ps6 (1)

ps6 (1) - Harvard-MIT Division of Health Sciences and...

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Harvard-MIT 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 Harvard-MIT Division of Health Sciences and Technology 6.022J/2.792J/BEH.371J/HST542J: Quantitative Physiology: Organ Transport Systems PROBLEM SET 6 Assigned: March 18, 2004 Due: April 1, 2004

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Problem 1 A pressure wave, P i , incident on an arterial (or bronchial) bifurcation will suffer a reﬂection, 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 formulæderived in the notes: Z 2 ρ 2 π R 3 = C u AC u h E where ρ is the ﬂuid density, A is the vessel cross-section, h is the vessel wall thickness, and E is the vessel modulus of elasticity. Figure 1: 2 R 1 2 R U 1 U 0 U 1 h 1 h 0 A. Assuming a symmetric bifurcation, ρ = constant, E 0 = E 1 , = , and no increase in R 1 R 0 total cross-sectional area across the bifurcation, calculate the reﬂection coefﬁcient. B. Suppose the vessels distal to the bifurcation are severely calciﬁed, so that E 1 E 0 . What will the reﬂection coeffecient be? Does this suggest a noninvasive method of detecting the presence of severe arterial disease? 2004/24 2 6.022j—2004: Problem Set 6
± ² ³ ´ 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 cross-section 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 ﬂuid will generally exhibit turbulent ﬂow. Energy will be lost in two ways: in

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ps6 (1) - Harvard-MIT Division of Health Sciences and...

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