Question

In an aortic aneurysm, a bulge forms where the walls of the aorta are weakened. If blood flowing through the aorta (radius 1.1 cm) enters an aneurysm with a radius of 3.0 cm, how much on average is the blood pressure higher inside the aneurysm than the pressure in the unenlarged part of the aorta? The average flow rate through the aorta is 110 cm³/s. Assume the blood is non-viscous and the patient is lying down so there is no change in height.

Answer 1

We have 2 sections of aorta :

Section 1: normal aorta with radius r₁ = 1.1 cm, Pressure : P₁

Section 2 : aorta with aneurysm having radius r₂ = 3.0 cm, Pressure : P₂

Flow rate is given : 110 cm³ /s

flow rate : f = Volume /time = A*v

A = cross sectional area

Cross sectional area at 1: A₁ = pi * r₁²

Cross sectional area at 2: A₂ = pi * r₂²

v = velocity of fluid

velocity at section 1: v₁

velocity at section 2: v₂

Now we calculate v₁ and v₂ from relation:

f = Av = A₁v₁

Thus, v₁ = f/A₁

Also, f = A₂v₂

Thus v₂ = f/A₂

Now to calculate pressure difference between two sections of aorta we use Bernoulli's principle:

The pressure, speed, and height (y) at two points in a steady-flowing, non-viscous, incompressible fluid are related by the equation:

(Here rho = density of the fluid)

P₁ + 1/2 rho * v₁² + rho * g * y₁ = P₂ + 1/2 rho * v₂² + rho * g * y₂

Now since patient is lying doen and there is no change of height, the equation becomes:

P₁ + 1/2 rho * v₁² = P₂ + 1/2 rho * v₂²

Pressure difference between two sections 1 and 2:

P₂ - P₁ = 1/2 rho * v₁² - 1/2 rho * v₂²

Substitute thw expressions for veocities from above:

P₂ - P₁ = 1/2 rho * ( v₁² - v₂² ) = 1/2 rho * ( (f / A₁)² - (f / A₂)² ) = 1/2 rho * (f² / pi²) * ( 1/r₁² - 1/r₂² )

= rho * 0.5 * (110² / 3.14²) ( (1/1.1²) - (1/3²) ) = rho * 438.9 gm /cm s² or ba

Thus pressure difference between aneuyrsm and noral sections of aorta :

P₂ - P₁ = 438.9 * rho gm / cm s² (CGS units)

Since the value of rho or density of fluid is not given in the question, we take the average value: 1.05 gm/mL

Thus Pressure at aneuyrsm is higher than pressure at normal aorta by:

P₂ - P₁ = 438.9 * 1.05 = 460.86 ( in the units of gm / cm s² or ba)