Continuous and discrete mathematical models of tumor-induced angiogenesis
type: Item
instance of: scholarly article
title: Continuous and discrete mathematical models of tumor-induced angiogenesis
zbMATH Open document ID: 0923.92011
published in: Bulletin of Mathematical Biology
publication date: 1999-11-01T00:00:00Z
review text: Angiogenesis is the formation of blood vessels whereby capillary sprouts are formed in response to chemical stimuli. Subsequently they grow, develop and organize themselves into a dendritic structure. A continuous and a discrete probabilistic mathematical model is presented for tumor-induced angiogenesis focussing on three important variables, the endothelial-cell density \(n\), the tumor angiogenic factor (TAF) concentration \(c\), and the fibronectin \(f\). The continuous model is presented as a system of three coupled nonlinear partial differential equations with a no-flux boundary condition where the density \(n\) is given as \[ \partial n/\partial t=\text{ random motility -- chemotaxis -- hepatotaxis} \] \[ \partial f/\partial t=\beta n-\gamma nf,\quad\partial c/\partial t=\eta n c. \] The aims of the model are to examine the relative importance of chemotaxis and haptotaxis in governing the migration of the endothelial cells, to examine capillary network formation with and without proliferation of endothelial cells, and to produce theoretical capillary network structures, from a model based on sound physiological principles with realistic parameter values, which are morphologically similar to those observed in vivo. It is possible to analyze these computed structures for quantitative information such as network expansion rates, loop formation and overall network architecture. Numerical solutions are obtained from a finite approximation of the system with biologically based boundary and initial conditions. The spatiotemporal evolution of the endothelial cell density and fibronectin uptake is determined from numerical simulation under a variety of chemotaxis/hepatotaxis scenarios producing a wide range of spatiotemporal behavior. On the basis of this continuous model and using the Euler finite difference approximation the authors formulate a discrete model and derive movement probabilities for an individual endothelial cell located at a sprout tip. New capillary …
reviewed by: Lutz Edler
Mathematics Subject Classification ID: 92C50
Mathematics Subject Classification ID: 92C15
Mathematics Subject Classification ID: 92B05
zbMATH DE Number: 1281787
zbMATH Keywords: Euler difference equation
zbMATH Keywords: nonlinear partial differential equations
zbMATH Keywords: angiogenesis
zbMATH Keywords: chemotaxis
Wikidata QID: Q52236194
author: Alexander R. A. Anderson
author: M. A. J. Chaplain
MaRDI profile type: Publication
full work available at URL: bulm.1998.0042
OpenAlex ID: W2048195900
DOI: 10.1006/BULM.1998.0042
description: scientific article; zbMATH DE number 1281787