MY SIMULATOR

Vessel Tree Lab

Murray's law, pulsatile flow and red-cell transit.

INTERACTIVE 3D

HR 72 bpmBP 120/80FPS 0

MEAN PRESSURE93 mmHg

BRANCH WSS0.8 Pa

MURRAY BALANCE-0.01

SLOW-MO⏱1x

Vessel Tree Lab: Murray's Law, Pulsatile Flow & Red-Cell Transit

Look inside a branching artery and vein. Change the heart, the blood and the radii of a bifurcation, and see the pressure wave, the flow split, the wall shear stress and the red cells in the lumen.

Educational and non-diagnostic. This is a simplified physics model with illustrative values. It is not medical advice and cannot be used to assess a person's health.

Preset

Heart, blood and vessel

Bifurcation report (parent → two daughters)

Pressure and inflow in one beat

Apparent viscosity vs vessel diameter

Missions

Independent numerical checks

Also run outside the browser: node src/check.js (Poiseuille integral, Murray exponent by numerical minimisation, bifurcation mass and shear balance, Windkessel decay and volume balance, viscosity limits, stenosis continuity).

Model values and simplifications: rigid cylindrical segments, Newtonian blood except for the apparent-viscosity fit, 2-element Windkessel for the whole arterial tree (R_p set from a 93 mmHg mean pressure, C = 1.5 ml/mmHg), the studied artery carries 5 % of cardiac output, systole lasts 0.35 √T. Constants marked [assumption] are illustrative. In the 3D view the cells are drawn much larger than real (7.8 µm) and vessels are not to scale.

Controls

Drag / one finger: orbit; wheel / pinch: zoom; Hero / Artery / Vein / Heart buttons (keys 1-4) move the camera.
Sliders change the heart (rate, stroke volume), the arteriolar resistance, the haematocrit, a stenosis in the artery, and the two daughter radii as fractions of the parent.
The playback slider slows the heartbeat down so you can follow the pulse; Pause freezes it.

The science

Poiseuille flow. In a long straight tube u(r) = 2U(1 − r²/R²), Q = πR⁴ΔP / (8μL), wall shear stress τ = 4μQ / (πR³). Red cells in the middle move fastest, near the wall slowest.

Murray's law. Minimising flow power plus the upkeep cost of the blood volume, P = 8μLQ²/(πr⁴) + bπr²L, gives Q ∝ r³, hence at a bifurcation r_p³ = r₁³ + r₂³ and equal wall shear stress everywhere. The report shows how far your radii are from this and the extra power they cost.

Fåhræus–Lindqvist. In narrow tubes the apparent viscosity of blood drops with diameter (cells migrate to the centre and leave a plasma layer). The in-vitro fit of Pries and colleagues (1992) is used [verify coefficients against the paper]; below about 10 µm the curve rises again.

Pulsation. A 2-element Windkessel, C dP/dt = Q_in(t) − P/R_p, turns the pulsed heart output (half-sine of duration 0.35 √T carrying the stroke volume) into the arterial pressure wave; in diastole P decays as e^−t/(R_pC).

Stenosis. Continuity vA = Q speeds the blood up in the narrowed throat; Bernoulli plus Poiseuille friction give the pressure drop ½ρ(v_s² − v₀²) + 8μL_sQ/(πR_s⁴).

Not modelled: elastic walls and wave reflections (Womersley), turbulence, non-Newtonian shear thinning in big vessels, branching asymmetry along a whole tree, regulation (baroreflex).