Cellular Potts model

The cellular Potts model is a lattice-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 model simulations of coarsening in metallic grains and soap froths, it has now been used to simulate foam, biological tissues, fluid flow and reaction-advection-diffusion-equations. In the CPM a generalized "cell" is a simply-connected 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 large-q Potts model, many subtle differences separate the CPM from Potts models and standard spin-based Monte Carlo schemes.

The primary rule base has three components:

1. rules for selecting putative lattice updates
2. a Hamiltonian or effective energy function that is used for calculating the probability of accepting lattice updates.
3. 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 off-lattice.[citation needed]

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 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 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, λvolume is the volume constraint, Vtarget 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.