Blood flow through each vessel segment is modeled with the Hagen–Poiseuille law, exactly as in lumped-parameter hemodynamic models used to study peripheral artery disease (PAD):
R = 8ηL / (πr⁴) (vascular resistance)
Q = ΔP / R (flow, Ohm's-law analog)
The narrowed native artery and the collateral vessels form a parallel resistor network bridging the diseased segment: 1/Rtotal = 1/Rstenosis + Σ 1/Rcollateral,i. Because resistance scales as 1/r⁴, a severe stenosis raises the main pathway's resistance enormously — and because it is fourth-power, even a modest collateral carries flow disproportionate to its size once the stenosis is tight enough.
- Stenosis / occlusion — narrows the diseased segment's lumen radius, driving its resistance up as 1/r⁴.
- Collateral vessels — how many parallel bypass channels the network has, each with its own resistance.
- Collateral caliber — the radius of each collateral vessel; because R ∝ 1/r⁴, doubling caliber cuts each collateral's resistance sixteen-fold.
- Perfusion pressure ΔP — the driving pressure gradient across the limb segment (proximal minus distal pressure).
- Stenosis-lost flow restored — compares total flow at the current settings against the same stenosis with zero collaterals, expressed as a fraction of the flow that stenosis alone took away from a healthy (unstenosed) artery. This is the quantity that determines whether a real PAD patient stays asymptomatic or develops claudication.
Particle color encodes local flow velocity (blue = slow, red = fast) — note the high-velocity jet through the stenosis, the same phenomenon that produces elevated peak systolic velocity on duplex ultrasound at a real stenosis. This 2D diagram solves the identical resistor-network equations as the 3D version of this simulator.