Stenosis, Poiseuille Flow, and Why In Vivo Flow Falls Less Than Predicted
In an isolated, perfused arteriole preparation, the internal radius is 50 µm and the pressure gradient along a 2 mm segment is held at 40 mmHg; flow is 12 µL/min. Investigators then wrap the segment in a cuff that reduces the radius uniformly to 40 µm, a 20% reduction, while holding pressure gradient, length, and perfusate viscosity (3.5 cP) constant. Flow falls to 4.9 µL/min. In a parallel in vivo experiment, the same 20% narrowing is produced in an interlobular artery of a rat kidney. There, blood flow through the narrowed artery falls only 25%, not 59%. During the experiment, distal arterioles dilate and the pressure gradient across the narrowed segment rises from 40 to 55 mmHg while pressure downstream of the stenosis falls. Perfusate density is 1.05 g/mL. The investigators claim the in vivo result disproves Poiseuille's law for small arteries.
1. Show quantitatively why the isolated vessel behaved as it did, and state the assumptions that make the r-to-the-fourth relationship valid.
Read the worked answer
Poiseuille's law gives Q = pi × deltaP × r^4/(8 × eta × L). The radius ratio is 40/50 = 0.8, and 0.8^4 = 0.4096, so predicted flow is 12 × 0.4096 = 4.9 µL/min, matching the measured value. Resistance, deltaP/Q, rises by 1/0.4096 = 2.44-fold. The fourth-power dependence follows because flow scales with cross-sectional area (r^2) and with the mean velocity of a parabolic profile (also r^2) at fixed pressure gradient. The assumptions are steady, laminar, fully developed flow of a Newtonian fluid in a rigid uniform cylinder with no slip at the wall and constant viscosity. A Reynolds number check supports the laminar assumption: Q = 2.0 × 10^-10 m^3/s, area = 1.96 × 10^-9 m^2, so v = 0.10 m/s and Re = 1050 × 0.10 × 50 × 10^-6/3.5 × 10^-3, about 1.5, far below the transition threshold.
2. Reconcile the 25% in vivo flow reduction with the 59% prediction. Is Poiseuille's law violated?
Read the worked answer
Poiseuille's law is not violated; one of its premises is. The law was applied with deltaP fixed, but in vivo deltaP across the stenosis changed. Holding r^4 constant at 0.4096 and raising deltaP by 55/40 = 1.375 gives a flow ratio of 1.375 × 0.4096 = 0.563, a 44% reduction, closer to the observed 25%. The remaining difference comes from the series arrangement of resistors. When distal arterioles dilate, total resistance falls and flow through the whole bed rises, while pressure immediately downstream of the stenosis falls further, enlarging the gradient across the stenosis. The stenosis therefore absorbs a larger share of a larger total pressure drop. Two secondary factors push in the same direction: blood is non-Newtonian, so viscosity in the high-shear throat is below the nominal 3.5 cP, and a cuffed segment is unlikely to be perfectly uniform. The correct conclusion is that autoregulation changed the boundary condition, not that the physics failed.
3. Predict how the Reynolds number in the narrowed segment changes in the in vivo experiment, and evaluate the intuition that narrowing must produce turbulence and a bruit.
Read the worked answer
Re = rho × v × d/eta, which can be rewritten as 4 × rho × Q/(pi × eta × d). Holding density and viscosity fixed, Re scales as Q/d. In vivo Q falls to 0.75 of baseline and d falls to 0.8, so Re changes by 0.75/0.8 = 0.94, a slight decrease. Velocity does rise (0.75/0.64 = 1.17-fold), but not enough to offset the smaller diameter. The intuition fails because flow and diameter are not independent once resistance rises. At these dimensions Re remains on the order of 1 to 2, three orders of magnitude below the transition to turbulence, so a bruit from this segment is not physically expected. A bruit would imply a much tighter stenosis with maintained flow, a post-stenotic jet, or vibration of the vessel wall. The isolated preparation shows the same pattern more strongly, since flow fell to 0.41 while diameter fell to 0.8.