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Acoustic Interaction Between a Coated Microbubble and a Rigid Boundary
Kostas Efthymiou & Nikos Pelekasis
Laboratory of Fluid Mechanics & Turbomachinery, Department of Mechanical Engineering, University of Thessaly, Volos, Greece Poster Published at 1st International Conference on Micro & Nanofluidics, 18-21 May 2014, Twente, The Netherlands
Motivation • Contrast perfusion imaging check the circulatory system by
means of contrast enhancers in the presence of ultrasound
(Sboros et al. 2002, Frinking & de Jong, Postema et al.,
Ultrasound Med. Bio. 1998, 2004)
• Sonoporation reinforcement of drug delivery to nearby cells
that stretch open by oscillating contrast agents
(Marmottant & Hilgenfeldt, Nature 2003)
• Micro-bubbles act as vectors for drug or gene delivery to
targeted cells
(Klibanov et al., adv. Drug Delivery Rev., 1999, Ferrara et al.
Annu. Rev. Biomed., 2007)
Need for specially designed contrast agents:
• Controlled pulsation and break-up for imaging and perfusion
measurements
• Chemical shell treatment for controlled wall adhesion for
targeted drug delivery
Need for models covering a wider range of CA behavior
(nonlinear material behavior, shape deformation, buckling,
interfacial mass transport etc., compression vs. expansion only
behavior, nonlinear resonance frequency-thresholding)
Need to understand experimental observations and standardize
measurements in order to characterize CA’s
Experiments have shown that the presence of a nearby wall affects the bubble’s
oscillations. In particular its maximum expansion
Asymmetric oscillations, toroidal bubble shapes during jet inception have been
observed
The bubble oscillates asymmetrically in the plane normal to the wall, while it
oscillates symmetrically in the plane parallel to the wall (i.e. deformation has an
orientation perpendicular to the wall)
(H. J. Vos et al., Ultrasound in Med. & Biol., 2008) (S. Zhao et al., Applied Physics,
2005)
Asymmetric oscillations of a microbubble near a wall
Methodology Define a global cylindrical coordinate system (ρ, z) with its origin at the crosss-
section of rigid wall with the axis of symmetry along with a local spherical
coordinate system (r, θ) with its origin at the centre of mass of the microbubble
The coordinates of the interface are computed via FEM using the kinematic
condition in r-direction and the tangential force balance on the interface
The velocity potential is computed via FEM using the dynamic condition on the
interface and the normal velocity of the interface is calculated via BEM using the
boundary integral equation
Integration in time of the kinematic and dynamic interfacial conditions is
performed via the 4th order explicit Runge – Kutta method
Assumptions 1) Axisymmetry, 2) Ideal, irrotational flow of high Reynolds number,
3) Incompressible surrounding fluid with a sinusoidal pressure change in the far field,
4) Ideal gas in the microbubble undergoing adiabatic pulsations, 5) Very thin
viscoelastic shell whose behavior is characterized by the constitutive law (e.g.Hooke,
Mooney-Rivlin or Skalak), 6) The shell exhibits bending modulus that determines
bending stresses along with curvature variations, 7) Shell parameters: area dilatation
modulus χ=3Gsδ, dilatational viscosity μs, degree of softness b for strain softening
shells or area compressibility C for strain hardening ones and the bending modulus kB
Uncoated Microbubble – Rigid Boundary
Step Change in Pressure Field
Sinusoidal Change in Pressure Field
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
r
z
Evolution of the microbubble in time
T=0.00
T=3.06
T=5.38
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
r
z
Evolution of the microbubble in time
T=0.00
T=6.35
T=14.45
0 5 10 15-0.5
0.0
0.5
1.0
1.5=2, d=1 P
0
P1
P2
P3
P4
Am
plit
ud
e
Time
0 5 10 15-1
0
1
2
3
4
5
6
7 =2, d=1 Kinetic
surface
Compression
Total
Ma
gn
itu
de
Time
0 1 2 3 4 5 6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
=2, d=1 P0
P1
P2
P3
P4
Am
plit
ud
e
Time
0 1 2 3 4 5 6 7-1
0
1
2
3
4
5
6
7 =2, d=1 Kinetic
surface
Compression
Total
Ma
gn
itu
de
Time
Close to the boundary Close to the boundary
Coated Microbubble – Rigid Boundary
Step Change in Pressure Field Sinusoidal Change in Pressure Field
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.5
0
0.5
1
1.5
2
2.5
3
r
z
Evolution of the microbubble in time
T=0
T=12.3
-2 -1.5 -1 -0.5 0 0.5 1 1.5 28
8.5
9
9.5
10
10.5
11
11.5
12
r
z
Evolution of the microbubble in time
T=0.00
T=23.30
0 5 10 15-0.2
0.0
0.2
0.4
0.6
0.8
1.0=2, d=1
P0
P1
P2
P3
P4
Am
plit
ud
e
Time
0 5 10 15 20 25 30
-0.2
0.0
0.2
0.4
0.6
0.8
1.0=2, d=5 P
0
P1
P2
P3
P4
Am
plit
ud
e
Time
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5197.5
198
198.5
199
199.5
200
200.5
201
201.5
202
202.5
r
z
Evolution of the microbubble in time
T=0.00
T=118.80
T=128.40
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
r
z
Evolution of the microbubble in time
T=0.00
T=125.10
T=127.80
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
r
z
Evolution of the microbubble in time
T=0.00
T=18.50
T=21.60
0 20 40 60 80 100 120 140
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
=2, d=100 P0
P1
P2
P3
P4
Am
plit
ud
e
Time
0 20 40 60 80 100 120 140-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2 =2, d=5 P0
P1
P2
P3
P4
Am
plit
ud
e
Time
0 5 10 15 20 25 30-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
=2, d=1 P0
P1
P2
P3
P4
Am
plit
ud
e
Time
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2198
198.5
199
199.5
200
200.5
201
201.5
202
r
z
Evolution of the microbubble in time
T=0.00
T=28.00
Far from boundary At middle distance Close to the boundary Far from boundary At middle distance Close to the boundary
Results
In all cases the microbubble approaches the rigid boundary. For smaller initial distance between the microbubble and the rigid
boundary, the attraction among them is greater
In the case of step change in pressure field the oscillations of the coated microbubble are dumped due to dilatational viscosity of
the shell – For the uncoated one the total energy is conserved whereas for sinusoidal change it oscillates in time with the period
of the acoustic pressure field
The amplitude of the volume pulsations is not significantly affected as the microbubble approaches the wall (the backscatter
cross-section is also not affected)
The translational velocity of the coated microbubble is constant for step change in pressure field while it is increasing for
sinusoidal change due to the interaction with the boundary (secondary Bjerknes force)
Once the volume pulsation stops the microbubble keeps propagating due to the absence of viscosity in the surrounding fluid
The initial distance between the coated microbubble and the wall affects the shape modes growth (e.g. in this particular case far
from the boundary shape mode 4 is emerging faster via harmonic resonance (four-lobed microbubble), while close to the
boundary shape mode 3 is emerging (three-lobed microbubble)
In the case of the uncoated microbubble at the late stages of the motion, the greater mobility of the fluid away from the
boundary causes the surface of the bubble there to collapse faster than elsewhere, leading to the formation of a liquid jet that
traverses the bubble and ultimately impacts upon the far side of the bubble
In the case of the coated microbubble, at the late stages of the motion, formation of jet is suppressed due to the presence of the
viscoelastic shell. Instead, in all cases, the microbubble exhibits a deformed shape without permitting extreme elongations
As time evolves bending energy arrests growth of the kinetic energy – During the final stages areas of very small radius of
curvature emerge during the compressive phase of the pulsation – Elastic energy is preferentially channeled towards bending
until a conical angle is expected to locally form at a finite time in which case the computations have to stop
Ongoing simulations with varying element size with local curvature are performed in order to accurately capture the above
pattern of break-up
14
0 33.6 , 20 , 80 , 0.051 , 3 10 ,
0 (neo-Hookean), 1.7 , 1.07, 101325 , 0, 1
s D B
f st l
NR m Pa s G MPa k N mm
b v MHz P Pa nm
Parameters for coated microbubble (phospholipid-type shell):
Parameters for air (uncoated) microbubble in water:
0 03.6 , 0.075 , 1.7 ,
1.7 , 1.4, 101325 , 0f st l
NR m v MHzm
v MHz P Pa
0 5 10 15 20 25 30 35-1
0
1
2
3
4
5
6
7=2, d=1 Kinetic
Surface
Compression
Strain
Bending
Total
Ma
gn
itu
de
Time
20 21 22 23 24 25 26 27
0.0
0.1
0.2
0.3
0.4
Kinetic
Surface
Strain
Bending
Ma
gn
itu
de
Time
0 50 100 150-1
0
1
2
3
4
5
6
7=2, d=5 Kinetic
Surface
Compression
Strain
Bending
Total
Ma
gn
itu
de
Time
120 122 124 126 128 130
0.0
0.1
0.2
0.3
0.4
Kinetic
Surface
Strain
Bending
Ma
gn
itu
de
Time
0 50 100 150
0
2
4
6
=2, d=100 Kinetic
Surface
Compression
Strain
Bending
Total
Ma
gn
itu
de
Time
130 131 132 133 134 135
0.0
0.1
0.2
0.3
0.4
Kinetic
Surface
Strain
Bending
Ma
gn
itu
de
Time
0 5 10 15 20 25 30
-0.2
0.0
0.2
0.4
0.6
0.8
1.0=2, d=100
P0
P1
P2
P3
P4
Am
plit
ute
Time
0 5 10 15 20 25 30 35
0
5
10
15
20
Ma
gn
itu
de
Time
Kinetic
Surface
Compression
Strain
Bending
Total
=2, d=100
0 10 20 30
0
5
10
15
20=2, d=5
Kinetic
Surface
Compression
Strain
Bending
Total
Ma
gn
itu
de
Time
0 5 10 15
0
5
10
15
20=2, d=1 Kinetic
Surface
Compression
Strain
Bending
Total
Ma
gn
itu
de
Time
Schematic representation of a coated
microbubble near a rigid boundary
depicting the global coordinate system
(ρ, z), the local coordinate system (r, θ),
the initial distance (d) between bubble
and wall, the disturbance of the
pressure field, where ε is the amplitude
of the latter and ξ is the Lagrangian
marker on the interface
26.0 26.5 27.0 27.5 28.0 28.5 29.0
0.00
0.11
0.22
0.33
17.29
17.48
17.67
17.86
Kinetic
Surface
Strain
Bending
Compression
Total
21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5
0.00
0.11
0.22
0.33
17.29
17.48
17.67
17.86
Kinetic
Surface
Strain
Bending
Compression
Total
12.00 12.25 12.50
0.00
0.11
0.22
0.33
17.29
17.48
17.67
17.86
Kinetic
Surface
Strain
Bending
Compression
Total
