The impact of exclusion processes on angiogenesis models
type: Item
instance of: scholarly article
title: The impact of exclusion processes on angiogenesis models
zbMATH Open document ID: 1406.35160
author: S. Pillay
author: H. M. Byrne
author: Philip K. Maini
published in: Journal of Mathematical Biology
publication date: 2018-12-21T00:00:00Z
review text: The authors study angiogenesis models and derive a class of nonlinear continuous PDE models from a lattice-based cellular automaton model by a mean-field approximation. More precisely, by letting \(N\) and \(E\) denote the densities of tip cells and endothelial cells, respectively, as well as \(c\) the concentration of the tumor angiogenic factor, the authors obtain in the spatially one-dimensional setting the model \[ \begin{cases} N_t = D(1-a_n N - B a_e E) N_{xx} - \chi \left( (N(1-N)c_x)_x + a_n N N_x c_x - B a_e E (N c_x)_x \right) \\ \qquad - \mu \left( a_n N^2 + B a_e NE \right) + \lambda c N(1-N-BE)^2, \\ E_t = \mu N(1-N + 2a_n N) - D(1-2a_n)N N_{xx} - \chi N \left( (1- a_n) c_x N_x + a_n (N c_x)_x \right).\end{cases}\tag{1} \] Here, \(a_n , a_e \in [0,1]\) account for tip cell and endothelical cell volume exclusion, respectively, in the following sense. If a tip cell tries to move to a site which is occupied by a tip cell or endothelial cell, then with probability \(1-a_n\) or \(1-a_e\) the move is aborted. The authors study on the one hand the model with \(B=0\), accounting for tip cell volume exclusion only, and on the other hand the model with \(B=1\), accounting for both types of volume exclusion. By numerical simulations the authors compare the behavior of these models with the corresponding cellular automaton model and with the PDE model from [\textit{H. M. Byrne} and \textit{M. A. J. Chaplain}, Bull. Math. Biol. 57, No. 3, 461--486 (1995; Zbl 0812.92011)]. It is discussed in how far the two variants of model (1) are able to capture the effects of volume exclusion and how appropriate parameter values can be estimated.
reviewed by: Christian Stinner
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 6996573
zbMATH Keywords: angiogenesis
zbMATH Keywords: cellular automaton
zbMATH Keywords: multiscale modeling
zbMATH Keywords: partial differential equations
zbMATH Keywords: volume exclusion
zbMATH Keywords: chemotaxis
describes a project that uses: MATLAB ODE suite
describes a project that uses: ode23s
describes a project that uses: Matlab
describes a project that uses: ode23
describes a project that uses: ode45
describes a project that uses: Ode15s
describes a project that uses: ode113
MaRDI profile type: Publication
full work available at URL: s00285-018-1214-1
OpenAlex ID: W2794335499
Wikidata QID: Q64948662
cites work: Continuous and discrete mathematical models of tumor-induced angiogenesis
cites work: Mathematical models for tumour angiogenesis: Numerical simulations and nonlinear wave solutions
cites work: A stochastic model for wound healing
cites work: Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development
cites work: Q4275296
cites work: On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds
cites work: An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds
cites work: On the derivation of approximations to cellular automata models and the assumption of independence
cites work: The importance of volume exclusion in modelling cellular migration
cites work: Q4265490
cites work: Mathematical modeling of tumor-induced angiogenesis
cites work: Modelling collective cell behaviour
cites work: Inference of cell-cell interactions from population density characteristics and cell trajectories on static and growing domains
cites work: A review of mathematical models for the formation of vascular networks
cites work: The MATLAB ODE Suite
cites work: Mesoscopic and continuum modelling of angiogenesis
cites work: Mathematical modelling of flow in 2D and 3D vascular networks: Applications to anti-angio\-genic and chemotherapeutic drug strategies
cites work: Q4151274
DOI: 10.1007/S00285-018-1214-1
Recommended article: A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: The Role of Randomness
Recommended article: On the mean field approximation of a stochastic model of tumour-induced angiogenesis
Recommended article: Stochastic modelling of tumour-induced angiogenesis
Recommended article: Q4677985
Recommended article: Mesoscopic and continuum modelling of angiogenesis
Recommended article: A discrete mathematical model for angiogenesis
Recommended article: A reinforced random walk model of tumour angiogenesis and anti-angiogenic strategies
Recommended article: Stochastic geometric models, and related statistical issues in tumour-induced angiogenesis
Recommended article: Angiogenesis model with Erlang distributed delays
description: scientific article; zbMATH DE number 6996573