A new interpretation of the Keller-Segel model based on multiphase modelling

Byrne, Helen M.; Owen, Markus R.

doi:10.1007/s00285-004-0276-4

Cited by 50 publications

(42 citation statements)

References 59 publications

(104 reference statements)

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“…Briefly, these are (i) arguments based on Fourier's law and Fick's law [45], (ii) biased random walk approaches [78], (iii) interacting particle systems [97], (iv) transport equations [2] or [35], and (v) stochastic processes [86]. A more recent derivation from multi-phase flow modelling has been proposed by Byrne and Owen [15].…”

Section: Derivation and Applications Of Chemotaxis Modelsmentioning

confidence: 99%

“…In by far the majority of applications a constant diffusion coefficient is assumed, yet it is far more likely that this term should depend nonlinearly on the signal concentration and/or the cell density, as can be seen from derivations of Keller-Segel type systems through the various approaches mentioned in the introduction [15,45,80,92,95,96]. An explicit example is given in the formulation of the density-dependent chemotactic sensitivity models above, where a diffusion coefficient of the form D(u) = D (q − uq u ) was derived.…”

Section: (M5) Nonlinear Diffusionmentioning

confidence: 99%

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A user’s guide to PDE models for chemotaxis

2008

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Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399-415, 1970; 30:225-234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display "auto-aggregation", has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller-Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

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Section: Derivation and Applications Of Chemotaxis Modelsmentioning

confidence: 99%

Section: (M5) Nonlinear Diffusionmentioning

confidence: 99%

A user’s guide to PDE models for chemotaxis

2008

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“…These resulting equations are closely related to the Keller-Segel model for chemotaxis [13,14,19,22,26,30,33,41,46,47].…”

Section: Formulation Of the Modelmentioning

confidence: 99%

Statistical Models of Criminal Behavior: The Effects of Law Enforcement Actions

Jones

2010

Math. Models Methods Appl. Sci.

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Abstract:We extend an agent-based model of crime-pattern formation initiated in Short et al. by incorporating the effects of law enforcement agents. We investigate the effect that these agents have on the spatial distribution and overall level of criminal activity in a simulated urban setting. Our focus is on a two-dimensional lattice model of residential burglaries, where each site (target) is characterized by a dynamic attractiveness to burglary and where criminal and law enforcement agents are represented by random walkers. The dynamics of the criminal agents and the target-attractiveness field are, with certain modifications, as described in Short et al. Here the dynamics of enforcement agents are affected by the attractiveness field via a biasing of the walk, the detailed rules of which define a deployment strategy. We observe that law enforcement agents, if properly deployed, will in fact reduce the total amount of crime, but their relative effectiveness depends on the number of agents deployed, the deployment strategy used, and spatial distribution of criminal activity. For certain policing strategies, continuum PDE models can be derived from the discrete systems. The continuum models are qualitatively similar to the discrete systems at large system sizes.

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“…It has been demonstrated in these works that simple multiphase models are able to reproduce many of the features that characterize solid tumour growth, such as the development of a necrotic core behind a proliferating rim of viable tumour cells. Models of similar structure have also been employed in the study of tumour capsules (Lubkin and Jackson (2002)), the effect of external loading on tumour growth ), development of tumour cords and fibrosis (Preziosi and Tosin (2009a), Tosin and Preziosi (2010)), and cell migration (Byrne and Owen (2004)). Multiphase models have also been applied in the field of tissue engineering, where, typically, viscous fluid flows through a fixed solid phase corresponding to a porous scaffold (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

A multi-scale analysis of drug transport and response for a multi-phase tumour model

2016

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In this article we consider the spatial homogenization of a multiphase model for avascular tumour growth and response to chemotherapeutic treatment. The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscopic tumour growth, coupled to transport of drug and nutrient, that explicitly incorporates details of the structure and dynamics of the tumour at the microscale. In order to derive these equations we employ an asymptotic homogenization of a microscopic description under the assumption of strong interphase drag, periodic microstructure, and strong separation of scales. The resulting macroscale model comprises a Darcy flow coupled to a system of reaction-advection partial differential equations. The coupled growth, response, and transport dynamics on the tissue scale are investigated via numerical experiments for simple academic test cases of microstructural information and tissue geometry, in which we observe drug-and nutrient-regulated growth and response consistent with the anticipated dynamics of the macroscale system.

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A new interpretation of the Keller-Segel model based on multiphase modelling

Cited by 50 publications

References 59 publications

A user’s guide to PDE models for chemotaxis

A user’s guide to PDE models for chemotaxis

Statistical Models of Criminal Behavior: The Effects of Law Enforcement Actions

A multi-scale analysis of drug transport and response for a multi-phase tumour model

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