On the derivation of approximations to cellular automata models and the assumption of independence

Davies, Kale; Green, J. E. F.; Bean, Nigel; Binder, Benjamin J.; Ross, Joshua V.

doi:10.1016/j.mbs.2014.04.004

Cited by 11 publications

(19 citation statements)

References 24 publications

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“…It has been shown that in the absence of reproduction this approximation provides the exact marginal probabilities of occupancy [18]. Numerous approximations which have been considered in the literature, mostly of the partial differential equation continuum type, are further approximations to this model [17,18,[27][28][29][30]. Consistent with the findings of these studies, the independence approximation is shown to be accurate only when the rate of reproduction is at least two orders of magnitude smaller than the rate of motility.…”

Section: Introductionmentioning

confidence: 60%

“…The CA framework consists of occupied sites on a lattice (agents) that represent individuals (e.g., plants or cells), and at any given time each site can only be occupied by a single agent, which is a spatial exclusion process [16,17]. We specify continuous-time and one-dimensional discrete space CA rules to simulate the motility and reproduction mechanisms in the invasion process [18].…”

Section: Introductionmentioning

confidence: 99%

“…The first approximation-the independence approximation-assumes independence (i.e., the mean-field assumption) between the occupancy of sites in the CA model. It has been shown that in the absence of reproduction this approximation provides the exact marginal probabilities of occupancy [18]. Numerous approximations which have been considered in the literature, mostly of the partial differential equation continuum type, are further approximations to this model [17,18,[27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

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Approximating spatially exclusive invasion processes

Ross

Binder

2014

Phys. Rev. E

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A number of biological processes, such as invasive plant species and cell migration, are composed of two key mechanisms: motility and reproduction. Due to the spatially exclusive interacting behavior of these processes a cellular automata (CA) model is specified to simulate a one-dimensional invasion process. Three (independence, Poisson, and 2D-Markov chain) approximations are considered that attempt to capture the average behavior of the CA. We show that our 2D-Markov chain approximation accurately predicts the state of the CA for a wide range of motility and reproduction rates.

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximating spatially exclusive invasion processes

Ross

Binder

2014

Phys. Rev. E

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“…Furthermore, as we use small, non-zero, finite values for h and in the CA model simulations, and in the definitions of the continuum model parameters, retaining the terms and neglecting higher-terms (used in Davies et al. 2014; Ross et al. 2015; Hywood et al.…”

Section: The Continuum Modelmentioning

confidence: 99%

“…2010; Baker and Simpson 2010; Davies et al. 2014). Further, the mean-field approximation may break down due to anastomosis, as the chances of an anastomosis event increases when agents are in close proximity to each other (Pillay et al.…”

Section: Mean-field Approximation and Behavior Of Pde Modelsmentioning

confidence: 99%

The impact of exclusion processes on angiogenesis models

2018

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Angiogenesis is the process by which new blood vessels form from existing vessels. During angiogenesis, tip cells migrate via diffusion and chemotaxis, new tip cells are introduced through branching, loops form via tip-to-tip and tip-to-sprout anastomosis, and a vessel network forms as endothelial cells, known as stalk cells, follow the paths of tip cells (a process known as the snail-trail). Using a mean-field approximation, we systematically derive one-dimensional non-linear continuum models from a lattice-based cellular automaton model of angiogenesis in the corneal assay, explicitly accounting for cell volume. We compare our continuum models and a well-known phenomenological snail-trail model that is linear in the diffusive, chemotactic and branching terms, with averaged cellular automaton simulation results to distinguish macroscale volume exclusion effects and determine whether linear models can capture them. We conclude that, in general, both linear and non-linear models can be used at low cell densities when single or multi-species exclusion effects are negligible at the macroscale. When cell densities increase, our non-linear model should be used to capture non-linear tip cell behavior that occurs when single-species exclusion effects are pronounced, and alternative models should be derived for non-negligible multi-species exclusion effects.

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Morphological Stability for in silico Models of Avascular Tumors

Blom,

Engblom

2024

Bull Math Biol

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The landscape of computational modeling in cancer systems biology is diverse, offering a spectrum of models and frameworks, each with its own trade-offs and advantages. Ideally, models are meant to be useful in refining hypotheses, to sharpen experimental procedures and, in the longer run, even for applications in personalized medicine. One of the greatest challenges is to balance model realism and detail with experimental data to eventually produce useful data-driven models. We contribute to this quest by developing a transparent, highly parsimonious, first principle in silico model of a growing avascular tumor. We initially formulate the physiological considerations and the specific model within a stochastic cell-based framework. We next formulate a corresponding mean-field model using partial differential equations which is amenable to mathematical analysis. Despite a few notable differences between the two models, we are in this way able to successfully detail the impact of all parameters in the stability of the growth process and on the eventual tumor fate of the stochastic model. This facilitates the deduction of Bayesian priors for a given situation, but also provides important insights into the underlying mechanism of tumor growth and progression. Although the resulting model framework is relatively simple and transparent, it can still reproduce the full range of known emergent behavior. We identify a novel model instability arising from nutrient starvation and we also discuss additional insight concerning possible model additions and the effects of those. Thanks to the framework’s flexibility, such additions can be readily included whenever the relevant data become available.

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On the derivation of approximations to cellular automata models and the assumption of independence

Cited by 11 publications

References 24 publications

Approximating spatially exclusive invasion processes

Approximating spatially exclusive invasion processes

The impact of exclusion processes on angiogenesis models

Morphological Stability for in silico Models of Avascular Tumors

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