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Chin. Phys. B Vol. 28, No. 11 (2019) 114301
Dynamics of an ultrasound contrast agent microbubble near
spherical boundary in ultrasound field∗
Ji-Wen Hu(胡继文)1,2, Lian-Mei Wang(王练妹)1, Sheng-You Qian(钱盛友)2,†, Wen-Yi Liu(刘文一)1,
Ya-Tao Liu(刘亚涛)1, and Wei-Rui Lei(雷卫瑞)1
1School of Mathematics and Physics, University of South China, Hengyang 421001, China
2School of Physics and Electronics, Hunan Normal University, Changsha 410081, China
(Received 27 August 2019; revised manuscript received 29 September 2019; published online 23 October 2019)
The goal of this article is to establish the conditions of excitation where one has to deal with ultrasound contrast
agent (UCA) microbubbles pulsating near biological tissues with spherical boundary in ultrasound field for targeted drug
delivery and cavitation-enhanced thrombolysis, etc., and contributes to understanding of mechanisms at play in such an
interaction. A modified model is presented for describing microbubble dynamics near a spherical boundary (including
convex boundary and concave boundary) with an arbitrary-sized aperture angle. The novelty of the model is such that an
oscillating microbubble is influenced by an additional pressure produced by the sound reflection from the boundary wall. It
is found that the amplitude of microbubble oscillation is positively correlated to the curve radius of the wall and negatively
correlated to the aperture angle of the wall and the sound reflection coefficient. Moreover, the natural frequency of the
microbubble oscillation for such a compliable wall increases with the wall compliance, but decreases with the reduction of
the wall size, indicating distinct increase of the natural frequency compared to a common rigid wall. The proposed model
may allow obtaining accurate information on the radiation force and signals that may be used to advantage in related as
drug delivery and contrast agent imaging.
Keywords: ultrasound contrast agent (UCA) microbubble, spherical boundary, ultrasound, natural frequency
PACS: 43.25.+y, 43.80.+p, 02.90.+p DOI: 10.1088/1674-1056/ab4d3f
1. Introduction
Research of acoustical response of ultrasound contrast
agent (UCA) microbubbles has attracted wide interests from
both the medical and acoustical communities, not only for
providing a better understanding of their complex dynamics,
but also for their potential use for targeted drug delivery and
molecular imaging.[1–3]Apart from these medical applica-
tions, UCA microbubble sonication is being recently utilized
in several other medical areas, such as thrombus dissolution,
gas embolotherapy, sonoporation, and micro-pumping.[4–6]In
each case, it requires consistent mathematical and physical
knowledge of sonicated microbubble oscillation near different
types of boundaries.
Some experimental studies[7–9]have investigated mi-
crobubble dynamics in confined spaces and have shown that
the proximity of a boundary can produce considerable changes
in the oscillation amplitude of a microbubble and its scat-
tered echo.[10,11]Moreover, there are some results indicat-
ing that microbubble collapse is mitigated as the driving fre-
quency increases, or the microbubble is closer to the rigid
boundary.[12]Theoretical attempts to model microbubble son-
ication in a confined configuration usually utilizes the bound-
ary integral method,[11]finite element method[13]or method
of images,[14,15]which describes the nonlinear oscillations of a
microbubble near rigid boundaries and on the effects of bound-
aries such as a single wall, two parallel walls, or a tube, and so
on. Most investigations focused on a cavitation microbubble
occurring near a flat boundary. In fact, microbubble cavitation
in a curved boundary is not uncommon in human body tis-
sues. It is well known that red blood cells, which contain large
amount of concave shape and are approximately 7.5–8.7 µm
in diameter[16]whereas the mean diameter of UCA microbub-
bles is generally less than 5 µm.[17]Medical imaging shows
the growing atherosclerotic plaque is ellipsoid or spherical in
shape,[18]and there are a lot of irregular filling cavities in the
clot aggregated by platelets and red blood cells crowded.[19,20]
Cavitation microbubbles often create many cavities or pits on
the surface of the target object during the removal of kidney
and bladder stones with focusing of the shockwave.[21]In ad-
dition, microbubble forms a jet directed toward the interface
due to interaction of cavitation microbubbles with cell mem-
branes and produces a porous wall during sonoporation,[22]
which potentially has a significant impact on membrane per-
meation observed in the experiment.[23]Accordingly, it is im-
perative to understand the bubble dynamics in soft tissues
with curved boundaries or cavities for such as targeted drug
delivery,[24]and so on.
A number of studies have been dedicated to the microbub-
∗Project supported by the National Natural Science Foundation of China (Grant Nos. 11774088 and 11474090), the Hunan-Provincial Natural Science Founda-
tion of China (Grant No. 13JJ3076), and the Science Research Program of Education Department of Hunan Province of China (Grant No. 14A127).
†Corresponding author. E-mail: syqian@foxmail.com
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