Angioplasty is a technique in which arteries partially blocked with plaque are dilated to increase blood flow. By what factor must the radius of an artery be increased in order to increase blood flow by a factor of 16?
To determine the factor by which the radius of an artery must be increased in order to enhance blood flow by a factor of 16, we need to use the relationship known as the Poiseuille's Law.
According to Poiseuille's Law, the volume flow rate (Q) through a cylindrical tube, such as an artery, is directly proportional to the fourth power of the radius (r) of the tube.
The formula for Poiseuille's Law is: Q = (π * ΔP * r^4) / (8 * η * L)
Where:
Q = Volume flow rate
ΔP = Pressure difference across the tube
r = Radius of the tube
η = Viscosity of the fluid (blood in this case)
L = Length of the tube (artery)
Since we want to increase the blood flow (Q) by a factor of 16, we can write the equation as:
16Q = (π * ΔP * (r')^4) / (8 * η * L)
Here, (r') is the increased radius we are looking for.
To find the factor by which the radius must be increased, we can simplify the equation:
16Q = (π * ΔP * (r^4)(16)) / (8 * η * L)
Cancelling out the common terms, the equation becomes:
16 = (r^4)(16)
Dividing both sides of the equation by 16, we have:
1 = r^4
Now, taking the fourth root of both sides gives us:
r = 1
This means that the radius of the artery must be increased by a factor of 1 (i.e., it remains the same) in order to increase blood flow by a factor of 16.
Therefore, no change in the radius of the artery is required to achieve the desired increase in blood flow.