Question

multiple choice question
a 3 - fold increase in the radius of a blood vessel will produce which of the following?

an 81 - fold increase in flow
no change in blood flow
a 21 - fold decrease in flow
a 3 - fold decrease in flow
a 3 - fold increase in flow

Explanation:

Step1: Recall Poiseuille's Law

Poiseuille's Law for blood flow (or fluid flow in a cylindrical vessel) is given by \( Q=\frac{\pi\Delta P r^{4}}{8\eta L} \), where \( Q \) is the flow rate, \( \Delta P \) is the pressure difference, \( r \) is the radius of the vessel, \( \eta \) is the viscosity of the fluid, and \( L \) is the length of the vessel.

Step2: Analyze the effect of radius change

Let the initial radius be \( r_1 \) and the final radius be \( r_2 = 3r_1 \) (since there is a 3 - fold increase). The initial flow rate \( Q_1=\frac{\pi\Delta P r_1^{4}}{8\eta L} \), and the final flow rate \( Q_2=\frac{\pi\Delta P r_2^{4}}{8\eta L} \). Substitute \( r_2 = 3r_1 \) into the formula for \( Q_2 \):

\( Q_2=\frac{\pi\Delta P(3r_1)^{4}}{8\eta L}=\frac{\pi\Delta P\times81r_1^{4}}{8\eta L}=81\times\frac{\pi\Delta P r_1^{4}}{8\eta L}=81Q_1 \)

This means that a 3 - fold increase in radius (since \( r_2 = 3r_1 \)) leads to an 81 - fold increase in flow rate (because the flow rate is proportional to the fourth power of the radius, \( Q\propto r^{4} \)).

Answer:

An 81 - fold increase in flow