Natural frequency of a gas bubble in a tube: Experimental and simulation results

Abstract

Use of ultrasonically excited microbubbles within blood vessels has been proposed for a variety of clinical applications. In this paper, an axisymmetric coupled boundary element and finite element code and experiments have been used to investigate the effects of a surrounding tube on a bubble’s response to acoustic excitation. A balloon model allowed measurement of spherical gas bubble response. Resonance frequencies match one-dimensional cylindrical model predictions for a bubble well within a rigid tube but deviate for a bubble near the tube end. Simulations also predict bubble translation along the tube axis and aspherical oscillations at higher amplitudes.

Introduction

Accurate determination of the natural frequency of an oscillating gas body, suspended in a liquid medium, surrounded by a compliant vessel has become a topic of importance in diagnostic and therapeutic ultrasound due to increased interest in medical applications of ultrasonically excited bubbles (Dayton and Rychak, 2007; de Jong et al., 2000; Ferrara et al., 2007; Klibanov, 1999; Sassaroli and Hynynen, 2005). One-dimensional (1D) linear models of a cylindrical bubble in a rigid tube have been studied theoretically and experimentally (Geng et al., 1999; Oguz and Prosperetti, 1998; Sassaroli and Hynynen, 2005; Leighton, 1995). Qin and Ferrara (2006) developed a model for a bubble within a compliant vessel and tissue layer using COMSOL MULTIPHYSICS 3.2 that predicts a bubble natural frequency in agreement with the one-dimensional cylindrical bubble model but predicts frequency increases as the tube stiffness decreases (Qin and Ferrara, 2007). The goal of the present study is to further investigate the effect of a surrounding compliant vessel on the natural frequency of a gas body, both experimentally and with a coupled boundary element method and finite element method (BEM-FEM) model.

Methods

Theory and simulation

Simulations of the three phase system, consisting of the gas bubble, surrounding liquid, and solid elastic tube, were done using a coupled BEM-FEM model developed in Miao and Gracewski, 2008. The model geometry is axisymmetric with an initially spherical gas bubble, located with its center on the axis of symmetry and a circular cylindrical tube with its generator along the axis of symmetry, as shown schematically in the inset of Fig. 1. The acoustic excitation is modeled as a pressure applied at infinity in the surrounding liquid similar to that used in the Rayleigh–Plesset and Gilmore models for spherically symmetric bubble dynamics (Leighton, 1994; Young, 1989).

Figure 1.

Bubble normalized natural frequency versus ratio of bubble radius to tube radius. Experimental results (solid markers) are compared to 1D model predictions (open markers) and simulation results (gray markers). Marker shape indicates whether the bubble is at the middle (circles), intermediate (triangles), or end (squares) of the tube. Solid and dashed lines connecting the data points are added so that the 1D model predictions and simulation results, respectively, can be readily identified. Experimental value for R₀∕r_tube=0.66 at tube end was not obtained. Inset shows a spherical gas bubble inside a circular tube immersed in liquid.

The gas inside the bubble is assumed to be spatially uniform and to obey the polytropic gas law (Prosperetti, 1991). A pressure jump is applied at the gas-liquid boundary equal to the surface tension times the local curvature (Miao and Gracewski, 2008; Hartland, 2004). The liquid is assumed to be incompressible, irrotational, and inviscid, and therefore modeled with the potential flow equations using the BEM. A linear elastic FEM was employed to solve the dynamic equations in the solid structure domain. Tractions and the normal velocity are assumed continuous across the fluid-solid boundary to couple the finite element and boundary element domains. To obtain the natural frequency of a bubble in a tube using the BEM-FEM model, an initial tensile pulse of half cycle sinusoid is applied at infinity to trigger the bubble’s harmonic oscillation. The free vibration period is determined from the equivalent radius (radius of a spherical bubble with equal volume) versus time plot after the excitation pulse is over. Except where indicated, bubbles in the simulations were excited by a half pulse of amplitude 1 kPa and frequency 1 kHz.

The results from the simulations and experiments were compared to the model for the resonance frequency of large bubbles developed in Oguz and Prosperetti, 1998, which is accurate when the initial radius, R₀, of the bubble is greater than ∼0.2 times the tube radius, r_tube. This model replaces the spherical gas bubble with a cylindrical one of the same volume, occupying the cross-section of the tube, as shown in the inset of Fig. 2. This simplifies the system to one-dimensional motion, where the gas bubble provides the effective stiffness, and the two liquid columns to either side provide the effective inertia, from which the natural frequency of the spring-mass system can readily be determined. The bubble position is specified by L₁ and L₂, the distances from the bubble center to the left and right ends of the tube, respectively, in a tube of length, L. The effective length of the liquid column on either side is obtained by subtracting half of the thickness h from L₁ and L₂, and adding a correction factor ΔL=0.62r_tube that accounts for the inertia of the liquid outside the tube (Levine and Schwinger, 1948). For this model, the natural frequency of the bubble is determined as

$${\left(\frac{f}{f_0}\right)}^2=\frac{r_{\text{tube}}^2}{4R_0}(\frac{1}{L_1-h∕2+\Delta L}+\frac{1}{L_2-h∕2+\Delta L}),$$ (1)

where f_o is the open volume resonance frequency of the bubble.

Figure 2.

Bubble normalized natural frequency versus its normalized position. The bubble radius is 1 cm, tube radius is 2 cm, and tube length is 20 cm. Solid line is the 1D model with large bubble assumption, and dotted line is the simulation result with surface tension of 20 N∕m in an essentially rigid tube (E=2 GPa). The simulation without surface tension (not shown) predicted frequencies about 0.5% lower than the 1D model, whereas the simulation with surface tension predicted frequencies on average 1.0% higher than the 1D model up to 2z∕L≈0.8. Inset shows the spherical gas bubble replaced by a cylindrical bubble of equal volume, with radius equal to the tube radius.

Experimental methods

Resonance frequency was measured for a range of bubble sizes, bubble axial positions, tube sizes, and tube materials. In the proposed biomedical applications, gas bubbles with a diameter of a few microns will be injected into the blood stream and pass through capillaries in the human body with diameters several times larger than that of the gas bubbles. The system of interest is scaled up for the laboratory testing, where a balloon of approximately 1.0–2.0 cm in diameter is placed within a tube of 2.5 cm diameter.

Tubes with a range of stiffnesses were used, including Plexiglas, high-density polyethylene (HDPE), and polyvinyl chloride (PVC). The tube material properties and dimensions are given in the Table 1. The cross-section of the PVC tube was slightly oval and the radius of the minor axis was used for the simulation and model calculations. If the material tensile properties could not be readily obtained from the manufacturer, they were determined by following ASTM D638-03 procedure—Standard Test Method for Tensile Properties of Plastics. The samples were prepared in the form of standard dumbbell-shaped type I specimens and pulled with an MTS Alliance RT∕50. The elastic modulus values in Table 1 were obtained using a strain rate of 0.001 in.∕s in the strain range from 0% to 2%. Poisson’s ratio for each material was determined to be approximately 0.499 from the elastic modulus values and longitudinal wave speed measurements across the thickness of samples, and this value was used for the simulation.

Table 1.

Material properties, dimensions of tubes used in the experiments, and corresponding balloons.

Material	Length (cm)	Radius (mm)	Thickness (mm)	Density (g∕cm3)	Elastic modulus (MPa)	R0∕rtube
Plexiglas	19.1	12.5	2.6	1.19	2200	0.60
HDPE	20.5	12.5	3.0	0.91	170	0.58, 0.66, 0.69
PVC	21.0	11.3	2.6	1.21	16	0.70, 0.74, 0.80

A spherical air-filled finger-cot (balloon) was used to experimentally model a gas bubble. Each balloon was inflated using a syringe to a gauge pressure of 60 cm of water. The resulting balloon radii ranged between 7.0 and 9.0 mm, corresponding to a membrane tension range of 20–26 N∕m. For this range, the nondimensionalized membrane tension σ∕(p₀R₀) is an order of magnitude smaller than for the surface tension of a 3 μm diameter bubble in water and the presence of the membrane increases the resonance frequency by less than 5% (Young, 2006). Therefore, the balloon was considered to be an adequate representation of a bubble. The measured balloon radii were compared to the values calculated from the balloon’s measured open volume resonance frequency without the tube using the Minnaert equation with surface tension σ (Minnaert, 1933),

$$f_0=\frac{1}{2\pi R_0}{(\frac{3\Gamma p_0}{\rho }(1+\frac{2\sigma }{p_0{R}_0})-\frac{2\sigma }{\rho R_0})}^{1∕2}.$$ (2)

where R₀ is the equilibrium bubble radius, Γ is the polytropic exponent, p₀ is the ambient liquid pressure, and ρ is the liquid density. The values for these constants used in the calculations are Γ=1.4, p₀=101 230 Pa, and ρ=1000 kg∕m³. Due mainly to folding of the balloon’s membrane where it was tied off, there were minor differences (<6%) between the balloon’s measured and calculated radii. The last column of Table 1 summarizes the balloon sizes used for each tube. The middle, intermediate, and end positions were approximately 2.0 cm, 6.0 cm, and 9.7 cm, respectively, from tube end.

A stainless steel cylindrical exposure chamber with a shaker (Labworks Inc., model ET-140) on the bottom was used to measure the bubble resonance frequency. The chamber, 25.5 cm in diameter and 35.5 cm in height, was filled with degassed, de-ionized water at room temperature. Each balloon was held in position with strings tied to a vertical flexible tube or the supporting fixture and located using a three-way positioner such that the balloon center was 10 cm below the water surface and centered in the exposure chamber. The shaker was excited by a digital signal generator (Hewlett-Packard HP33120A) and power amplifier (Labworks Inc. PA141). Frequency was swept over a specified range, from 80 to 500 Hz, well below the lowest tank resonance frequency of approximately 950 Hz. A hydrophone (B&K 8103) was used to measure the pressure near a balloon and the resonance frequency of a balloon was identified by a peak in the pressure versus frequency plot.

Results and Discussion

Measured and predicted values of the normalized frequency ratio, f∕f₀, are presented in Fig. 1 for all of the balloon radii and positions used experimentally. The results in Fig. 1 are ordered by the ratio of balloon radius to the tube radius. A bubble’s natural frequency in a stiff tube decreases as the bubble’s position is moved from tube end to tube center, as well as when the bubble size increases. There is a good agreement between the results from the experiments, 1D model predictions, and the BEM-FEM simulations, especially when the bubble is located near the tube center.

The experimental variability was higher for balloons positioned near the tube’s end than at intermediate positions or in the middle of the tube, due to the higher sensitivity of frequency on axial position. In Fig. 2, the frequency ratio versus bubble position along the tube is plotted to illustrate this higher sensitivity. This plot also shows that the 1D model results are close (∼1% difference) to the BEM-FEM simulation results when the bubble is near the middle of the tube. Near the tube ends, Eq. 1 becomes less accurate because the correction factor ΔL dominates either L₁ or L₂. In comparison, the BEM-FEM simulation predicts the bubble frequency to asymptotically converge to that for a bubble in an open volume, as the bubble moves away from the tube. Simulations predict that the effect of the tube presence is still noticeable (f∕f₀=0.93) for a distance of one bubble radius away from the tube end.

The animation in Mm. 1 discussed in Fig. 3 shows that a bubble initially positioned at the end of the tube is drawn into the tube as the bubble oscillates. The bubble with 1 cm radius is initially positioned such that one-half is inside the tube with 2 cm radius and 20 cm length. The elastic modulus was chosen to be 2 GPa, so that the tube is essentially rigid. The bubble is excited by a half pulse with frequency 500 Hz and amplitude 10 kPa. The bubble frequency decreases as the bubble moves into the tube, consistent with the resonance frequency trend shown in Fig. 2. Though similar behavior is observed for a bubble initially at an intermediate position along the tube, the magnitude of translational motion is less pronounced, because the translational velocity decreases as the initial bubble position approaches the middle of the tube. Slight asphericity can also be observed in the shape of bubble oscillation, even with this small amplitude.

Figure 3.

Bubble position at successive peaks of the oscillation versus time curve corresponding to Mm. 1, showing that the bubble translates into the tube as it oscillates. The bubble radius is 1 cm, tube radius is 2 cm, tube length is 20 cm, and the tube is essentially rigid (E=2 GPa). Excitation is a tensile half pulse with frequency 500 Hz and amplitude 10 kPa.

Mm. 1.

Download video file ^{(4.1MB, avi)}

[Animation of a 1 cm radius bubble initially at the tube opening, showing the bubble translating into the tube as it oscillates. This is a file of type “avi” (4.11 Mbytes).]

The asphericity of the bubble oscillations increases with oscillation amplitude, as shown in the animation in Mm. 2 discussed in Fig. 4 for a bubble that experiences a 45% change in volume upon expansion. Simulation parameters are the same as in Fig. 3, except that the excitation pulse amplitude is 100 kPa and the bubble is positioned at the middle of the tube, so it does not translate. The aspect ratios (z dimension∕r dimension) shown in Fig. 4 are 1.07 and 0.69 for the first expansion and collapse, respectively. The displacements of the bubble top and bottom surfaces (at r=0) along the axis of the tube are 73% larger than the radial displacements of the bubble sides (at z=0). In the animation in Mm. 2, the bubble shape alternates between more aspherical and nearly spherical for each successive oscillation. For example, the asphericity decreases during the second expansion and collapse, with aspect ratios equal to 1.01 and 1.00, respectively, but increases again for the third oscillation. The particular behavior of the bubble oscillations depends on the bubble to tube size ratio, the oscillation amplitude, and the tube stiffness. The asphericity also increases as R₀∕r_tube increases (results not shown).

Figure 4.

Bubble shape at time t=0 and at its first maximum and minimum volume corresponding to Mm. 2, showing aspherical bubble oscillations. The bubble radius is 1 cm, tube radius is 2 cm, tube length is 20 cm, and the tube is essentially rigid (E=2 GPa). Excitation is a tensile half pulse with frequency 500 Hz and amplitude 100 kPa.

Mm. 2.

Download video file ^{(3.5MB, avi)}

[Animation of a 1 cm radius bubble centered in a tube, showing aspherical bubble oscillations. This is a file of type “avi” (3.5 Mbytes).]

Summary

In this paper, experiments and simulations were used to investigate the effect of a surrounding tube on the resonance frequency and free vibration response of a bubble. To the authors’ knowledge, this is the first published experimental measurement of resonance frequency of a spherical gas body in a tube. Simulations were obtained with a coupled BEM-FEM code that was developed specifically to investigate the response of acoustically excited bubbles near deformable structures. Simulated and experimentally measured resonance frequencies decreased as the bubble moved into the tube or increased in size. For a bubble near the center of the tube, the resonance frequency was consistent with a 1D cylindrical bubble model for a large bubble in a rigid tube. For a bubble near the end of the tube, the experimental variation in resonance frequency increased due to inaccuracies in bubble location and the higher sensitivity of frequency on position near the end of the tube. The BEM-FEM simulation results predict that the resonance frequency asymptotically approaches the open volume value as the bubble moves away from the tube end into the open volume. The effect of tube elastic modulus on bubble frequency is the subject of further investigation; however, varying the magnitude of the tube elastic modulus from a few kPa to few GPa in the BEM-FEM simulations had a negligible effect on the predicted bubble frequency (results not shown). Simulation of the free vibration response of the bubble also predicts that a bubble will translate toward the center of the tube due to aspherical oscillations. This is similar to the bubble translation observed in simulations by Ory et al. (2000) for the much larger expansion and collapse of a vapor bubble in a narrow tube. The asphericity increases with vibration amplitude, tending to be more elongated in the axial direction on expansion and in the radial direction on collapse.