Given that a coronary blood vessel has an axisymmetric contraction because of a atherosclerotic plaque build up (stenosis), the corresponding diastolic and systolic velocities at the stenosis throat can be found using the Bernoulli equation, as well as the pressure gradient between the upstream and the stenosis throat for diastolic and systolic conditions.The Bernoulli equation for the diastolic condition is: 0.5 x 1.057 x (10^2) / 3^2 = 0.35 N/m^2
The Bernoulli equation for the systolic condition is: 0.5 x 1.057 x (15^2) / 1^2 = 11.3 N/m^2
Therefore, the pressure gradient between the upstream and the stenosis throat for diastolic condition is 80 mmHg, and the pressure gradient between the upstream and the stenosis throat for systolic condition is 131.3 mmHg.The additional pressure that the pumping heart has to develop over the normal diastolic and the systolic pressures (80 mmHg and 120 mmHg correspondingly) due to the presence of the stenosis is 51.3 mmHg.
Given conditions of upstream velocities of 10 cm/s during diastole and 15 cm/s during systole, correspondingly, we need to find the corresponding diastolic and systolic velocities at the stenosis throat and the pressure gradient between the upstream and the stenosis throat for diastolic and systolic conditions. Also, the additional pressure that the pumping heart has to develop over the normal diastolic and systolic pressures due to the presence of the stenosis is to be found.
The diameter of the regular vessel is given as (D=3mm) and the diameter of the stenosis is (d=1mm).
Axisymmetric contraction: A coronary blood vessel has an axisymmetric contraction because of the atherosclerotic plaque build-up (stenosis).Due to the stenosis, the area of the throat is less than the area of the normal vessel. Therefore, the velocities at the throat increase to maintain the continuity equation.
Let the velocities at the throat be and Vs for diastolic and systolic conditions respectively.Using the continuity equation, the ratio of the velocities at the throat and the diameter of the vessel can be expressed as follows:AdVd = AsVsAd= π/4D²As= π/4d²Substituting the given values in the equation, we get,Vd = (π/4D²) * (As/Ad) * VdVs = (π/4D²) * (As/Ad) * Vsis the area of stenosis and A is the area of the regular vessel. We know that A = π/4D², As = π/4d².Substituting the given values, we get,Vd = (1/3) * (15) = 5 cm/sVs = (1/3) * (10) = 3.33 cm/sPressure gradientThe pressure gradient between the upstream and the stenosis throat for diastolic and systolic conditions is given as follows:ΔPd = ρ (Vd - Vup)²/2ΔPs = ρ (Vs - Vup)²/2where ρ is the density of blood and Vup is the upstream velocity.Substituting the given values, we get,ΔPd = (1.057 g/cm³) (5 - 10)²/2 = 26.425 PaΔPs = (1.057 g/cm³) (3.33 - 15)²/2 = 240.206 PaAdditional pressureThe additional pressure that the pumping heart has to develop over the normal diastolic and systolic pressures due to the presence of the stenosis is given as follows:ΔPadd, diastolic = ρ (Vup)²/2 + ΔPd - 80 mmHgΔPadd, systolic = ρ (Vup)²/2 + ΔPs - 120 mmHgwhere ΔPadd is the additional pressure.Substituting the given values, we get,ΔPadd, diastolic = (1.057 g/cm³) (10)²/2 + 26.425 Pa - 80 mmHg = 54.03 mmHgΔPadd, systolic = (1.057 g/cm³) (15)²/2 + 240.206 Pa - 120 mmHg = 123.79 mmHgTherefore, the corresponding diastolic and systolic velocities at the stenosis throat are 5 cm/s and 3.33 cm/s respectively. The pressure gradient between the upstream and the stenosis throat for diastolic and systolic conditions are 26.425 Pa and 240.206 Pa respectively. The additional pressure that the pumping heart has to develop over the normal diastolic and systolic pressures due to the presence of the stenosis is 54.03 mmHg and 123.79 mmHg respectively.