Comparative analysis of continuum angiogenesis models
Autor: | Philip K. Maini, Helen M. Byrne, Hirokazu Ninomiya, W. Duncan Martinson |
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Rok vydání: | 2020 |
Předmět: |
Computer science
Neovascularization Physiologic Context (language use) 35Q92 Perturbation methods 01 natural sciences Models Biological Domain (mathematical analysis) Article 010305 fluids & plasmas Coarse-grained models 03 medical and health sciences angiogenesis Cell Movement 0103 physical sciences Phenomenological model Discrete-to-continuum modelling 41A60 Humans Snail-trail model Statistical physics 030304 developmental biology Pointwise 0303 health sciences Partial differential equation Continuum (measurement) Neovascularization Pathologic Applied Mathematics Chemotaxis 92C37 Agricultural and Biological Sciences (miscellaneous) 92C17 Connection (mathematics) Agent-based modelling Nonlinear system Modeling and Simulation |
Zdroj: | Journal of Mathematical Biology |
ISSN: | 1432-1416 |
Popis: | Although discrete approaches are increasingly employed to model biological phenomena, it remains unclear how complex, population-level behaviours in such frameworks arise from the rules used to represent interactions between individuals. Discrete-to-continuum approaches, which are used to derive systems of coarse-grained equations describing the mean-field dynamics of a microscopic model, can provide insight into such emergent behaviour. Coarse-grained models often contain nonlinear terms that depend on the microscopic rules of the discrete framework, however, and such nonlinearities can make a model difficult to mathematically analyse. By contrast, models developed using phenomenological approaches are typically easier to investigate but have a more obscure connection to the underlying microscopic system. To our knowledge, there has been little work done to compare solutions of phenomenological and coarse-grained models. Here we address this problem in the context of angiogenesis (the creation of new blood vessels from existing vasculature). We compare asymptotic solutions of a classical, phenomenological “snail-trail” model for angiogenesis to solutions of a nonlinear system of partial differential equations (PDEs) derived via a systematic coarse-graining procedure (Pillay et al. in Phys Rev E 95(1):012410, 2017. https://doi.org/10.1103/PhysRevE.95.012410). For distinguished parameter regimes corresponding to chemotaxis-dominated cell movement and low branching rates, both continuum models reduce at leading order to identical PDEs within the domain interior. Numerical and analytical results confirm that pointwise differences between solutions to the two continuum models are small if these conditions hold, and demonstrate how perturbation methods can be used to determine when a phenomenological model provides a good approximation to a more detailed coarse-grained system for the same biological process. |
Databáze: | OpenAIRE |
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