Human blood vessel can be modeled as a cylindrical pipe with the inner radius r . As an biomedical engineer, you are supposed to work in a research project of addressing arteriosclerosis, which is believed to be mainly caused by blood clot around the inner surface of the blood vessel (refer to Figure 1). Now, please address following questions.

1. For the stable circulation, the blood flux in a vessel should be maintained at a constant level. Your assistant reported that the blood clot is getting thicker for a patient with heart disease as time goes by after age 20. Given the initial radius of the blood vessel of r0 = 0.5 cm and the growth rate of the blood clot as dh/dt = 0.02 mm/year , please provide a plot showing a relationship between the patient age (range: 20-80 years old) and the blood pressure normalized to the initial blood pressure ( p / p0 ). For this problem, you can use simple Hagen-Poiseuille flow model for the blood flow inside the horizontal blood vessel.

2. It is known that the elevation of the blood pressure is very dangerous for the patients heart, and should be addressed with a relevant medical devices such as a stent. It is recommend to use the stent if p / p0 =1.6. For a patient with p / p0 = 2.0, you recommend to use the stent to expand the blood vessel in a radial direction (i.e., radius of the expanded vessel = r ' ) to make p / p0 = 1.2 . Please calculate the ratio between the radius of the blood vessel after the stent installation and r0 . For this problem, you need to consider the volume conservation of the blood clot in the course of the vessel expansion.

3. For a severe patient with p / p0 = 2.0 , it is also recommended to install a complementary artificial cylindrical-shape blood vessel to make a constant blood flux with p / p0 = 1.0 . For a patient (age 70), please calculate the radius of the narrowed blood vessel due to the blood clot. what kind of method would you recommend to use? If necessary, please calculate the radius of the artificial vessel. Is the calculated radius of the complementary vessel greater than the narrowed vessel? What do you think about this symptom due to the clot? Why is it critical to prevent or circumvent the blood clot problem? (max. 500 words)