If the diameter of an arterial segment is halved without a change in perfusion pressure, what is the expected blood flow rate?

In this scenario, when the diameter of an arterial segment is halved, we can analyze the impact on blood flow rate using the principles of fluid dynamics, particularly Poiseuille's law. According to this law, blood flow rate (Q) through a vessel is directly proportional to the fourth power of the radius (r) of the vessel and the pressure gradient (ΔP), and inversely proportional to the viscosity (η) and the length (L) of the vessel. Mathematically, it can be expressed as:

Q = (π * r^4 * ΔP) / (8 * η * L)

When the diameter is halved, the radius is also halved. Given that the radius is related to the diameter as r = d/2, halving the diameter means the new radius is r/2. When we substitute this into the equation, we see that:

(Q_new) = (π * (r/2)^4 * ΔP) / (8 * η * L)

Since (r/2)^4 = r^4 / 16, we can express the new flow rate:

Q_new = (π * (r^4 / 16) * ΔP) / (8