The cellular Potts model is a latticebased computational modeling method to simulate the collective behavior of cellular structures. Other names for the CPM are extended largeq Potts model and Glazier and Graner model. First developed by James Glazier and Francois Graner in 1992 as an extension of largeq Potts model simulations of coarsening in metallic grains and soap froths, it has now been used to simulate foam, biological tissues, fluid flow and reactionadvectiondiffusionequations. In the CPM a generalized "cell" is a simplyconnected domain of pixels with the same cell id (formerly spin). A generalized cell may be a single soap bubble, 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 automaton (CA). Although it also closely resembles certain Monte Carlo methods, such as the largeq Potts model, many subtle differences separate the CPM from Potts models and standard spinbased Monte Carlo schemes.
The primary rule base has three components:

rules for selecting putative lattice updates

a Hamiltonian or effective energy function that is used for calculating the probability 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 adhesion, signalling, volume and surface area control, chemotaxis 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 offlattice.
The central component of the CPM is the definition of the Hamiltonian. The Hamiltonian is determined by the configuration of the cell lattice and perhaps other sublattices containing information such as the concentrations of chemicals. The original CPM Hamiltonian included adhesion energies, and volume and surface area constraints. We present a simple example for illustration:
\begin{align} H = & \sum_{ i, j \text{ neighbors}} J\left(\tau(\sigma(i)), \tau(\sigma( j))\right) \left(1  \delta(\sigma( i), \sigma( j))\right) \\ + & \sum_{ i} \lambda_\text{volume}[V(\sigma( i))  V_\text{target}(\sigma( i))]^2\\ + & \sum_{ i} \lambda_\text{surface}[S(\sigma(i))  S_\text{target}(\sigma( i))]^2 .\\ \end{align}
Where for cell σ, λ_{volume} is the volume constraint, V_{target} is the target volume, and for neighbouring lattice sites i and j, J is the boundary coefficient between two cells (σ,σ') of given types τ(σ),τ(σ'), and the boundary energy coefficients are symmetric: J[τ(σ),τ(σ')]=J[τ(σ'),τ(σ)], and the Kronecker delta is δ_{(x,y)}={1,x=y; 0,x≠y}.
Many extensions to the original CPM Hamiltonian control cell behaviors including chemotaxis, elongation and haptotaxis.
References

Graner, François; Glazier, James A. (1992). "Simulation of Biological Cell Sorting Using a TwoDimensional Extended Potts Model".

Chen, Nan; Glazier, James A.; Izaguirre, Jesus A.; Alber, Mark S. (2007). "A parallel implementation of the Cellular Potts Model for simulation of cellbased morphogenesis". Computer Physics Communications 176 (11–12): 670–681.
External links

James Glazier (professional website)

CompuCell3D, a CPM simulation environment: Sourceforge

SimTK

Notre Dame development site

Artificial Life model of multicellular morphogenesis with autonomously generated gradients for positional information using the Cellular Potts model
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