The 3D version of this sim tracks liver mass as a single whole-organ number obeying a logistic ODE. This 2D companion instead simulates the actual spatial process: a cross-section of liver tissue where local hepatocyte density M(x,y) both proliferates and diffuses outward from the remaining remnant into the resected region — a genuine reaction-diffusion (Fisher–KPP) system, the same class of equation used to model real invading-cell fronts in wound healing and tumor growth.
∂M/∂t = D·∇²M + k·H(t)·M·(1 − M) (hepatostat feedback ON)
∂M/∂t = D·∇²M + k·H(t)·M·0.55 (feedback OFF — unchecked)
H(t) = 1 + gf·e^(−t/τ) (HGF/EGF pulse, same as 3D)
Because M both spreads (D·∇²M) and grows locally, the regenerating tissue advances as a genuine traveling wave — not an instant reveal. Classical Fisher–KPP theory predicts this front's asymptotic speed analytically: v = 2·√(D·k·H), and this simulation's own numerics were checked against that exact formula (see verification script) before publishing — the front you watch sweep across the tissue is really solving that equation, not an animation timed to look right.
- Resection extent — sets the area of the remnant "seed" region (scaled so area ∝ radius², the 2D analogue of the 3D sim's volume ∝ radius³ reveal).
- Growth-factor strength — scales H(t), which raises the local reaction rate k·H and therefore speeds up the wave itself (v ∝ √H), not just the final mass.
- Hepatostat feedback — the (1 − M) brake stops local growth once a cell's neighborhood reaches full density; switch it off to watch density blow past 100% (capped visually at 130%) with no size-sensing checkpoint.
- Heatmap color: dark = resected/empty tissue, tan = restored quiescent hepatocyte density, bright green = the actively advancing proliferative front (where local growth rate is highest right now); the dashed outline marks the full pre-resection liver boundary — the hepatostat's 100% target.
Because regrowth here is a genuine spatial wave rather than an instantaneous global reveal, the whole-organ average mass lags slightly behind the pure 0-D ODE the 3D sim solves — a real and measurable consequence of the fact that regeneration must physically propagate through tissue, not happen everywhere simultaneously.