Lattice-Gas Cellular Automaton Models for Biology: From Fluids to Cells
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Lattice-Gas Cellular Automaton Models for Biology: From Fluids to Cells Bastien Chopard • Rafik Ouared • Andreas Deutsch Haralambos Hatzikirou • Dieter Wolf-Gladrow
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Received: 24 June 2010 / Accepted: 5 July 2010 / Published online: 14 August 2010 Springer Science+Business Media B.V. 2010
Abstract Lattice-gas cellular automaton (LGCA) and lattice Boltzmann (LB) models are promising models for studying emergent behaviour of transport and interaction processes in biological systems. In this chapter, we will emphasise the use of LGCA/LB models and the derivation and analysis of LGCA models ranging from the classical example dynamics of fluid flow to clotting phenomena in cerebral aneurysms and the invasion of tumour cells. Keywords Lattice-gas cellular automaton Lattice Boltzmann model Discrete dynamical system Tumour invasion Blood clotting Collective behaviour
1 Introduction Cellular automata (CA) are discrete, local dynamical systems. They have been introduced by J. v. Neumann and S. Ulam in the 1950s as collective model of individual reproduction. During the last years it has become clear that CA have a much broader potential as models for physical, chemical and biological systems. In particular, CA models have now been proposed for a large number of biological applications in which one is interested in the emergence of collective macroscopic
B. Chopard (&) R. Ouared University of Geneva, Geneva, Switzerland e-mail: [email protected] A. Deutsch H. Hatzikirou Center for Information Services and High Performance Computing, Technische Universita¨t Dresden, Dresden, Germany D. Wolf-Gladrow Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany
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behaviours arising from the microscopic interaction of individual components— being it molecules, cells or organisms. However, there are drawbacks: currently, there exists a huge jungle of different rules for often the same or similar processes. Therefore, there is need for a specification and classification of CA rules. Wolfram made a promising start with the classification of 1D automata. Furthermore, analysis of CA models is still rather limited and often restricted to visual superficial inspection of simulation outcomes. In the same spirit as CA, lattice-gas cellular automaton (LGCA) and lattice Boltzmann (LB) models are promising models for studying transport and interaction processes in biological systems (Chopard and Droz 1998; Deutsch and Dormann 2005; Wolf-Gladrow Dieter 2000). In these models the update rule is split into two parts which are called collision (interaction) and propagation, respectively. The collision rule of LGCA can be compared with the update rule for CA in that it assigns new states to each cell based on the states of the sites in a local neighborhood. After the collision step the state of each node is propagated to a neighboring node. This split of the update rule guarantees propagation of quantities while keeping the rules simple. The desired behaviour, e.g. spatio-temp
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