Question

Under control conditions, flow through a blood vessel is 100 ml/min with a pressure gradient of 50 mm Hg. What would be the approximate flow through the vessel after increasing the vessel diameter by 100%, assuming the pressure gradient is maintained at 50 mmHg?

Answer 1

Answer: flow = 1600 ml/min

Hi!

This problem si solved using Poiseuille law, which predicts the flow φ of a viscous fluid through a tube:

$\phi = \frac{\pi r^4}{8\eta} \frac{\Delta P}{L}$

$L = length\\r = radius\\\Delta P = \text{pressure gradient}\\\eta = viscosity$

We can calculate the ratio of initial (1) and final (2) flow. As we keep the same length, and viscosity, and pressure gradient the ratio is:

$\frac{\phi_2}{\phi_1} = (\frac{r_2}{r_1} )^4$

$r_2 = 2r_1$   (100% increase in radius)

$\phi_2 = \phi_1 (\frac{r2}{r1}^4) = 100\frac{ml}{min} 2^4$ = 1600 ml/min