The 3D version tracks each antigen particle as an individual random walker. This 2D version instead solves the continuum limit of that same process directly: a concentration field C(x,y,t) on a finite-volume grid, obeying the advection–diffusion–decay (Fokker–Planck) equation
∂C/∂t = D·∇²C + ∇·(spring·x·C) − k·C
Diffusion D and clearance k are the same Fickian/first-order terms as the 3D model; the drift term (spring·x, pulling mass back toward the injection origin) is the field-equation counterpart of the per-particle restoring nudge that models an adjuvant physically anchoring antigen in place — the depot effect. Fluxes are computed at each cell face (upwind for advection, centered for diffusion) so total mass is conserved exactly except for the explicit −k·C decay term, which is verified numerically against the analytic M(t)=M₀·e−kt solution.
Chemotaxis here reads the actual solved gradient of the field: each recruited cell samples ∇C at its own grid cell via centered finite differences and steers along that direction at a fixed speed scaled by χ, directly implementing v_cell = χ·∇C/|∇C| — literally "climb the concentration gradient" rather than the 3D model's shortcut of heading straight for a precomputed centroid. Cells that reach the depot deplete grid mass locally (phagocytosis) at a capped rate, converting captured mass into a "carried antigen" count; once a cell carries enough, it primes and drifts to the lymph node instead.
- Adjuvant strength — lowers D and k and raises the restoring drift together (stronger depot, slower clearance, more sustained recruitment) — identical parameter mapping to the 3D model.
- Antigen dose — sets the total initial mass ∫C dA of the depot.
- Chemotactic sensitivity χ — scales how fast recruited cells move once they detect the gradient.
- Simulation speed — the PDE is sub-stepped internally at a stability-safe Δt regardless of display speed, so raising simulation speed never destabilizes the solver.