Cells as spheroidal agents
==========================

waxMorph represents a tissue as a three-dimensional population of spheroidal
cellular agents.

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