If radius of a vessel is doubled then the blood flow is increased by?
The formula is Q = (πΔP r^4)/(8ηL), where Q is flow, ΔP is the pressure difference, r is radius, η is viscosity, and L is length. So yes, radius has a huge impact because it's to the fourth power. So if radius doubles, flow increases by 16 times. So the correct answer should be 16-fold increase.
Now, looking at the options, even though they aren't provided, the correct answer would be 16 times. The other options might be common mistakes. For example, someone might think it's squared (4x), or linear (2x), or maybe even 8x if they confused radius with diameter. Let me break down why each wrong option is wrong. If someone thought the relationship was squared, they might pick 4x. If they thought linear, 2x. If they thought radius is half of diameter, maybe they thought doubling radius would double diameter and use that in the equation, leading to 8x. But no, the radius is the key here. The clinical pearl here is that even small changes in vessel radius have a massive effect on blood flow, which is why vasodilation/vasoconstriction are so important in regulating blood flow.
**Core Concept**
Blood flow through a vessel is governed by Poiseuille’s Law, which states that flow rate (Q) is proportional to the fourth power of the vessel radius (r⁴). This principle is critical in hemodynamics and explains how vascular resistance and blood flow are regulated.
**Why the Correct Answer is Right**
Doubling the radius (r₁ → 2r₁) increases flow by (2⁴) = 16-fold. Since resistance (R) is inversely proportional to r⁴ (R ∝ 1/r⁴), reduced resistance leads to increased flow. For example, if initial flow is Q₁, doubling radius results in Q₂ = 16Q₁ under constant pressure gradient and vessel length.
**Why Each Wrong Option is Incorrect**
**Option A: 2-fold** – Assumes linear relationship (r), but flow depends on r⁴.
**Option B: 4-fold** – Reflects squared relationship (r²), correct for laminar flow but not for Poiseuille’s Law.
**Option C: 8-fold** – Incorrectly applies cube of radius (r³), not the fourth power.
**Clinical Pearl / High-Yield Fact**
Vessel radius changes significantly alter blood flow. For example, a 50% reduction in radius (due to atherosclerosis) decreases flow by 50%⁴ = 6.25%, highlighting why even mild stenosis can cause severe ischemia.
**Correct Answer: 16-fold increase**.
**Correct Answer: D