The 3D version of this simulator treats the liver as one well-stirred ("venous equilibrium") compartment — every molecule sees the same average concentration. This 2D version instead solves the parallel-tube model (Pang & Rowland, 1977): the sinusoid is a tube of normalized length L=1 along which concentration falls continuously as blood flows past a uniformly distributed bed of CYP450 enzyme.
Local ODE (position x runs 0→1 along the sinusoid):
Q · dC/dx = −(Vmax/L) · C(x) / (Km + C(x))
Parallel-tube extraction: E_PT = 1 − C(L)/Cu (C(L) from numeric integration)
Well-stirred (reference): CLint = Vmax/(Km+Cu); E_VE = CLint/(Q+CLint)
- The concentration curve above is obtained by integrating the ODE with RK4 across 120 steps — a genuine spatial simulation, not a flattened camera view of a 3D scene.
- Low dose (Cu ≪ Km): the ODE becomes linear, C(x) decays exponentially, and E_PT ≈ E_VE — the two models agree when kinetics are linear.
- High dose (Cu ≫ Km, saturating): downstream segments of the sinusoid see a lower local concentration than the inlet, so they operate at a less-saturated point on the Michaelis-Menten curve than the well-stirred model assumes for the whole liver — this is why the spatially-resolved model can predict a higher extraction ratio than the lumped one at the same nominal Vmax, Km, Q, Cu, a difference that is a well-documented finding in hepatic clearance pharmacokinetics.
- The heat strip beneath the curve shows local enzyme occupancy θ(x) = C(x)/(Km+C(x)) — green near the inlet, shifting toward red only where local concentration keeps the enzyme bed saturated.
- Flowing particles are individual molecules; each is removed at position x with a probability derived from the same local hazard rate that the ODE integrates, so their aggregate survival fraction should track the deterministic F_PT curve as more of them flow through.