Original contributionSubharmonic Response of Encapsulated Microbubbles: Conditions for Existence and Amplification
Introduction
Encapsulated microbubbles, known also as ultrasound contrast agents (UCA), are used for imaging of blood vessels. UCA are small gas bubbles with diameters of a few micrometers, encapsulated by a thin coating (e.g., albumin or lipid) and filled with a gas of high molecular weight and low diffusivity. The gas may be named as an osmotic gas (Kabalnov et al. 1998a). UCA are usually injected i.v. and traverse the lung capillaries before reaching the heart and the internal organs, where imaging is usually performed. Substantial improvement in contrast can be obtained by using the second harmonic, the super-harmonic, or the subharmonic signal emitted by the microbubbles, with preference given to the latter (Krishna et al 1999, Shi et al 1999, Shankar et al 1997, Shankar et al 1999). Subharmonic signals emitted by UCA were also shown to have a strong correlation with the ambient hydrostatic pressure - stronger than the correlation with the first or the second harmonics. This property suggested using UCA for noninvasive blood pressure measurements (Shi et al 1999, Krasovitski et al 2004, Adam et al 2005).
The diameters of the UCA bubbles vary in time mostly due to two processes. The first is a relatively slow diffusion of the osmotic gases gradually reducing the bubble size. This diffusion process is mostly affected by the permeability of the shell to the osmotic gas and the air, the mechanical properties of the shell, the ambient pressure and gas content in the surrounding liquid (Kabalnov et al 1998a, Kabalnov et al 1998b, Krasovitski et al 2004).
The second type of process affecting the time variation of bubble size is the rapid oscillations of the bubble diameter in response to the high frequency ultrasonic pressure oscillations.
When the resonant frequency of the bubble is close to half the applied frequency a subharmonic response may appear. Theoretical prediction of the conditions of existence of the subharmonics was first obtained by Eller and Flynn (1969) and later by Prosperetti (1974). Both studies considered free gas bubbles in noncompressible liquid. Prosperetti (1974) used the asymptotic method of Bogoliubov-Krylov (1961) to obtain an expression for the subharmonic amplitude. Subharmonic threshold is defined by the minimal effective pressure that provides existence of the expression (Prosperetti 1974). Shankar et al. (1999) expanded the solution reported by Prosperetti to the case of encapsulated bubbles. The shell response in their solution was taken into account using the simplified technique suggested by de Jong (1993). According to Eller and Flynn (1969), the subharmonic threshold is defined as the condition where a pure harmonic solution of the equation of motion of the bubble becomes unstable. This instability is expected to lead to generation of a subharmonic component.
Here, we expand the techniques that were previously developed for free bubbles, to determine the subharmonic intensity and its range of existence (threshold) for a cloud of encapsulated microbubbles. Changes in the properties of the microbubble caused by gas diffusion are also considered.
Section snippets
Subharmonic intensity and threshold
Consider an encapsulated microbubble that is submerged in liquid and exposed to a uniform ultrasound field. The bubble initially contains an osmotic gas. The bubble dynamic equation may be written as (Church 1995, Krasovitski et al 2004): Here R = instantaneous bubble radius; t = time; ρl = density of surrounding liquid; Re = bubble equilibrium radius at the moment of ultrasound application; R0 = bubble initial
Ultrasound contrast agent (UCA)
The ultrasound contrast agent Definity™ (Bristol-Myers Squibb Medical Imaging, N. Billerica, MA, USA) comprises of perfluropropane microbubbles coated with a flexible lipid shell. According to the manufacturer’s description, the mean diameter range is 1.1 μm to 3.3 μm, with only 2% of bubbles larger than 10 μm, and a maximal diameter of 20 μm.
To obtain more information about the size distribution of the bubbles and the mechanical properties of the lipid shell several preliminary tests were
Attenuation measurements
Once the bubble size distribution of the Definity™ UCA is known, as depicted in Fig. 1, some shell properties of the microbubbles may be evaluated. To that extent, the attenuation of ultrasound beam was measured at different frequencies while passing through the layer of the bubbly solution (within the inner chamber in Fig. 2). The results of the attenuation experiments (mean ± SD for six experiments) are shown in Fig. 4.
One may notice that the maximum attenuation occurs at frequencies of about
Acknowledgments
This research was supported by a grant from the Star Foundation (Michigan) and by the Center for Absorption in Science, Ministry of Immigrant Absorption.
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2020, Physics Letters, Section A: General, Atomic and Solid State PhysicsCitation Excerpt :Pioneering theoretical work of Eller [64] and Prosperetti [65–68] investigated the pressure threshold for the generation of subharmonics for uncoated free bubbles. Later, studies have theoretically and experimentally investigated the pressure threshold for the generation of SHs in the encapsulated bubbles [69–72]. The focus of these studies was on the determination of the conditions required to achieve the lowest pressure threshold that can produce 1/2 order SHs.