Mesoscopic and continuum modelling of angiogenesis
- 952 Downloads
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.
KeywordsAngiogenesis Stochastic models Master equation Mesoscopic models Reaction–diffusion system
Mathematics Subject Classification (2000)92C17 Cell movement (chemotaxis, etc.) 60J70 Applications of Brownian motions 35Q92 PDEs in connection with biology and other natural sciences
This publication was based on work supported in part by Award No KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). TA and PG gratefully acknowledge the Spanish Ministry for Science and Innovation (MICINN) for funding under grant MTM2011-29342 and Generalitat de Catalunya for funding under grant 2009SGR345. PKM was partially supported by the National Cancer Institute, National Institutes of Health grant U54CA143970.
- Billy F, Ribba B, Saut O, Morre-Trouilhet H, Colin T, Bresch D, Boissel JP, Grenier E, Flandrois JP (2009) A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy. J Theor Biol 260(4):545–562CrossRefMathSciNetGoogle Scholar
- Das A, Lauffenburger D, Asada H, Kamm RD (2010) A hybrid continuum-discrete modelling approach to predict and control angiogenesis: analysis of combinatorial growth factor and matrix effects on vessel-sprouting morphology. Philos Trans R Soc A Math Phys Eng Sci 368(1921):2937–2960CrossRefMATHMathSciNetGoogle Scholar
- d’Onofrio A, Gandolfi A (1999) Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al (1999). Math Biosci 191(2):159–184Google Scholar
- Erban R, Chapman J, Maini P (2007) A practical guide to stochastic simulations of reaction–diffusion processes. arXiv:0704.1908 (preprint)
- Falconer K (2007) Fractal geometry: mathematical foundations and applications. Wiley, New YorkGoogle Scholar
- Hahnfeldt P, Panigrahy D, Folkman J, Hlatky L (1999) Tumor development under angiogenic signaling a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res 59(19):4770–4775Google Scholar
- Mac Gabhann F, Popel AS (2004) Model of competitive binding of vascular endothelial growth factor and placental growth factor to vegf receptors on endothelial cells. Am J Physiol Heart Circ Physiol 286(1):H153–H164Google Scholar
- Mac Gabhann F, Ji JW, Popel AS (2006) Computational model of vascular endothelial growth factor spatial distribution in muscle and pro-angiogenic cell therapy. PLoS Comput Biol 2(9):e127Google Scholar
- Van Kampen NG (1992) Stochastic Processes in Physics and Chemistry, vol 1. North Holland, AmsterdamGoogle Scholar