Cells as spheroidal agents

waxMorph represents tissues as three-dimensional populations of spheroidal cellular agents.

Each cell \(i\) carries a center position \(x_i \in \mathbb{R}^3\), a radius \(r_i \in \mathbb{R}_+\), and a unit polarity vector \(p_i \in \mathbb{R}^3\) with \(\lVert p_i \rVert_2 = 1\). The cell volume follows from the radius \(v_i = \tfrac{4}{3}\pi r_i^3\), and the concentration \(c_{i, \mathcal{M}}\) of a molecule \(\mathcal{M}\) is its abundance divided by \(v_i\). The tissue state is the collection of all cells, \(S(t) = \{s_i(t)\}\), over the \(N(t)\) cells present at time \(t\).

In code, a minimal state comprises four arrays, with an optional fifth for cell type.

X

Center positions, shape [N, 3].

P

Polarity vectors, shape [N, 3]. Polarity enters the edge features and, in the forward simulator, the epithelial geometry constraints.

R

Radii, shape [N]. Radii induce the contact neighborhoods and set the characteristic length scale for packing and mechanics.

c

Signaling molecule concentrations, shape [N, num_molecules]. These form the node features for the emulator and diffuse over the contact graph.

CT

Optional integer cell-type labels. The forward simulator uses cell type to distinguish epithelial and mesenchymal interaction parameters.