Cellular Potts model

The cellular Potts model (CPM) 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 the 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 generlized cellular automata (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.

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 Hamiltonain included adhesion energies, and volume and surface area constraints. We present it here without definition as an illustration and will discuss it in greater detail later: $$ \begin{align} H = & \sum_{\vec i, \vec j \text{neighbors}} J\left(\tau(\sigma(\vec i)), \tau(\sigma(\vec j))\right) \left(1 - \delta(\sigma(\vec i), \sigma(\vec j))\right) \\ + & \sum_{\vec i} \lambda_\text{volume}[V(\sigma(\vec i)) - V_\text{target}(\sigma(\vec i))]^2\\ + & \sum_{\vec i} \lambda_\text{surface}[S(\sigma(\vec i)) - S_\text{target}(\sigma(\vec i))]^2\\ \end{align} $$.

Many extensions to the original CPM Hamiltonian control cell behaviors including chemotaxis, elongation and haptotaxis.