Suppose that there is a partial blockage of the aorta, which normally pumps about 5 L of blood per
minute. If the diameter of the aorta were reduced by 30%, what increase in blood pressure gradient would
be needed to obtain the normal flow rate? (Assume that Poiseuille’s law applies.)
a. The pressure gradient would have to increase by a factor of 3.3.
b. The pressure gradient would have to increase by a factor of 1.4.
c. The pressure gradient would have to increase by a factor of 4.2.
d. The pressure gradient would have to increase by a factor of 123.
e. The pressure gradient would have to increase by a factor of 2.
Notes: I honestly don't even know where to start? Do I just take 30% off 5?
Q=π•P₁•r₁⁴/8•η• L = π•P₂•r₂⁴/8•η• L
P₂=(r₁/r₂)⁴P₁= (100%/70%)⁴• P₁=4.2• P₁.
c. The pressure gradient would have to increase by a factor of 4.2.
To solve this problem, we need to apply Poiseuille's law, which states that the flow rate (Q) through a cylindrical tube is proportional to the fourth power of the radius (r) and the pressure gradient (∆P), and inversely proportional to the viscosity (η) and the length (L) of the tube.
Mathematically, Poiseuille's law is represented as: Q = (∆P * π * r^4) / (8 * η * L)
In this case, we want to find the increase in pressure gradient required to maintain the normal flow rate (Q) through the partially blocked aorta.
Let's denote the diameter of the normal aorta as D and the diameter of the partially blocked aorta as D'. The ratio of D' to D can be calculated as: D' / D = (100% - 30%) / 100% = 70% / 100% = 0.7
Since diameter and radius are proportional (radius = diameter/2), we can calculate the ratio of the radii as: r' / r = (D' / 2) / (D / 2) = (0.7 * D) / (D) = 0.7
Given that Q should remain constant, we have (using the fourth power relationship):
(Q' / Q) = (r' / r)^4 = (0.7)^4 ≈ 0.24
To obtain the normal flow rate, the pressure gradient (∆P') must increase by a certain factor to compensate for the reduced radius. Let's denote this factor as x.
So, (∆P' / ∆P) = x
Multiplying both sides of the equation by (Q' / Q), we have:
(x * ∆P / ∆P) = (Q' / Q) = 0.24
Therefore, x ≈ 0.24
Now, we need to determine the factor by which the pressure gradient should increase. This can be calculated as:
∆P' = x * ∆P = 0.24 * ∆P
Comparing this with the answer options provided:
a. The pressure gradient would have to increase by a factor of 3.3: This is not consistent with our calculation.
b. The pressure gradient would have to increase by a factor of 1.4: This is not consistent with our calculation.
c. The pressure gradient would have to increase by a factor of 4.2: This is not consistent with our calculation.
d. The pressure gradient would have to increase by a factor of 123: This is not consistent with our calculation.
e. The pressure gradient would have to increase by a factor of 2: This is not consistent with our calculation.
Therefore, none of the provided answer options match the result of our calculation.
To determine the increase in blood pressure gradient needed to obtain the normal flow rate, we can use Poiseuille's Law, which states that the flow rate through a tube is proportional to the fourth power of the radius.
The formula for Poiseuille's Law is:
Flow rate = (π * ΔP * r^4) / (8 * η * L)
Where:
- Flow rate is the volume of fluid passing through in a given time (in this case, 5 L/min).
- ΔP is the pressure difference across the aorta (what we need to find).
- r is the radius of the aorta.
- η is the viscosity of the fluid (assumed constant).
- L is the length of the aorta (assumed constant).
Since the question states that the diameter of the aorta is reduced by 30%, we can calculate the new radius by multiplying the original radius by 0.7 (1 - 0.3).
Let's consider the original flow rate as Q1 and the modified flow rate as Q2. The question asks for the increase in pressure gradient needed to obtain the normal flow rate (Q1).
We can set up a proportionality equation:
Q1 / Q2 = (r1^4) / (r2^4)
Substituting the values:
Q1 / 5 = (r1^4) / (0.7r1)^4
Simplifying:
Q1 / 5 = 1 / (0.7^4)
Q1 = 5 * 0.7^4
Calculating the value:
Q1 ≈ 5 * 0.24
Q1 ≈ 1.2 L/min
Since Q1 is the desired flow rate (the normal flow rate), we need to find the pressure gradient that will give this flow rate.
Therefore, the pressure gradient would have to increase by a factor of 1.2.
Hence, the correct answer is e. The pressure gradient would have to increase by a factor of 2.