Experimental and numerical evidence of intensified non-linearity at the micro and nano scale: The lipid coated acoustic bubble
AJ Sojahrood, H. Haghi, T.M. Porter, R. Karshafian, M.C. Kolios
EExperimental and numerical evidence of intensifiednon-linearity at the micro and nano scale: The lipidcoated acoustic bubble
AJ. S
OJAHROOD a , b * ,H. H AGHI a , b , T.M. P ORTER c R. K
ARSHAFIAN a , b M ICHAEL
C. K
OLIOS a , b a Department of Physics, Ryerson University, Toronto, Ontario, Canada. b Institute for Biomedical Engineering and Science Technology,A Partnership Between Ryerson University and St. Michael’s Hospital, Toronto, Canada. c Department of Biomedical Engineering, The University of Texas at Austin, Texas, USA
February 26, 2021
Abstract
A lipid coated bubble (LCB) oscillator is a very interesting non-smooth oscillator with many importantapplications ranging from industry and chemistry to medicine. However, due to the complex behaviorof the coating intermixed with the nonlinear behavior of the bubble itself, the dynamics of the LCB arenot well understood. In this work, lipid coated Definity microbubbles (MBs) were sonicated with 25MHz 30 cycle pulses with pressure amplitudes between 70kPa-300kPa. Here, we report higher ordersubharmonics in the scattered signals of single MBs at low amplitude high frequency ultrasound excitations.Experimental observations reveal the generation of period 2(P2), P3, and two different P4 oscillationsat low excitation amplitude. Despite the reduced damping of the uncoated bubble system, such enhancednonlinear oscillations has not been observed and can not be theoretically explained for the uncoated bubble.To investigate the mechanism of the enhanced nonlinearity, the bifurcation structure of the lipid coated MBsis studied for a wide range of MBs sizes and shell parameters. Consistent with the experimental results, weshow that this unique oscillator can exhibit chaotic oscillations and higher order subharmonics at excitationamplitudes considerably below those predicted by the uncoated oscillator. Buckling or rupture of the shelland the dynamic variation of the shell elasticity causes the intensified non-linearity at low excitations. Thesimulated scattered pressure by single MBs are in good agreement with the experimental signals.
I. I
NTRODUCTION
Acoustically excited bubbles are known to be nonlinear oscillators [1, 2] with "infinite complexity" [1]. Theaddition of a lipid stabilizing shell increases this complexity, thus, even after over a decade of study, thedynamics of ultrasonically excited lipid coated microbubbles (MBs) are not fully understood. Interestingly,lipid coated MBs have been shown to exhibit 1/2 order subharmonic (SH) oscillations even when the excitationamplitude is low ( <
30 kPa [3–5]). Despite the increased inherent damping due to the coating, such lowthreshold values contradict the predictions of the theoretical models as these values are even below thethresholds expected for uncoated free MBs [6, 7].The lipid coating may cause compression dominated oscillations [8] or limit the MB oscillations to onlyabove a certain pressure threshold [9]. It has been shown in [3] that the low pressure threshold for 1/2 orderSH emissions is due to the compression only behavior of the MBs due to the buckling of the shell. Overveldeet al. [10] showed that the lipid coating may enhance the nonlinear MB response at acoustic pressures aslow as 10 kPa . In addition, even a small ( ≈ kPa ) increase in the acoustic pressure amplitude leads to a * Corresponding author email: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] F e b ignificant decrease in the main resonance frequency [10] resulting to a pronounced skewness of the resonancecurve. The origin of the “thresholding” [9] behavior has been linked to the shift in resonance [10]. Nonlinearresonance behavior of the lipid coating has also been observed at higher frequencies (5-15 MHz [11]), (8-12MHz [12]) and (11-25 MHz [13]). Through theoretical analysis of the Marmottant model for lipid coatedMBs [14], Prosperetti [6] attributed the lower SH threshold of the lipid MBs to the variation in the mechanicalproperties of the coating in the neighborhood of a certain MB radius (e.g. the occurrence of buckling).The coating may also result in expansion dominated behavior in liposome-loaded MBs [15]. Expansiondominated oscillations occur when the initial surface tension of the lipid coated MB is close to that of thewater [13, 15]. In this regime, and in contrast to compression dominated behavior, the MB expands morethan it compresses. Expansion-dominated behavior was used to explain the enhanced non-linearity at higherfrequencies (25 MHz) [13]. The Marmottant model effectively captures the behavior of the MB includingexpansion-dominated behavior [13, 14, 16], compression only behavior [8], thresholding [9] and enhancednon-linear oscillations at low excitation pressures [3, 4, 10, 16–18]. Despite the numerous studies whichemployed the methods of nonlinear dynamics and chaos to investigate the dynamics of acoustically excitedMBs [1, 2, 19–43], the detailed bifurcation structure and nonlinear dynamics of the lipid coated MBs have notbeen studied.Lipid coated MBs are being routinely used in diagnostic ultrasound [4, 44–47]. Moreover, they have beenused in pioneering non-invasive treatments of brain disorders and tumors in humans [48]. Currently thereare several investigations on the potential use of lipid coated micro-bubbles (MBs) in high resolution andhigh contrast imaging procedures [49] as well as non-invasive ultrasonic treatments and localized drug/genedelivery [50]. Despite the promising results of these investigations, the complex dynamics of the systemmakes it difficult to optimize the behavior of lipid coated MBs. Moreover, from the nonlinear dynamics pointof view, the lipid coated MB oscillator is an interesting topic of investigation due to the highly nonlinearnature of the system. The complex behavior of the uncoated MB is interwoven with the nonlinear behavior ofthe lipid coating which enables unique dynamical properties for this oscillator.In this work we study the bifurcation structure of the lipid coated MBs as a function of size and frequency atdifferent pressure values. Numerical results are accompanied by experimental observations of single MBscattering events at low pressure excitation. We show for the first time, both experimentally and numericallythat in addition to 1/2 order SHs, higher order SHs (e.g. 1/3, 1/4) can be generated at low excitation amplitudes.Analysis of the bifurcation structure of the system reveals a unique property of the lipid coated MB which isthe generation of chaos at excitation pressures as low as 5kPa. II. M
ETHODS i. Experimental method
Very dilute solutions of Definity MBs were sonicated with 25 MHz continuous pulse trains using the Vevo770 ultrasound imaging machine (VisualSonics Inc. Toronto, Ontario). The pulse length was held constant at30 cycles while the applied acoustic pressure was varied over the range of ≈ − kPa . The backscatteredsignals from individual MBs were extracted and different nonlinear modes of oscillations were identified.Acquisition of signals from single MB were similar to the approach in [51]. Fig. 1 shows a schematic of anacquired signal from a single MB event ( ≈ kPa and f = MHz ).2a) (b)
Figure 1: a) Schematic of the Vevo 770 (Visualsonics) machine which was used in the experiments to detect the signalsfrom single MB events in the region of interest (ROI). b) Left: ROI selected for an ultrasound pulse train at 25MHz and 250 kPa of pressure (each large subdivision is ≈ µ m), and Right: Signal (red) from a singleperiod-3 MB event. The frequency spectrum of the received signal is shown in blue exhibiting 1/3 order SHs at8.33 and 2/3 order SHs 16.66 MHz. i. Numerical procedure ii.1 Marmottant Model Dynamics of coated MBs undergoing buckling and rupture can be effectively modeled using the Marmottontequation [14]: ρ (cid:18) R ¨ R + ˙ R (cid:19) = (cid:20) P + σ ( R ) R (cid:21) ( RR ) − k (cid:18) − kc ˙ R (cid:19) − P − σ ( R ) R − µ ˙ RR − k s ˙ RR − P a ( t ) (1)In this equation, R is radius at time t, R is the initial MB radius, ˙ R is the wall velocity of the bubble, ¨ R is thewall acceleration, ρ is the liquid density (998 kgm ), c is the sound speed (1481 m/s), P is the atmosphericpressure, σ ( R ) is the surface tension at radius R, µ is the liquid viscosity (0.001 Pa.s), k s is the coatingviscosity and P a ( t ) is the amplitude of the acoustic excitation ( P a ( t ) = P a sin ( π f t ) ) where P a and f are theamplitude and frequency of the applied acoustic pressure. The numerical values in the parentheses are forpure water at 293 K. The gas inside the MB is C3F8 and water is the host medium.The surface tension σ ( R ) is a function of radius and is given by: σ ( R ) = i f R ≤ R b χ ( R R b − ) i f R b ≤ R ≤ R r σ water i f R > R r (2) σ water is the water surface tension (0.072 N/m), R b = R (cid:113) + σ χ is the buckling radius, R r = R b (cid:113) + σ r χ is therupture radius, χ is the shell elasticity, σ is the initial surface tension at R = R , and σ r is the rupture surfacetension. Similar to [13] in this paper σ r = σ water . Shear thinning of the coating is included in the Marmottantmodel using [52]: k s = k + α | ˙ R | R ; (3)where k is the shell viscous parameter and α is the characteristic time constant associated with the shearrate. In this work shell parameters of χ = N / m , k = × − kgs − and α = × − s are usedfor the De f inity
MBs. These values are adopted from the estimated parameters in [52–55].Due to the negligible thermal damping of C3F8 even at high amplitude oscillations [56, 57] thermal dampingis neglected in this paper. ii.2 Keller-Miksis model
The dynamics of the uncoated MBs were also visualized alongside the lipid coated MBs to highlight thecontributions of the coating to the nonlinear bheavior of the bubble. To model the uncoated MB dynamics theKeller-Miksis model [58] is used: ρ [( − ˙ Rc ) R ¨ R + ˙ R ( − R c )] = ( + ˙ Rc )( G ) + Rc ddt ( G ) (4)where G = (cid:16) P + σ water R (cid:17) ( RR ) − k − µ L ˙ RR − σ R − P − P A sin ( π f t ) .4 i.3 Scattered pressure by MBs The pressure scattered (re-radiated) by the oscillating MB can be calculated using [59, 60]: P sc = ρ Rr ( R ¨ R + ˙ R ) (5)here r is the distance from the MB center. The scattered pressure ( P sc ) at 15 cm (approximate path length ofthe MBs in experiments) is calculated for 30 cycle pulses to match experimental conditions. The calculated P sc is then convolved with the one way transducer response accounting for attenuation effects in water(0.000221 dBmmMHz [63]). Moreover, to better compare with experimental observations, the sampling frequencyfor simulations is chosen to be equal to the sampling frequency in experiments (460 MHz ). ii.4 Investigation tools The results of the numerical simulations were visualized using a comprehensive bifurcation analysis method[42]. In this method the bifurcation structure of the normalized MB oscillations ( RR ) are plotted in tandemversus a control parameter using two different bifurcation methods (Poincaré section at each driving periodand method of peaks). The bifurcation diagrams of the normalized MB oscillations ( R / R ) are calculatedusing both methods . When the two results are plotted alongside each other, it is easier to uncover moreimportant details about the superharmonic (SuH) and ultraharmonic (UH) oscillations, as well as the SHand chaotic oscillations. This gives insight into the nonlinear behavior over a wide range of parameters,and enables the detection of SuH and UH oscillations alongside SH and chaotic oscillations [42, 61, 62].This approach reveals the intricate details of the oscillations. In this paper the bifurcation diagrams of thenormalized radial oscillations of the De f inity
MBs were plotted versus the MB initial diameter for fixedfrequencies of 25 MHz and for the range of the pressure values studied in the experiments.
III. R
ESULTS i. Bifurcation structure
In i.1 we present the bifurcation structure of an uncoated C3F8 MB and a Definity MB with 2 µ m initial sizeas a function of frequency at P a = kPa for different values of initial surface tension σ . We show that thelipid buckling or rupture of the lipid shell enhances the unexpected generation of the nonlinear behavior.Despite the reduced damping, the unocated MB of the same size does not exhibit nonlinear oscillations. Ini.2 we study the bifurcation structure of the uncoated and Definity MBs as a function of the initial size atdifferent σ . The sonication frequency is fixed at 25 MHz (freqeuncy of the experiments) and P a = kPa .We show that depending on the bubble size and σ , different nonlinear regimes can be enhanced. i.1 Bifurcation of RR as a function of frequency for uncoated and coated MBs with different σ ( R ) In order to visualize the dynamics of the MBs at low pressures, the bifurcation structure of a 2 µ m MB isplotted as a function of frequency when P a = kPa (Fig. 2). The blue graph represents the results using themaxima method and the red graph represents the results using the conventional method (Poincaré sectionat each period). The uncoated MB (Fig. 2a) exhibits a P1 signal over 1-30 MHz with one maximum andresonant oscillations at ≈ MHz . Contrary to Fig. 2a, the lipid MB with σ ( R ) = th − nd order) are seen for f < ≈ MHz . Forexample the 5th order SuH at 1MHz is a P1 signal with 5 maxima. P1 resonance occurs at f (cid:117) MHz with P2 oscillations over a wide frequency range of ≈ MHz < f < MHz with a small window ofP4-2 and chaos. We call this P4-2 oscillations as it occurs when P2 oscillations undergo a period doubling(PD) to P4 [22, 43]. P3 occurs between ≈ MHz < f < MHz . P4-1 occurs in the frequency rangebetween 18.2
MHz < f < MHz (highlighted in subplot within Fig. 2b). We call this P4-1 regime as it5a) (b)
Frequency (MHz) R / R P a =5kPa, Diameter=2 m , =0.01N/m PDf r Frequency (MHz) R / R P a =5kPa, Diameter=2 m , =0.03N/m (c) (d) Frequency (MHz) R / R P a =5kPa, Diameter=2 m , =0.062N/m PDf r (e) (f) Figure 2:
The bifurcation structure (blue represent the maxima and the red represents the conventional method) ofthe RR of a µ m MB as a function of frequency for P a = kPa for the: a) uncoated MB and for the lipidMB with b) σ ( R ) = N / m, c) σ ( R ) = N / m, d) σ ( R ) = N / m, e) σ ( R ) = N / m and f) σ ( R ) = N / m. σ ( R ) = N / m (Fig. 2c) exhibits P1 oscillations with one maximum. The pressuredependent resonance frequency ( PD f r [41]) occurs at f ≈ MHz . The MB behavior is of P1 with onemaximum for all the studied frequencies.The lipid MB with σ ( R ) = N / m (Fig. 2d) exhibits P1 behavior with 1 maximum and a resonanceat ≈ MHz . The behavior of the MB with σ = N / m (Fig. 2e) is similar to σ = N / m with PD f r (cid:117) MHz .The MB with σ ( R ) = N / m exhibits a similar behavior to the MB with σ ( R ) = N / m demonstrating6 th − nd order SuHs between the freqeuncy range of 1-3.7 MHz. Some of the SuH between 1 MHz < f < MHz are highlighted in a subplot within Fig. 2f). P2 occurs between 9.3
MHz < f < MHz . P3 occursbetween 16.5
MHz < f < MHz (highlighted as a subplot inside Fig. 2f), between 19.7 < f < MHz and between 20.3
MHz < f < MHz . P4-1 regime occurs between 23.4
MHz < f < MHz (highlightedwithin a subplot in Fig. 2f). The MB also demonstrates P1 resonance frequency at ≈ MHz .Notably, despite the higher damping due to the coating, the coated MB undergoing shell rupture exhibits agreater oscillation amplitude (Figs. 2a and 2f). i.2 Bifurcation RR as a function of size (initial diameter) for uncoated and coated MBs with different σ ( R ) In order to investigate the nonlinear behavior of the commercially available
De f inity
MBs for the experimentalexposure conditions, the bifurcation structure of the RR is studied as a function of MB size when P a = kPa and f = MHz . This is because of the polydisperse nature of the
De f inity
MBs [55], and since in theexperiments we are limiting our analysis to the transducer focal zone with small variations in pressure and thefixed sonication frequency. The size distribution in the simulations replicates the distribution of the native
De f inity [55] Thus, the RR plot versus MB size will provide insight relevant to the experimental conditions inthis study.Fig. 3a, shows the bifurcation structure of an uncoated MB as a function of size. MB with sizes between0.27 µ m -0.28 µ m exhibit 2nd order SuH (P1 oscillation with 2 maxima as highlighted in a subplot) and MBswith sizes 0.54 µ m are resonant. Fig. 3a shows that at f = MHz and P a = kPa the uncoated MB cannotproduce SHs.Fig. 3b, shows the bifurcation structure of the De f inity @ MBs with σ ( R ) = N / m . In stark contrast to theuncoated MB (Fig. 3a), an abundance of nonlinear behavior is observed. This includes 4th, 3rd and 2nd orderSuHs for MB sizes smaller than 0.345 µ m (some are highlighted in a subplot within Fig. 3b), P1 resonance for ≈ µ m MBs , P2, P4-2 and chaotic behavior for MB sizes of ≈ µ m < R < µ m , P3 oscillationsfor ≈ µ m < R < µ m , and intermittent P4-1 oscillations for ≈ µ m < R < µ m (P4-1 ishighlighted in a subplot within Fig. 3b). De f inity
MBs with σ ( R ) = N / m (Fig. 3c), exhibit enhanced nonlinear behavior including 5 th − nd order SuHs (highlighted in a subplot), P2, P4-2, P3 and chaos. Fig. 3d represents the behavior of MBs with σ ( R ) = N / m . 2nd order SuH (highlighted in a subplot), PD f r , P2 and P3 oscillations are observed.Fig. 3e-f represent the MBs with initial surface tension close to that of water and thus with a higher tendencyfor rupture and expansion dominated behavior. For σ ( R ) = N / m and for MB sizes 0.2 µ m < R < µ m , 3rd and 2nd order SuH and 5/2 UH regimes are observed. 5/2 UH is a P2 with 4 maxima and ishighlighted in a subplot within Fig. 3e. PD f r , P2, P4-2 and P3 (highlighted in a subplot) oscillations areobserved for MB sizes 2 R > µ m . When σ ( R ) = N / m (Fig. 3f), in addition to the nonlinearbehavior we observe in Fig. 3e, we observe a 4th order SuH regime (highlighted in a subplot as a P1 with 4maxima) and P4-1 and the absence of 5/2 UHs.Results indicate that the nonlinear behavior of the MBs is highly sensitive to the initial surface tension aswell as the MB size. The closer the surface tension to 0 or that of water ( σ water = N / m ), the greater isthe tendency of the MB to exhibit nonlinear behavior. Notably, P4-1 oscillations were only observed when σ ( R ) = N / m (Fig. 3). 7 .2 0.7 1.2 1.7 2.2size( m)0.911.11.21.31.41.5 R / R P a =0.25MPa,f=25MHz, uncoated bubble (a) (b) R / R P a =0.25MPa,f=25MHz , =0.01N/m (c) (d) R / R P a =0.25MPa,f=25MHz , =0.062N/m size( m) R / R P a =0.25MPa,f=25MHz , =0.072N/m (e) (f) Figure 3:
The bifurcation structure of the RR (blue represent the maxima and the red represents the conventional method)as a function of size (MB diameter) at P a = kPa and f = MHz for the: a) uncoated MB and for thelipid MBs with b) σ ( R ) = N / m, c) σ ( R ) = N / m, d) σ ( R ) = N / m, e) σ ( R ) = N / m and f) σ ( R ) = N / m. Figure 4:
The bifurcation structure of RR as a function of σ ( R ) at f = MHz and P a = kPa for a MB size of: a)0.92 µ m & b) 1.89 µ m The MBs with σ ( R ) = σ ( R ) = σ ( R ) on the MB behavior, the bifurcation structure of the RR of the MB is plotted as a function of σ ( R ) for two different MB sizes in Fig. 4. The bifurcation structureof a MB with an initial diameter of 0.92 µ m is depicted in Fig. 4a. The nonlinear behavior occurs onlyfor the two extreme ends of the σ ( R ) . P2 occurs for σ ( R ) < N / m and σ ( R ) > N / m withP4-2 happening for 0.0032 N / m < σ ( R ) < N / m and σ ( R ) > N / m . For a MB with initialdiameter of 1.89 µ m , the same general behavior is observed. For initial surface tension values between0.0127 < σ ( R ) < N / m we observe P1 behavior with 1 maximum. As we approach to the lowerand higher σ ( R ) , nonlinear behavior manifests itself in the bifurcation diagrams. P4-1 oscillations occursfor 0.0035 N / m < σ ( R ) < N / m and σ ( R ) > N / m . P3 occurs for 0.009 N / m < σ ( R ) < N / m and 0.058 N / m < σ ( R ) < N / m . i.3 Experiments In experiments at P a ≈ kPa and f = MHz we observed 5 main types of backscattered signals in the datacollected from single MB events. A representative of each category is shown in Fig. 5a (P1), Fig. 5d (P2),Fig. 5g (P3) , Fig. 5j (P4-2) and Fig. 5m (P4-1). The results of the numerical simulations are presented in thesecond column and the frequency spectrum of the experimental signals and the numerical simulations areplotted in the third column (blue:experiments, red:simulations). Numerical simulations are for the
De f inity
MBs with σ ( R ) = N / m with the corresponding sizes chosen from the bifurcation diagram (Fig. 3f) tomatch the observed behavior in the experiments.Fig. 5a displays a typical P1 signal observed in experiments. The calculated P sc for a 2 µ m De f inity MB isdisplayed in Fig. 5b (in red color for distinction) and the power spectrum of the signals in Fig. 5a and 5b areshown in Fig. 5c. The scattered pressure has one maximum and the frequency spectrum has a peak at 25MHz.A representative signal of the P2 oscillations is displayed in the second row of Fig. 5. Both experimental andsimulated (initial size of 0.955 µ m ) signals have two maxima revealing a P2 oscillation regime. The powerspectra in Fig. 5f consist of a SH peak at 12.5MHz and a 3/2 UH peak at 32.5 MHz.A representative of the P3 signal is shown in the third row of Fig. 5. The experimental and simulated (initial9
10 20 30 40 50
Frequency (MHz) -70-60-50-40-30-20-100 A m p li t ud e ( d B ) P1 frequency spectrum at 250 kPa experimentsimulation (a) (b) (c)
Frequency (MHz) -70-60-50-40-30-20-100 A m p li t ud e ( d B ) P2 frequency spectrum at 250 kPa experimentsimulation (d) (e) (f)
Frequency (MHz) -70-60-50-40-30-20-100 A m p li t ud e ( d B ) P3 frequency spectrumb at 250 kPa experimentsimulation (g) (h) (i)
Frequency (MHz) -70-60-50-40-30-20-100 A m p li t ud e ( d B ) P4-2 frequnecy spectrum at 250 kPa experimentsimulation (j) (k) (l)
Frequency (MHz) -70-60-50-40-30-20-100 A m p li t ud e ( d B ) P4-1 frequency spectrum at 250 kPa experimentsimulation (m) (n) (o)
Figure 5:
Demonstration of 5 main oscillation regimes acquired experimentally (blue) and simulated (red) choosing MBsizes based on the feature similarity in Fig. 3f. Representative experimental data and simulations of: 1st rowP1, 2nd row P2, 3rd row P3, 4th row P4-2 and 5th row P4-1.
100 200 300 400 500
Pressure (kPa) R / R f=25MPa, diameter=0.92 m, =0.072 N/m P2Pd P4-2 (a)(b)
Figure 6:
The bifurcation structure of the RR of the MB as a function of pressure for excitation with f = MHz for alipid MB with σ ( R ) = N / m and a diameter of : a) 0.92 µ m & b) 1.89 µ m size of 1.39 µ m ) signals have 3 maxima and the order of the maxima are consistent between experiments andsimulations. The power spectra in Fig. 5i show a good agreement between experiments and simulations withSHs at (1/3 order) 8.33 MHz, (2/3 order) 16.66 MHz and UHs at (4/3 order) 33.33 MHz and (5/3 order) 41.66MHz.P4-2 oscillations are shown in the 4th row of Fig. 5. There is a good agreement between the experimentaland the simulated signals (initial size of 0.92 µ m ). Both signals have 4 peaks in two envelopes and eachenvelope repeats itself once every two acoustic cycles. In each envelope there are two peaks and the peaksrepeat themselves in an amplitude order of (largest, small, large, smallest). The frequency spectra of thesignals are shown in Fig. 5l. There are 3 SHs at (1/4 order) 6.25 MHz, (1/2 order) 12.5 MHz and (3/4) orderat 18.75 MHz. The 1/2 order SH is the strongest detected SH and due to the weakness of the 1/4 SH this peakis hardly detectable. This is because the transducer sensitivity drops sharply away from the center frequencyand especially below 12.5 MHz (transducer bandwidth is 100%). While the numerically simulated P sc in theabsence of convolution with transducer response had a clear peak at 6.25 MHz, however, after the signal isconvolved with the transducer response, the signal drops below the noise level of -70 dB in our experiments.The last row of Fig. 5 depicts the case of the P4-1 oscillations. Simulations are for a MB with initial sizeof 1.89 µ m . The signals have one envelope with 4 maxima that repeats itself once every 4 acoustic cycles.Amplitudes repeat themselves in the order of smallest, largest, large and small. Both experimental andsimulated signals demonstrate the same pattern of peaks and their orders. The power spectra in Fig. 5o showsa good agreement between the orders of the SHs and their locations. There are 3 SHs at (1/4 order) 6.25MHz, (1/2 order) 12.5 MHz and (3/4 order) at 18.75 MHz. The 3/4 order SH is the strongest detected SH. Itshould be noted that 1/4 order SH is the strongest peak in the calculated P sc in the absence of convolutionwith transducer response (see Fig. 7f). Due to the reduced sensitivity of the transducer at 6.25 MHz, the11a) (b) (c) freqeuncy (MHz) -70-60-50-40-30-20-100 A m p li t ud e ( d B ) P4-1 frequnecy spectrum f/4 f/2 3f/4 f (d) (e) (f)
Figure 7:
Characteristics of the two P4 oscillations identified: a) P4-2 radial oscillations, b) P4-2 phase portrait, c)power spectrum of the P4-2 P sc , d)P4-1 radial oscillations, e)P4-2 phase portrait and f) power spectrum ofP4-2 P sc . Here P sc is not convolved with the transducer response. (red circles shows the location of the R every4 acoustic cycles) detected strength of the 1/4 order SH diminishes strongly and it drops below all the other SHs. i.4 Difference between P4-1 and P4-2 Fig. 6a shows the bifurcation structure of a 0.92 µ m De f inity MB as a function of pressure when f = MHz .At low pressures there are linear oscillations with period doubling (Pd) at ≈ kPa . P2 oscillations undergofurther Pd to P4-2 oscillations at ≈ kPa . The process of P4-2 generation and disappearance is througha bubbling bifurcation. In case of the 1.89 µ m De f inity @ (Fig. 6b), P4 oscillations are generated through adirect period quadrupling via a saddle node bifurcation similar to [25]. This is the reason why we namedthis a P4-1 oscillations. Models for uncoated MBs or coated MBs with pure viscoelastic behavior predictvery high pressures for the generation of P4-1 oscillations; however, here we show, for the first time, that thedynamic variation of the shell elasticity including buckling and rupture enhances the generation of the P4-1oscillations at very low acoustic pressures ( P a ≈ kPa in Fig. 6b).Fig. 7 compares the radial oscillations, phase portraits and the power spectra of the P sc for both P4 oscillationsat f = MHz and P a = kPa . P4-2 radial oscillations consist of two envelops, with each envelope having2 maxima or one with 2 maxima and the other with a maxima and a critical point. These envelopes repeatthemselves once every two acoustic cycles in Fig. 7a. The phase portrait of the P4-2 oscillations consistsof a loop undergoing two internal loops with the largest loop undergoing another internal loop. The powerspectrum depicts SHs with strength order of 1 / > / > /
4. P4-1 oscillations in Fig. 7d have one envelopewith 4 maxima which repeats itself once every 4 acoustic cycles. The phase portrait consist of a main loopthat has undergone 3 bends to create 3 internal loops. The frequency spectrum of P sc depicts SHs in thestrength order of 1 / > / > /
4. It should be noted that due to the lower sensitivity of the transducer aswe move away from central frequency, the strength order of the SHs that are detected in experiments weredifferent. After, convolving the simulations results with the one way transducer response, experiments and12imulations were in good agreement. To our best knowledge, this is the first time that the two types of P4oscillations are detected experimentally and characterized numerically for a MB oscillator.
IV. D
ISCUSSION
A MB oscillator is an extremely complex system that has beneficial applications in a wide range of fieldsincluding material science and sonochemistry [64–66], food technology [67] underwater acoustics [68, 69]and medical ultrasound (ranging from imaging blood vessels [70], drug delivery [50] to thrombolysis [71]and the treatment of brain through intact skull [48, 72]). In addition to these important applications, thecomplex dynamical properties of the MB system make it a very interesting subject in the field of nonlineardynamics. It is well known that an ultrasonically excited MB is a highly nonlinear oscillator. Due to theimportance of the understanding of the MB behavior in several applications, numerous studies have employedthe methods of nonlinear dynamics and chaos to study the complex behavior of the system. Pioneering worksof [19, 66, 73] have revealed several nonlinear and chaotic properties of the MB oscillations (both numericallyand experimentally). Recent extensive studies on the nonlinear behavior of MBs in water [22–26, 42, 43],coated MBs [25, 26, 41], MBs in highly viscous media [27–32], MBs sonicated with asymmetrical drivingacoustic forces [32–40] and MBs in non-Newtonian fluids [36] have revealed many nonlinear features in theMB behavior. Occurrence of P2, P3, P4-2, P4-1 and higher periods, as well as chaotic oscillations, has beendemonstrated in these works. Moreover, the effect of nonlinear dynamics of MBs on the propagation of soundwaves in a bubbly medium is under recent investigation [25, 74, 75].Despite these studies that employed the methods of chaos physics to investigate the nonlinear dynamics ofthe uncoated and coated MBs with viscoelastic behavior, the effect of the lipid coating on the dynamics of theMB especially in the realm of nonlinear dynamics and chaos has not been systematically investigated.In this study we investigated the bifurcation structure of the lipid coated MBs and used the numerical resultsto help interpret unique signals that we observed experimentally. In stark contradiction to the results ofclassical theory of uncoated MBs, and despite the increased damping of the coated MBs, lipid coated MBsexhibited higher order nonlinear behavior at low excitation amplitudes (shown here both experimentally andnumerically). The numerical and experimental findings can be summarized as follows:a- Theoretically, we have shown that even at pressures as low as 5 kPa , 6 th − nd order SuHs, P4-2, P2, P3P4-1 and chaotic regimes manifest themselves in the MB behavior. To our best knowledge the existence ofhigher order SHs and chaotic behavior at such low excitation amplitudes is first reported here.b- The initial surface tension of the MB plays a critical role in the enhanced nonlinear behavior. We haveshown that the closer σ ( R ) is to 0 (leading to buckling and compression only behavior) or to σ water (leadingto shell rupture and expansion dominated behavior), the lower the excitation threshold for nonlinear behaviorand the higher the order of non-linearity.c- Despite the increased damping of the lipid coated MBs we show that, the MBs with surface tension ≥ N / m may have higher radial oscillation amplitude compared to the uncoated bubble.d- We have experimentally shown that single De f inity
MBs, can exhibit, P2, P3, P4-2 and P4-1 oscillations athigh frequencies (25 MHz) and low pressures (250 kPa). These results can not be predicted using conventionalcoated MB models (with pure viscoelastic behavior) and they even contradict predictions of uncoated MBmodels with less damping effects.e- Through numerical simulations of Marmottant model [14] and visualization of the results using bifurcationdiagrams we showed that
De f inity
MBs can exhibit enhanced nonlinear behavior. Using this model andassuming MBs with initial surface tension close to 0 N/m or σ water could be used to explain experimentalobservations of higher order nonlinear oscillationsf- The 5 main regimes of oscillations were identified as P1, P2, P3, P4-2 and P4-1. Simulation results of thescattered pressure were in good agreement with experimental observations both in terms of the shape of theamplitude versus time signal and also its frequency content.g- For the first time, the two different P4 oscillations of the MB system were identified and characterizedexperimentally and numerically. P4-2 oscillations are the result of two consecutive well known period13oublings while P4-1 oscillations occur through a single period quadrupling via a saddle node bifurcation. Thedistinct features of the signal shapes and their unique frequency spectrum were identified both experimentallyand numerically. P4-1 oscillations require larger MBs compared to P4-2 oscillations.Previous studies have shown that lipid coated MBs can exhibit 1/2 order subharmonic oscillations evenwhen the excitation amplitude is low ( <
30 kPa [3–5]) where such low pressure thresholds are below thethresholds expected even for uncoated free MBs [6, 7]. The low pressure threshold for SH emissions hasbeen attributed to the buckling of the coating and compression only behavior [3]. Compression dominatedoscillations [8] occur when the coating buckles and the effective surface tension on the MB drops to valuesclose to zero. In such an instance, the MB compresses more than it expands. In addition to compression onlybehavior, lipid coating may also result in expansion dominated behavior where the MB expands more than itcompresses [13, 15]. Expansion-dominated behavior occurs when the shell ruptures. This effect was used toexplain the enhanced non-linearity at at higher frequencies (25 MHz) [13, 76]. Theoretical analysis of theMarmottant model for lipid coated MBs [14] by Prosperetti [6] attributed the lower SH threshold of the lipidMBs to the variation in the mechanical properties of the coating in the neighborhood of a certain MB radius(e.g. occurrence of buckling). In this work we show that there is a symmetry for enhanced non-linearityin the bifurcation structure of the RR of the MB as a function of σ ( R ) . Both buckling and rupture can beresponsible for enhanced non-linearity, where the closer the σ ( R ) to the buckling state (0 N/m) or rupturethreshold (0.072 N/m), the lower the excitation threshold required for the generation of nonlinear oscillations.Moreover, the closer the σ ( R ) to these two limit values, the higher the order of the nonlinearity.Using the estimated parameters for the De f inity
MB in [55] and considering the shear thinning [52], theobserved experimental behavior was only replicated for MBs with initial surface tension close to the two limitvalues of 0 and 0.072 N/m. However, it should be noted that during the sonication of a polydisperse solutionof lipid MBs different values in initial surface tension and coating properties (coating elasticity and viscosity)are expected. It is been reported that even for MBs of the same size, the lipid coating can be differentfrom MB to MB and are shown to be heterogeneous for MBs smaller than 10 µ m [77, 78]. Despite thebetter homogenity of lipid distribution in lipid coated MBs similar to De f inity [77, 78], the small differencesin the lipid distribution in the coating influences the effective coating properties, thus changing the MBresponse [79–81]. Moreover, its shown that the coating elasticity and coating viscosity changes with the MBsize [82–84]. Despite assuming the same coating properties for all MB sizes in this work, we were still be ableto replicate the peculiar higher order nonlinearities in experiments. Moreover, we used the simplest modelfor lipid coated MBs and we neglected the possible stiffness softening [85] or higher viscoelastic effects.Implementation of these effects are outside of the focus of this study but can be used to better characterizingthe coating. In addition, simulation results only implemented a monofreqeuncy ultrasound source, and theeffects of nonlinear propagation of sound waves in the medium are neglected. The generation of the SHs andUHs were not due to the nonlinear propagation of waves as nonlinear propagation manifests itself throughgeneration of only harmonics.Effects of the shape oscillations on the MBs response were also neglected in this paper. Holt and Crumobserved significant effects of shape oscillations on the nonlinear behavior of the larger MBs with initialradii between 20 µ m < R < µ m [86]. Versluis et al. [87] using high speed optical observations identifiedtime-resolved shape oscillations of mode n= 2 to 6 in the behavior of single air bubbles with radii between10 µ m and 45 µ m . [87] concludes that close to resonance, bubbles were found to be most vulnerabletoward shape instabilities. The effect of non-spherical bubble oscillations on nonlinear bubble behavior isstudied in [88] through GPU accelerated large parameter investigations. The active cavitation threshold hasbeen shown to depend on the shape instability of the bubble [88]. [88] also shows that shape instabilitycan affect the subharmonic threshold and nonlinear behavior of bubbles. Nonspherical oscillations ofultrasound contrast agent coated MBs are investigated in [89] through high speed optical observations.They showed that non-spherical bubble oscillations are significantly present in medically relevant rangesof bubble radii and applied acoustic pressures. Non-spherical oscillations develop preferentially at theresonance radius and may be present during SH oscillations [89]. Thus, for a more accurate modeling ofthe MB oscillations, more sophisticated theoretical modeling of bubble coating, accounting for membrane14hear and bending is required [89]. Recently Guédra et al. [90] showed that at sufficiently large pressureamplitudes, energy transfer from surface to volume oscillations may trigger subharmonic spherical oscillationsat smaller pressure amplitudes than predicted by models of spherical only bubble oscillations. Guédraand Inserra studied the nonlinear interactions between spherical and nonspherical modes in [91]. Theyshowed that bubble shape oscillations could be sustained for excitation amplitudes below the classicalparametric threshold. Experimentally, nonlinear interactions between the spherical, translational, and shapeoscillations of micrometer-size bubbles have been reported in detail in [92]. Liu et al [93] numericallystudied the shape oscillations of encapsulated bubbles. They showed that, in case of very small encapsulatedmicrobubbles, the shape oscillation is less likely to occur since the surface tension suppresses the developmentsof nonspherical shape modes. Their model however, does not take into account the dynamic variations in theeffective surface tension that occurs in case of lipid coated bubbles. Liu and Wang [94] showed that shapemodes of an encapsulated microbubbles in a viscous Newtonian liquid are stable. In case of encapsulatedbubbles with stiffness hardening or softening behavior, Tisiglifis and Plekasis [95] numerically showed thatparametric instability is possible and result in shape oscillations as a result of subharmonic or harmonicresonance. Generation of the shape modes for encapsulated bubbles exhibitng breakup and buckling have beennurmeically investigated for higher bubble oscillation amplitude [96]. Implementation of the nonspehericalbubble oscillations is beyond the scope of current paper, however, it may help to achieve a better agreementbetween numerical simulations and experimental observations in case of larger bubbles (e.g. Figure 5m-o).Since experimental study was done in very dilute solutions of MBs to record single scattering events, bubble-bubble interaction was not included in the model. Bubble-bubble interaction have been shown to lower thethreshold of subharmonic emissions and chaos [43, 97–99]. However even at very high concentrations [43]the changes to the pressure threshold of subharmonics is minimal compared to the signification decrease inthe onset of higher order nonlinearities when the shell undergoes buckling and rupture.Generation of higher order SHs at low pressures may have potential in high resolution SH imaging due to theirhigher frequencies, higher contrast to tissue and signal to noise ratio. A SH of order 2/3 or 3/4 can be detectedmore effectively by the transducer as they are closer to the transducer center frequency when compared to 1/2order SHs. Moreover, the higher scattered pressures, faster oscillations and the lower frequency contents ofthe oscillations of the higher order SHs may enhance the nondestructive shear stress on cells for enhanceddrug delivery or in cleaning applications. Mixing applications are another category of applications that cantake advantage of higher order SHs at high frequencies. V. C
ONCLUSION
We have shown experimentally and for the first time that higher order SHs (e.g. 1/3,1/4,..) can be generated inthe oscillations of lipid coated MBs when insonated at high frequencies and low excitation amplitudes. Thebifurcation structure of a simple model of lipid coated MBs were studied as function of frequency and size toexplain the experimental observations. We showed that compression only behavior or expansion dominatedoscillations due to buckling and rupture of the coating and dynamic variation of the effective surface tensioncan explain the observed enhanced non-linearity in MBs oscillations.
Acknowledgments
The work is supported by the Natural Sciences and Engineering Research Council of Canada (DiscoveryGrant RGPIN-2017-06496), NSERC and the Canadian Institutes of Health Research ( Collaborative HealthResearch Projects ) and the Terry Fox New Frontiers Program Project Grant in Ultrasound and MRI forCancer Therapy (project R EFERENCES [1]
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