Hepatic Lobule Zonation: 2D Reaction-Diffusion Map

Sinusoidal Perfusion

Sinusoidal blood flow 1.0×

Wnt/β-catenin signalling strength 1.0×

Hepatotoxin Challenge

Toxin dose (CYP450-activated, e.g. APAP) 0

Hepatic glutathione (GSH) reserve 100

Lobule Readouts

O2 — zone 1 (periportal)

65 mmHg

O2 — zone 3 (pericentral)

6 mmHg

Necrotic hepatocytes

Zone 1Zone 2Zone 3

How it works

This is the 2D counterpart to the rotatable 3D lobule — same biology, genuinely different engine. Instead of evaluating a closed-form snapshot formula every frame, this simulator numerically time-steps a real advection–reaction PDE for oxygen along the sinusoid, so the O2 field actually has state and relaxes toward its new steady profile whenever you move the flow slider (watch for the "relaxing…" flag), rather than snapping instantly:

∂O2/∂t = −v·∂O2/∂x − k·O2,   O2(x=0,t) = 65 mmHg (portal, fixed)
v = blood flow,  k = 2.4 (O2 consumption-rate constant)

Independently, Wnt/β-catenin is modelled as a classic morphogen-gradient problem: secreted at fixed concentration at the central vein (x=1) and swept/degraded as it spreads toward the portal tract — the textbook source–transport–decay picture used for real diffusible morphogen gradients:

∂Wnt/∂t = +w·∂Wnt/∂x − λ·Wnt,   Wnt(x=1,t) = 100 (central vein, fixed)
w = signalling strength,  λ = 3.0 (decay-rate constant)

Both PDEs are integrated with explicit upwind finite differences (Δx=0.01, sub-stepped every animation frame at a CFL-stable Δt). At steady state they reduce to exactly the same exponential curves used by the 3D version — O2(x)=65·exp(−2.4x/v) and Wnt(x)=100·exp(−3(1−x)/w) — verified numerically to within <1 mmHg / <1.1 unit of the analytic curve — but getting there is now a genuine transient, not an instantaneous lookup.

Zonation index Z(x) = 0.5·(1 − O2(x)/65) + 0.5·(Wnt(x)/100)
CYP450 activity(x) = Z(x)^1.3
net injury(x) = clamp(toxin/100·CYP(x) − 0.55·GSH/100, 0, 1)

A hepatocyte with net injury above threshold necroses — reproducing zone-3 centrilobular necrosis, the classic biopsy pattern after an acetaminophen (APAP) overdose, and why glutathione-repleting N-acetylcysteine is the antidote.

⏱ relaxing toward new steady state…

Zone 1 — periportal Zone 3 — pericentral Necrotic hepatocyte Portal triad Central vein