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Unformatted text preview: HarvardMIT Division of Health Sciences and Technology HST.542J: Quantitative Physiology: Organ Transport Systems Instructors: Roger Mark and Jose Venegas MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departments of Electrical Engineering, Mechanical Engineering, and the HarvardMIT Division of Health Sciences and Technology 6.022J/2.792J/BEH.371J/HST542J: Quantitative Physiology: Organ Transport Systems PROBLEM SET 6 SOLUTIONS April 1, 2004 Problem 1 A pressure wave, P i , incident on an arterial (or bronchial) bifurcation will suffer a reection, P r , of magnitude P r Z L Z o P i = Z L + Z o where Z o is the upstream arterial impedance, and Z L is the impedance of the bifurcation. The impedance, Z , of a vessel of radius R may be calculated from the formulderived in the notes: 2 R 3 Z 2 = AC u C u hE where is the uid density, A is the vessel crosssection, h is the vessel wall thickness, and E is the vessel modulus of elasticity. Figure 1: 2 R 1 2 R 0 U 0 U 1 U 1 h 1 h A. Assuming a symmetric bifurcation, = constant, E 0 E 1 , , and no increase in = R 1 = R 0 total crosssectional area across the bifurcation, calculate the reection coefficient. 2 R 3 C u hE A R 2 = 1 1 1 Z L = Z l + Z l 2 = Z l same = 6.022j2004: Solutions to Problem Set 6 2 Z 2 hE = AC u = R 2 2 R 3 hE = 2 2 R 5 1 / 2 1 / 2 1 h 1 E 1 h 0 E 0 2 R 5 R 5 P r Z L Z o 1 0 1 / 2 1 / 2 P i = Z L + Z o = 1 h 1 E 1 h 0 E 0 2 R 5 + R 5 1 0 h 1 h If E 1 E 0 and , = R 1 = R 0 P r 1 1 1 1 0 1 R R 1 0 2 1 2 R 2 R 2 2 P i = 1 1 1 = 1 R 0 2 2 R 1 2 + R 2 2 R 1 + 1 1 R 0 2 For 2 A 1 A , R 1 = = 2 R . R 1 = 2 P r 0 P i = B. Suppose the vessels distal to the bifurcation are severely calcified, so that E 1 E . What will the reection coeffecient be? Does this suggest a noninvasive method of detecting the presence of severe arterial disease? P r Z L Z , so > 1 and reection is positive for positive wave in. P i 2004/24 3 6.022j2004: Solutions to Problem Set 6 Problem 2 This problem deals with the estimation of the pressure drop to be expected across a vascular steno sis. Figure 2 is a sketch of the crosssection of a stenotic artery with a concentric plaque. An idealized model is shown in Figure 3, demonstrating a narrowed region followed by a sudden ex pansion where the uid will generally exhibit turbulent ow. Energy will be lost in two ways: in viscous ow in the narrow region and in turbulent loss in the expansion. In this problem we will concentrate on the latter. Figure 2: Figure 3: u 1 , A 1 , P 1 u 2 , A 2 , P 2 u 3 , A 3 , P 3 1 2 3 The first task will be to estimate the pressure drop from point 2 to point 3. Position 3 is chosen far enough downstream to be in a region of uniform ow, of velocity u 3 . In order to solve this problem we must make use of the linear momentum theorem....
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 Fall '06
 MatthewLang
 Physiology

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