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TECHNICAL PAPERS: Fluids/Heat/Transport

Microbubble Expansion in a Flexible Tube

[+] Author and Article Information
Tao Ye

Biomedical Engineering Department, The University of Michigan, Ann Arbor, MI 48109

Joseph L. Bull

Biomedical Engineering Department, The University of Michigan, Ann Arbor, MI 48109joebull@umich.edu

J Biomech Eng 128(4), 554-563 (Jan 30, 2006) (10 pages) doi:10.1115/1.2206200 History: Received July 27, 2005; Revised January 30, 2006

We have utilized a computational model of the expansion of a microbubble in a liquid-filled flexible tube to investigate the potential for acoustic vaporization of perfluorocarbon droplets to damage blood vessels during a novel gas embolotherapy technique for the potential treatment of tumors. This model uses a fixed grid, multi-domain, interface tracking, direct numerical simulation method that treats all interfaces and boundaries as sharp discontinuities for high accuracy. In the current work, we examined effects of initial bubble size on the flows and wall stresses that result from droplet vaporization. The remaining dimensionless parameters that govern the system response (Reynolds, Weber, and Strouhal numbers, initial bubble pressure, and wall stiffness and tension) were selected to model an arteriole. The results for a flexible tube are significantly different from those for a rigid tube. Two major flow regimes occur due to the combined effect of bubble and tube deformation: in flow at the tube ends and out flow near the bubble surface. The flexibility of the tube largely dissipates the extreme pressure that develops in the rigid tube model. Both the magnitude and the overall expansion time of the rapidly changing pressure are greatly reduced in the flexible tube. Smaller initial bubble diameters, relative to the vessel diameter, result in lower wall stresses. This study indicates that wall flexibility can significantly influence the wall stresses that result from acoustic vaporization of intravascular perfluorocarbon droplets, and suggests that acoustic activation of droplets in larger, more flexible vessels may be less likely to damage or rupture vessels than activation in smaller and stiffer vessels.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of a bubble in a flexible tube. The bubble is located at the center of the tube and its initially high pressure drives the subsequent bubble expansion. The initial bubble size ranges from 10% to 90% of the tube diameter, D. The tube wall is flexible over the entire length, Lt. The tube is characterized by radial stiffness, ksp, and longitudinal tension, σw. The tube is filled with a viscous incompressible liquid that is initially at rest. A constant ambient pressure, P∞*, is specified at the tube exits.

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Figure 2

(a) Bubble and wall shapes, and streamlines at t=0.004, 0.4, 0.8, 1.2, 1.6, 2.0, and 2.4, for case 1 in Table 1, Re=427.59, We=6.93, St=10.47, Ωs=5×10−9, Ωt=0.05, Ph=176.39(Ph*=20bar), di=0.1. One dimensionless time unit corresponds to 10.5×10−6s. The bubble is initially spherical with a diameter of 0.1. The bubble shape evolves in time. The horizontal axis is the longitudinal direction, z, of the tube. Only half of the physical domain, 0⩽z⩽3.5, is shown because of symmetry. (b) The same plots for the rigid tube case. The dimensionless parameters are identical to those in (a) except for Ωs and Ωt that do not exist in rigid tube case.

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Figure 3

Comparison of results for pressure (a & b) and shear stress (c & d) on the tube wall between flexible wall (a & c) and rigid wall (b & d) models. The two models share an identical set of parameters: Re=427.59, We=6.93, St=10.47, Ph=176.39, and di=0.1. The flexible wall model has two additional parameters characterizing the wall flexibility. The horizontal axis is the longitudinal direction, z, of the tube. Only half of the physical domain, 0⩽z⩽3.5, is shown because of symmetry. The dimensional stress is 11338.8N∕m2 per dimensionless unit.

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Figure 4

Bubble and wall shapes, and streamlines at t=0.004, 0.4, 0.8, 1.2, and 1.6 for case 3 in Table 1, Re=427.59, We=6.93, St=10.47, Ωs=5×10−9, Ωt=0.05, Ph=176.39, and di=0.5. Only half of the physical domain, 0⩽z⩽3.5, is shown because of symmetry.

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Figure 5

Bubble and wall shapes, and streamlines at t=0.004, 0.04, 0.08, 0.12, and 0.16 for case 5 in Table 1, Re=427.59, We=6.93, St=10.47, Ωs=5×10−9, Ωt=0.05, Ph=176.39, and di=0.9. Only half of the physical domain, 0⩽z⩽3.5, is shown because of symmetry.

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Figure 6

(a) Pressure and (b) shear stress on the tube wall at various times for case 3 in Table 1, Re=427.59, We=6.93, St=10.47, Ωs=5×10−9, Ωt=0.05, Ph=176.39, and di=0.5. Only half of the physical domain, 0⩽z⩽3.5, is shown because of symmetry. The dimensional stress is 11338.8N∕m2 per dimensionless unit.

Grahic Jump Location
Figure 7

(a) Pressure and (b) shear stress on the tube wall at various times for case 5 in Table 1, Re=427.59, We=6.93, St=10.47, Ωs=5×10−9, Ωt=0.05, Ph=176.39, and di=0.9. Only half of the physical domain, 0⩽z⩽3.5, is shown because of symmetry. The dimensional stress is 11338.8N∕m2 per dimensionless unit.

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Figure 8

The maximum pressure on the tube wall versus di at t=0.004. Re=427.59, We=6.93, St=10.47, Ωs=5×10−9, Ωt=0.05, Ph=176.39.

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