The
cellular Potts model is a
latticeIn physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...
-based computational modeling method to simulate the collective behavior of cellular structures. Other names for the CPM are
extended large-q Potts model and
Glazier and Graner model. First developed by James Glazier and Francois Graner in 1992 as an extension of large-q
Potts modelIn statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
simulations of coarsening in metallic grains and soap froths, it has now been used to simulate
foamThe most general definition of foam is a substance that is formed by trapping many gas bubbles in a liquid or solid. It can also refer to anything that is analogous to such a phenomenon, such as quantum foam. Often the term is used in reference to polyurethane foam , XPS foam, Polystyrene, or many...
, biological tissues, fluid flow and reaction-advection-diffusion-equations.
The
cellular Potts model is a
latticeIn physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...
-based computational modeling method to simulate the collective behavior of cellular structures. Other names for the CPM are
extended large-q Potts model and
Glazier and Graner model. First developed by James Glazier and Francois Graner in 1992 as an extension of large-q
Potts modelIn statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
simulations of coarsening in metallic grains and soap froths, it has now been used to simulate
foamThe most general definition of foam is a substance that is formed by trapping many gas bubbles in a liquid or solid. It can also refer to anything that is analogous to such a phenomenon, such as quantum foam. Often the term is used in reference to polyurethane foam , XPS foam, Polystyrene, or many...
, biological tissues, fluid flow and reaction-advection-diffusion-equations. In the CPM a generalized "cell" is a simply-connected
domain-General:* some kind of territory , such as a demesne or a realm* a field of study* public domain, a body of works and knowledge without proprietary interest...
of pixels with the same
cell id (formerly
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
). A generalized cell may be a single
soap bubbleA soap bubble is a very thin film of soapy water that forms a sphere with an iridescent surface. Soap bubbles usually last for only a few moments before bursting, either on their own or on contact with another object. Before they pop on their own, the bubble itself usually starts to thin, then it...
, an entire biological cell, part of a biological cell, or even a region of fluid.
The CPM is evolved by updating the cell lattice one pixel at a time based on a set of probabilistic rules. In this sense, the CPM can be thought of as a generalized
cellular automatonA cellular automaton is a discrete model studied in computability theory, mathematics, physics, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off". The grid can be in any finite number of...
(
CA). Although it also closely resembles certain Monte Carlo methods, such as the large-q
Potts modelIn statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
, many subtle differences separate the CPM from Potts models and standard spin-based Monte Carlo schemes.
The primary rule base has three components:
- rules for selecting putative lattice updates
- a Hamiltonian
Hamiltonian may refer toIn mathematics:* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian * Hamiltonian matrix...
or effective energy function that is used for calculating the probabilityProbability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy...
of accepting lattice updates.
- additional rules not included in 1. or 2..
The CPM can also be thought of as an
agent based method in which cell agents evolve, interact via behaviors such as
adhesionCellular adhesion is the binding of a cell to a surface, extracellular matrix or another cell using cell adhesion molecules such as selectins, integrins, and cadherins.- Process :...
, signalling, volume and surface area control,
chemotaxisChemotaxis is the phenomenon in which bodily cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment. This is important for bacteria to find food by swimming towards the highest concentration of food molecules, or...
and proliferation. Over time, the CPM has evolved from a specific model to a general framework with many extensions and even related methods that are entirely or partially off-lattice.
The central component of the CPM is the definition of the
HamiltonianHamiltonian may refer toIn mathematics:* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian * Hamiltonian matrix...
. The Hamiltonian is determined by the configuration of the cell lattice and perhaps other sub-lattices containing information such as the concentrations of chemicals. The original CPM Hamiltonian included adhesion energies, and volume and surface area constraints. We present it here without definition as an illustration:
.
Many extensions to the original CPM Hamiltonian control cell behaviors including
chemotaxisChemotaxis is the phenomenon in which bodily cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment. This is important for bacteria to find food by swimming towards the highest concentration of food molecules, or...
, elongation and
haptotaxisHaptotaxis is the directional motility or outgrowth of cells, e.g. in the case of axonal outgrowth, usually up a gradient of cellular adhesion sites or substrate-bound chemoattractants...
.
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