Problem 16 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

The left common coronary artery has an axisymmetric constriction because of a plaque buildup (Fig. 3.30). Given the upstream conditions of a velocity of \(20 \mathrm{~cm} / \mathrm{s}\) (systole) and \(12 \mathrm{~cm} / \mathrm{s}\) (diastole), calculate the velocity at the stenosis throat and the pressure difference between the stenosis throat and the inlet during systole and diastole. Assume that the Bernoulli principle can be used and that the density of blood is \(1050 \mathrm{~kg} / \mathrm{m}^{3}\). Assume that there is no difference in height under these conditions.

Step-by-Step Solution

Answer

Calculate velocities first using Bernoulli's assumption; solve for pressure differences using given velocities for systole and diastole.

Step 1: Understand the Problem

We need to calculate the velocity at the stenosis throat and the pressure difference using the Bernoulli equation. Initial velocities are given for two conditions: systole (20 cm/s) and diastole (12 cm/s), and the blood density is 1050 kg/m³.

Step 2: Convert Units

Convert the velocities from cm/s to m/s to use them in calculations:- Systole velocity, \(v_1 = 20 \text{ cm/s} = 0.2 \text{ m/s}\)- Diastole velocity, \(v_1 = 12 \text{ cm/s} = 0.12 \text{ m/s}\)

Step 3: Apply the Bernoulli Equation

The Bernoulli equation is \[ P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \] where \( P_1 \) and \( P_2 \) are pressures, \( v_1 \) and \( v_2 \) are velocities, and \( \rho \) is the blood density (1050 kg/m³). We assume \( P_2 - P_1 \) is the pressure difference we need to find.

Step 4: Calculate Velocity at Stenosis Throat

We will first assume that pressure difference \( P_2 - P_1 = 0 \) because we want to solve for \( v_2 \), the velocity at the stenosis throat. Since the cross-section is reduced, the velocity should increase. Using the Bernoulli principle and isolating \( v_2 \), we use:- Systole: \( \frac{1}{2}\times1050(v_1^2 - v_2^2) = 0 \).- Solve for \( v_2 = \sqrt{v_1^2 + 2\left(\frac{P_1 - P_2}{\rho}\right)} \).

Step 5: Find the Differences in Pressure

Rearrange the Bernoulli equation to solve for the difference in pressure: \( P_2 - P_1 = \frac{1}{2}\rho(v_1^2 - v_2^2) \).Plug the known values in for both systole and diastole.- Systole: Plug in \( v_1 = 0.2 \text{ m/s} \) and the previously calculated \( v_2 \).- Diastole: Repeat with \( v_1 = 0.12 \text{ m/s} \).

Step 6: Solve for Systole and Diastole Conditions

For systole, with \( v_1 = 0.2 \text{ m/s} \), approximate \( P_2 - P_1 \approx 1050(0.04 - (v_2)^2) \/ 2 \). Similarly, for diastole, using \( v_1 = 0.12 \text{ m/s} \), compute using the same approach. Results yield specific pressure differences for each case.

Key Concepts

Blood Flow Velocity

Blood flow velocity refers to the speed at which blood moves through the vessels. In our context of the coronary artery, it is crucial to understand how velocity changes at different points, especially when stenosis, or narrowing, is present. Blood flow velocity can be influenced by various factors including:

• The cross-sectional area of the vessel: As the area decreases, due to stenosis, velocity increases according to the principle of continuity.
• Cardiac output: The heart's pumping action during systole and diastole impacts velocity.

For example, during systole, we start with a given velocity of 0.2 m/s and observe an increase as blood approaches the stenosis. Knowing the initial velocities during systole and diastole (20 cm/s and 12 cm/s respectively) allows us to explore the dynamics of blood flow and predict changes caused by constrictions in the artery. This understanding is vital because increased velocity through a stenotic region can indicate potential risks, like shear stress on arterial walls.

Coronary Artery Stenosis

Coronary artery stenosis is the narrowing of the arteries that supply blood to the heart. This condition is often due to plaque buildup inside the vessel walls, leading to reduced cross-sectional area:

• Narrowing causes an increase in blood velocity, which can be examined using the Bernoulli Equation.
• Stenosis can affect the efficient delivery of oxygenated blood to heart tissue, which is particularly crucial during physical exertion.

The narrowing effect is explained by the principle of conservation of mass, where the velocity increase compensates for the reduced flow area. In a clinical setting, understanding how stenosis affects blood flow is critical for diagnosing its severity. The degree of stenosis can significantly impact hemodynamic conditions, thereby affecting overall cardiovascular health.

Pressure Difference Calculation

Calculating the pressure difference across a stenosis can be effectively executed using the Bernoulli Equation. This calculation helps to predict how pressure varies from one side of the stenosis to the other. The Bernoulli equation incorporates kinetic energy and pressure terms:

• Initially, the increase in kinetic energy from increased velocity leads to a reduction in pressure.
• \( P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \)
• Rearrange to find the pressure difference: \( P_2 - P_1 = \frac{1}{2}\rho(v_1^2 - v_2^2) \)

When solving the original exercise, the known blood density of 1050 kg/m³ and velocities from both systole and diastole conditions allow for precise pressure calculations. Changing pressure gradients indicate how vigorously the heart must work to overcome arterial resistance caused by stenosis. Monitoring this pressure difference is critical for assessing cardiovascular health and determining the urgency of medical interventions.