Relationship between the reaction–diffusion master equation and particle tracking models

2008 ◽  
Vol 41 (6) ◽  
pp. 065003 ◽  
Author(s):  
Samuel A Isaacson
Keyword(s):  
Master Equation ◽  
Particle Tracking ◽  
Reaction Diffusion

scholarly journals Integrodifference master equation describing actively growing blood vessels in angiogenesis

2020 ◽  
Vol 21 (7-8) ◽  
pp. 705-713 ◽  
Author(s):  
Luis L. Bonilla ◽  
Manuel Carretero ◽  
Filippo Terragni
Keyword(s):  
Master Equation ◽  
Blood Vessels ◽  
Transition Probabilities ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Birth Process ◽  
Sink Term ◽  
System Of Particles ◽  
Vessel Network ◽  
Tip Cells

AbstractWe study a system of particles in a two-dimensional geometry that move according to a reinforced random walk with transition probabilities dependent on the solutions of reaction-diffusion equations (RDEs) for the underlying fields. A birth process and a history-dependent killing process are also considered. This system models tumor-induced angiogenesis, the process of formation of blood vessels induced by a growth factor (GF) released by a tumor. Particles represent vessel tip cells, whose trajectories constitute the growing vessel network. New vessels appear and may fuse with existing ones during their evolution. Thus, the system is described by tracking the density of active tips, calculated as an ensemble average over many realizations of the stochastic process. Such density satisfies a novel discrete master equation with source and sink terms. The sink term is proportional to a space-dependent and suitably fitted killing coefficient. Results are illustrated studying two influential angiogenesis models.

scholarly journals Hierarchical algorithm for the reaction-diffusion master equation

2020 ◽  
Vol 152 (3) ◽  
pp. 034104
Author(s):  
Stefan Hellander ◽  
Andreas Hellander
Keyword(s):  
Master Equation ◽  
Reaction Diffusion ◽  
Hierarchical Algorithm

scholarly journals A convergent reaction-diffusion master equation

2013 ◽  
Vol 139 (5) ◽  
pp. 054101 ◽  
Author(s):  
Samuel A. Isaacson
Keyword(s):  
Master Equation ◽  
Reaction Diffusion
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