ABSTRACT

All cellular Potts model approaches include a list of objects, a description of their interactions, and rules for their dynamics. The CPM domains are d-dimensional lattices Ω ⊆ Rd, where d = 1, 2, 3. The term lattice defines a regular repeated graph, formed by identical d-dimensional closed grid sites x ∈ Rd, and characterized by periodic or fixed boundary conditions in each direction. The volumetric extension of Ω is equal to the total number of its sites, that therefore represent the basic unit of length of the system. Each site x ∈ Ω is uniquely identified by its location and is labeled by an integer, σ(x) ∈ N, where σ can be interpreted as a degenerate spin value coming from the original Ising approach [85, 199, 315]; see Figure 1.1(A). With an abuse of notation, x also usually denotes the closed elementary spatial region (e.g., the voxel) centered in x. As classically adopted in CPM models, the border of a lattice site x is denoted by ∂x, one of its neighbors by x′, while its overall neighborhood by Ω

′ x, i.e., Ω

′ x = {x′ ∈ Ω : x′ is a neighbor of x}.