne oxygen into the tissue and its uptake at the cellular level. Model simulations will be presented to illustrate the effect that spatial heterogeneity induced by blood flow through the vascular network has on the tumour’s growth dynamics and explain how the model may be used to compare the efficacy of different anti-cancer treatment protocols

A cellular automaton model for tumour growth in inhomogeneous environment

Most of the existing mathematical models for tumour growth and tumour-induced angiogenesis neglect blood flow. This is an important factor on which both nutrient and metabolite supply depend. In this paper we aim to address this shortcoming by developing a mathematical model which shows how blood flow and red blood cell heterogeneity influence the growth of systems of normal and cancerous cells. The model is developed in two stages. First we determine the distribution of oxygen in a native vascular network, incorporating into our model features of blood flow and vascular dynamics such as structural adaptation, complex rheology and red blood cell circulation. Once we have calculated the oxygen distribution, we then study the dynamics of a colony of normal and cancerous cells, placed in such a heterogeneous environment. During this second stage, we assume that the vascular network does not evolve and is independent of the dynamics of the surrounding tissue. The cells are considered as elements of a cellular automaton, whose evolution rules are inspired by the different behaviour of normal and cancer cells. Our aim is to show that blood flow and red blood cell heterogeneity play major roles in the development of such colonies, even when the red blood cells are flowing through the vasculature of normal, healthy tissue

Cancer disease: integrative modelling approaches

Cancer is a complex disease in which a variety of phenomena interact over a wide range of spatial and temporal scales. In this article a theoretical framework will be introduced that is capable of linking together such processes to produce a detailed model of vascular tumour growth. The model is formulated as a hybrid cellular automaton and contains submodels that describe subcellular, cellular and tissue level features. Model simulations will be presented to illustrate the effect that coupling between these different elements has on the tumour's evolution and its response to chemotherapy

Mathematical modelling of angiogenesis and vascular adaptation

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The use of hybrid cellular automaton models for improving cancer therapy, In Proceedings, Cellular Automata: 6th International Conference on Cellular Automata for Research and Industry, ACRI 2004, Amsterdam, The Netherlands, eds P.M.A. Sloot, B. Chopard, A.G. Hoekstra

The Hybrid Cellular Automata (HCA) modelling framework can be an efficient approach to a number of biological problems, particularly those which involve the integration of multiple spatial and temporal scales. As such, HCA may become a key modelling tool in the development of the so-called intergrative biology. In this paper, we first discuss HCA on a general level and then present results obtained when this approach was implemented in cancer research