Atherosclerosis is a cardiovascular disease characterized by the deposition of materials such as lipids, cholesterol, and proteins such as fibrins in the arteries, as seen in Figure 1 below. This leads to increased resistance to blood flow and causes the stress on the heart to increase.
The main risk of atherosclerosis is that it greatly increases the probability of blood clots forming in arteries. Should such clots occur in the carotid or coronary arteries, they can result in strokes or myocardial infarctions, which can be fatal2. This paper seeks to analyze how physical principles can be used to elucidate the pathophysiology of atherosclerosis and its effects on the human body.
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The subscripts 1 and 2 refer to two defined locations in a flow tube respectively3. Bernoulli’s Equation can be interpreted in terms of the Law of Energy Conservation. Looking at Eq. (2), P refers to the potential energy stored in the fluid due to pressure difference, ρgy refers to the gravitational potential energy per unit volume of the fluid and 1/2ρv^2 refers to the kinetic energy density of the fluid. In an ideal fluid and in the absence of friction against the flow tube, the total energy of the fluid system must be constant4.
P+ ρgy+1/2 mv^2=k, where k is a constant (3)
Based on the Conservation of Energy of the fluid, Eq. (3) is another interpretation of Bernoulli’s Equation.
Both the Continuity Equation and Bernoulli’s Equation assume ideality, whereby there are no retarding forces in the fluid and all fluid molecules consequentially possess the same velocity. For a real fluid, however, viscosity friction exists between fluid molecules and this causes different molecules in the fluid to have differing velocities.
∆V/∆t=(πR^4 (p_1- p_2 ))/8ηL (4)
∆V/∆t represents the volume flow rate, R is the radius of the tube, p_1- p_2 is the pressure gradient, η is the fluid viscosity and L is the length of the tube. Looking at Eq. (4), the most significant relation is that the volume flow rate and the pressure gradient are both related to the fourth power of the tube radius3. Hence, small changes in the radius can have large implications for flow rate and pressure gradient. Also, based on Eq. (4), it can be seen that as a viscous fluid flows, the pressure always decreases. This pressure gradient is necessary because in a real fluid, viscosity friction exists between the molecules which acts to retard fluid flow. Hence, the pressure difference multiplied by the cross-sectional area of the tube provides the force needed to sustain the flow of the viscous fluid4.
In real fluids, there is a certain critical flow velocity, beyond which the fluid flow becomes turbulent. Turbulent flow is characterized by whirls and haphazard motion (refer to Figure 2) and leads to a dramatic increase in the viscous friction and hence the resistance to flow6. The critical velocity of a fluid can be approximated by the following equation:
Looking at Eq. (5), v_c is the critical flow velocity, η is the viscosity of the fluid, ρ is the density of the fluid, D is the diameter of the flow tube and R is the Reynold’s number, an experimental value, whereby for most fluids, it normally has a value of approximately 2000 and 30004.
Analysis of the effects of atherosclerosis using fluid mechanics
The effect of reduced vessel diameter on blood flow rate
As covered in the introduction, atherosclerosis refers to the excessive deposition of plaque onto the walls of arteries which carry oxygenated blood....