2. We employed the Law of Laplace for thin/thick-walled spheres to delineate the relation between wall tension or wall stress and inner pressure regarding cardiac muscle contraction. Meanwhile, we can also employ the Law of Laplace for thin/thick-walled cylinder to delineate the relation between wall tension or wall stress and inner pressure regarding blood vessel contraction, as shown in the right figure. The inner radius of the cylindrical tube is r1, the outer radius of the cylindrical tube is r2, and the length of the cylindrical tube is L.

a) We first assume the blood vessel as a thin-walled cylindrical tube, i.e. r1 = r2 = r, or the thickness of tube w = r2 - r1 can be assumed to be zero. As we analyzed in the lecture, we draw an imaginary plane that bisects the cylindrical tube, and perform the load analysis on this imaginary plane based on mechanical equilibrium. Please derive the expression of inner pressure P in terms of wall tension T and relevant dimensional quantity.

b) We then assume the blood vessel as a thick-walled cylindrical tube, i.e. the thickness of tube w = r2 - r1 cannot be assumed to be zero. Using the imaginary plane as above, we perform the load analysis on it based on mechanical equilibrium. Please derive the expression of inner pressure P in terms of wall stress and relevant dimensional quantities.

Note: you need to show derivational process to obtain the full credit for a) and b).

c) Using the expression that you derive in a) based on Law of Laplace for thin-walled cylinder, please compute the wall tension T for different types of blood vessels, in terms of the average blood pressure and dimensional quantity as shown in the following table. Note: 1 mmHg = 133.322 Pa.

	Aorta	Artery	Arteriole	Capillary	Venule	Vein	Vena Cava
Inner radiusr1	12 mm	2 mm	15 m	3 m	10 m	2.5 mm	15 mm
Blood pressure	95 mmHg	90 mmHg	65 mmHg	15 mmHg	12 mmHg	10 mmHg	5 mmHg
Wall tensionT	151.987 N/m