Subharmonic Threshold Research Articles

Stability of thermal convection in two superimposed miscible viscous fluids under gravitational modulation

Abstract The thermal stability of a system of two superimposed miscible viscous fluid layers, under gravitational modulation, is investigated using linear stability analysis. It is well known in the literature that in the absence of gravitational modulation, there exist two convection regimes depending on the buoyancy number: the single-cell regime, where convection is oscillating and occupies the entire volume of the two fluid layers, and the stratified regime where stationary convection occurs in each fluid layer. Using Chebyshev's spectral collocation method and Floquet's theory, the numerical results show that under gravitational modulation, single-cell convection oscillates at half the frequency of the external oscillation (subharmonic threshold), while the stratified regime oscillates at a frequency equal to that of the forcing (harmonic threshold). The critical buoyancy number corresponding to the transition between the two convection regimes depends, in addition to the viscosity ratio and the depth of the two layers, on certain dimensionless frequencies permeating this transition and also on the amplitude of oscillation via the Froude number. In the plane of the critical Rayleigh number versus dimensionless modulation frequency, the branch associated with stratified convection does not depend on the buoyancy number for equal viscosities whereas this is the case for the whole-convection regime. For different viscosities, both convection regimes depend on the buoyancy number. At last, in the stratified regime (harmonic threshold), the critical value of the viscosity contrast, corresponding to the passage from an active layer to a passive layer and vice versa, increases with the modulation frequency.

Ambient Pressure Sensitivity of the Subharmonic Response of Coated Microbubbles: Effects of Acoustic Excitation Parameters.

The sensitivity of the acoustic response of microbubbles, specifically a strong correlation between their subharmonic response and the ambient pressure, has motivated development of a non-invasive subharmonic-aided pressure estimation (SHAPE) method. However, this correlation has previously been found to vary depending on the microbubble type, the acoustic excitation and the hydrostatic pressure range. In this study, the ambient pressure sensitivity of microbubble response was investigated. The fundamental, subharmonic, second harmonic and ultraharmonic responses from an in-house lipid-coated microbubble were measured for excitations with peak negative pressures (PNPs) of 50-700 kPa and frequencies of 2, 3 and 4 MHz in the ambient overpressure range 0-25 kPa (0-187 mmHg) in an in vitro setup. The subharmonic response typically has three stages-occurrence, growth and saturation-with increasing excitation PNP. We find distinct decreasing and increasing variations of the subharmonic signal with overpressure that are closely related to the threshold of subharmonic generation in a lipid-shelled microbubble. Above the excitation threshold, that is, in the growth-saturation phase, subharmonic signals decreased linearly with slopes as high as -0.56 dB/kPa with ambient pressure increase; below the threshold excitation (at atmospheric pressure), increasing overpressure triggers subharmonic generation, indicating a lowering of subharmonic threshold, and therefore leads to an increase in subharmonic with overpressure, the maximum enhancement being ∼11 dB for 15 kPa overpressure at 2 MHz and 100 kPa PNP. This study indicates the possible development of novel and improved SHAPE methodologies.

Analysis and Modeling of Nonlinear Effects in Constant-Frequency Peak-Current Control

In this paper, constant-frequency peak-current control is analyzed focusing on the operation above the subharmonic threshold limit. The analysis is performed by mixing analytical and numerical approaches. Two levels of normalization are introduced: on the converter level and on the switching cell level, resulting in unified analysis regardless of the converter type. A function that maps the inductor current value at the beginning of a switching period to its value at the end of the switching period is derived. The analysis is performed by iterating this mapping, leading to information of the inductor current periodicity and the switching cell averaged output current. It is shown that before reaching chaotic state a converter passes through a sequence of bifurcations involving discontinuous conduction modes characterized by higher order periodicity. Boundaries of the region where the higher order discontinuous conduction modes occur are derived. Obtained dependence of the switching cell output current average on the operating parameters is used to derive a small signal model. The model parameters expose huge variations in the areas of deep subharmonic operation. The results are experimentally verified.

Increased resonant frequency of microslit filtered contrast agents

Imaging using ultrasound contrast agents is highly dependent on the resonant frequency of the population. According to the Marmottant model, modern lipid coated agents have three regimes of resonant frequencies corresponding to the three regimes of bubble dynamics, the buckled, elastic, and ruptured states. The transition from buckled to elastic regime corresponds to the largest change in resonant frequency due to the addition of a second term inversely proportional to the bubbles’ radius. For a 1.75 μm bubble, this corresponds to a change from approximately 1.7 MHz, to approximately 5 MHz, leading to higher image resolution, and better separation between the fundamental and subharmonic components. Here we present a novel ultrathin silicon nitride membrane for this purpose. The membrane itself contains thousands of 1.75 × 50 μm slits and is housed in a centrifuge device. While the exact mechanism is unknown, bubbles that are forced across the membrane during centrifugation show a higher resonant frequency than their strictly size isolated counterparts. In addition, the centrifuged agent also had a lower subharmonic threshold compared to native and size isolated agents, independent of concentration. The possibility of tuning the device to precise frequencies for optimized imaging is also examined.Imaging using ultrasound contrast agents is highly dependent on the resonant frequency of the population. According to the Marmottant model, modern lipid coated agents have three regimes of resonant frequencies corresponding to the three regimes of bubble dynamics, the buckled, elastic, and ruptured states. The transition from buckled to elastic regime corresponds to the largest change in resonant frequency due to the addition of a second term inversely proportional to the bubbles’ radius. For a 1.75 μm bubble, this corresponds to a change from approximately 1.7 MHz, to approximately 5 MHz, leading to higher image resolution, and better separation between the fundamental and subharmonic components. Here we present a novel ultrathin silicon nitride membrane for this purpose. The membrane itself contains thousands of 1.75 × 50 μm slits and is housed in a centrifuge device. While the exact mechanism is unknown, bubbles that are forced across the membrane during centrifugation show a higher resonant frequency...

Is the subharmonic threshold always lower than the fundamental one in plane Poiseuille flow?

We revisit the secondary instability of a Tollmein-Schlichting wave in plane Poiseuille flow at Re = 5000 and find that the fundamental threshold can be lower than the subharmonic one in the wavenumber space, in contrast to the prevalent view in the literature that only instability of subharmonic modes occurs at very low disturbance amplitudes and fundamental modes need a higher disturbance amplitude to become unstable. This is because of overlooking the even class of fundamental modes, which happen to produce the lower threshold at those wavenumbers.

Characterisation of ultrasound contrast agents for subharmonic imaging for use in quantitative dynamic contrast enhanced ultrasound

Dynamic contrast enhanced ultrasound (DCE-US) using ultrasound contrast agents (UCA) is a non-invasive method of imaging the microvasculature, employing harmonic signal detection. UCAs are currently used clinically for qualitative visualisation purposes predominately. There is current interest in detecting the subharmonic component, known to be generated by UCAs but not tissue, which may increase sensitivity, although optimal transmission parameters remain to be determined. The aim of this work was to establish an acoustic characterisation system for use in establishing optimal subharmonic parameters. A reflection-based broadband acoustic system for UCA characterisation was developed and used to measure the attenuation of the UCA Definity™ (Lantheus, USA) across the diagnostic frequency range 1–10 MHz. A Coulter counter (aperture size 20 μm) was used to establish the size distribution (1.3–10 μm) and concentration of the UCA (1.2–1.4 × 109 bubbles/ml). Using a pulse-echo substitution method and single element transducer (Olympus, USA) the attenuation was measured of a solution of Definity diluted in saline, made up to an ‘in vivo’ concentration of 800,000 bubbles/ml. Pressure levels and frequency content, which influence the production of subharmonic signal, were also investigated using a through-transmission set-up with a needle hydrophone (Precision Acoustics, UK). A through-transmission narrowband characterization system for use in subharmonic characterization was also developed and pressure levels above the subharmonic threshold of ∼50 kPa were measured. An attenuation of 1.1 dB/cm was measured at 5 MHz, which is close to that reported in literature [1] . The developed broadband and narrowband systems are enabling an in-depth investigation of optimal transmit settings to maximise the subharmonic signal. Acknowledgement Irish Research Council, Government of Ireland Postgraduate Scholarship.

A derivation of the stable cavitation threshold accounting for bubble-bubble interactions

The subharmonic emission of sound coming from the nonlinear response of a bubble population is the most used indicator for stable cavitation. When driven at twice their resonance frequency, bubbles can exhibit subharmonic spherical oscillations if the acoustic pressure amplitude exceeds a threshold value. Although various theoretical derivations exist for the subharmonic emission by free or coated bubbles, they all rest on the single bubble model. In this paper, we propose an analytical expression of the subharmonic threshold for interacting bubbles in a homogeneous, monodisperse cloud. This theory predicts a shift of the subharmonic resonance frequency and a decrease of the corresponding pressure threshold due to the interactions. For a given sonication frequency, these results show that an optimal value of the interaction strength (i.e. the number density of bubbles) can be found for which the subharmonic threshold is minimum, which is consistent with recently published experiments conducted on ultrasound contrast agents.

Subharmonic Response of Polymer Contrast Agents Based on the Empirical Mode Decomposition.

The subharmonic threshold for ultrasound contrast agents has been defined as a 20-25 dB difference between the fundamental and subharmonic (2/1) spectral components of the backscatter signal. However, this Fourier-based criterion assumes a linear time-invariant signal. A more appropriate criterion for short cycle and frequency-modulated waveforms is proposed with an adaptive signal-processing approach based on the empirical mode decomposition (EMD) method. The signal is decomposed into an orthogonal basis known as intrinsic mode functions (IMFs) and a subharmonic threshold is defined with respect to the energy ratio of the subharmonic IMF component to that of the incident signal. The method is applied to backscatter data acquired from two polymer-shelled contrast agents, Philips (#38, mean diameter 2.0 [Formula: see text]) and Point Biomedical (#12027, mean diameter 3.9 [Formula: see text]). The acoustic backscatter signals are investigated for a single contrast agent subjected to monofrequency (20 MHz, 20 cycles) and chirp (15-25 MHz, 20 cycles) forcing for incident pressures ranging from 0.5 to 2.4 MPa. In comparison to the spectral peak difference (20 dB) criterion, the EMD method is more sensitive in determining subharmonic signals.

Characterisation of Liposome-Loaded Microbubble Populations for Subharmonic Imaging

Therapeutic microbubbles could make an important contribution to the diagnosis and treatment of cancer. Acoustic characterisation was performed on microfluidic generated microbubble populations that either were bare or had liposomes attached. Through the use of broadband attenuation techniques (3–8 MHz), the shell stiffness was measured to be 0.72 ± 0.01 and 0.78 ± 0.05 N/m and shell friction was 0.37 ± 0.05 and 0.74 ± 0.05 × 10−6 kg/s for bare and liposome-loaded microbubbles, respectively. Acoustic scatter revealed that liposome-loaded microbubbles had a lower subharmonic threshold, occurring from a peak negative pressure of 50 kPa, compared with 200 kPa for equivalent bare microbubbles. It was found that liposome loading had a negligible effect on the destruction threshold for this microbubble type, because at a mechanical index >0.4 (570 kPa), 80% of both populations were destroyed.

Subharmonic threshold quantification of ultrasound contrast agents

The measurement and prediction of the subharmonic threshold of ultrasound contrast agents is of significant interest, particularly for high frequency applications. Theoretical analytical predictions typically follow from a weakly nonlinear analysis assuming continuous, single frequency forcing. Furthermore, numerical simulation and experimental definitions are often based on the quantification of spectral response of the nonlinear harmonics. Limitations of these approaches are investigated with respect to pulsed forcing and associated non-stationary (chirp) excitations. Novel quantification and definitions are proposed with respect to instantaneous frequency and relative energy content of an empirical mode decomposition of the scattered pressure. Teager-Kasiser energy operator allows for additional quantification. The methodology is examined with respect to experimental polymer contrast agent data.

Subharmonic response and threshold of polymer ultrasound contrast agents

Analytical expressions for subharmonic threshold previously developed for free gas bubbles often been generalized and used for ultrasound contrast agents. However, many of those formulations were developed under the assumptions of mono-frequency steady state, continuous acoustic forcing which are typically not applicable to diagnostic and therapeutic ultrasound. Nonstationary subharmonic responses are investigated by analytical and numerical methods. The subharmonic threshold is investigated as means to differentiate between the proposed constitutive formulations of polymer shells in terms the shear modulus parameter. The potential influence of other agents in close proximity on the subharmonic response and threshold is highlighted with respect to the phase interaction between coupled agents.

A general derivation of the subharmonic threshold for non-linear bubble oscillations

The paper describes an approximate but rather general derivation of the acoustic threshold for a subharmonic component to be possible in the sound scattered by an insonified gas bubble. The general result is illustrated with several specific models for the mechanical behavior of the surface coating of bubbles used as acoustic contrast agents. The approximate results are found to be in satisfactory agreement with fully non-linear numerical results in the literature. The amplitude of the first harmonic is also found by the same method. A fundamental feature identified by the analysis is that the subharmonic threshold can be considerably lowered with respect to that of an uncoated free bubble if the mechanical response of the coating varies rapidly in the neighborhood of certain specific values of the bubble radius, e.g., because of buckling.

The response of phospholipid-encapsulated microbubbles to chirp-coded excitation: Implications for high-frequency nonlinear imaging

The current excitation strategy for harmonic and subharmonic imaging (HI and SHI) uses short sine-bursts. However, alternate pulsing strategies may be useful for enhancing nonlinear emissions from ultrasound contrast agents. The goal of this study was to corroborate the hypothesis that chirp-coded excitation can improve the performance of high-frequency HI and SHI. A secondary goal was to understand the mechanisms that govern the response of ultrasound contrast agents to chirp-coded and sine-burst excitation schemes. Numerical simulations and acoustic measurements were conducted to evaluate the response of a commercial contrast agent (Targestar-P(®)) to chirp-coded and sine-burst excitation (10 MHz frequency, peak pressures 290 kPa). The results of the acoustic measurements revealed an improvement in signal-to-noise ratio by 4 to 14 dB, and a two- to threefold reduction in the subharmonic threshold with chirp-coded excitation. Simulations conducted with the Marmottant model suggest that an increase in expansion-dominated radial excursion of microbubbles was the mechanism responsible for the stronger nonlinear response. Additionally, chirp-coded excitation detected the nonlinear response for a wider range of agent concentrations than sine-bursts. Therefore, chirp-coded excitation could be a viable approach for enhancing the performance of HI and SHI.

Determination of the Interfacial Rheological Properties of a Poly(DL-lactic acid)-Encapsulated Contrast Agent Using In Vitro Attenuation and Scattering

The stabilizing encapsulation of a microbubble-based ultrasound contrast agent (UCA) critically affects its acoustic properties. Polymers, which behave differently from materials commonly used (i.e., lipids or proteins) for monolayer encapsulation, have the potential for better stability and improved control of encapsulation properties. Air-filled microbubbles coated with poly(DL-lactic acid) (PLA) are characterized here using in vitro acoustic experiments and several models of encapsulation. The interfacial rheological properties of the encapsulation are determined according to each model using attenuation of ultrasound through a suspension of microbubbles. Then the model predictions are compared with scattered non-linear (sub- and second harmonic) responses. For this microbubble population (average diameter, 1.9 μm), the peak in attenuation measurement indicates a weighted-average resonance frequency of 2.5–3 MHz, which, in contrast to other encapsulated microbubbles, is lower than the resonance frequency of a free bubble of similar size (diameter, 1.9 μm). This apparently contradictory result stems from the extremely low surface dilational elasticity (around 0.01–0.07 N/m) and the reduced surface tension of the poly(DL-lactic acid) encapsulation, as well as the polydispersity of the bubble population. All models considered here are shown to behave similarly even in the non-linear regime because of the low surface dilational elasticity value. Pressure-dependent scattering measurements at two different excitation frequencies (2.25 and 3 MHz) revealed strongly non-linear behavior with 25–30 dB and 5–20 dB enhancements in fundamental and second-harmonic responses, respectively, for a contrast agent concentration of 1.33 μg/mL in the suspension. Sub-harmonic responses are registered above a relatively low generation threshold of 100–150 kPa, with up to 20 dB enhancement beyond that pressure. Numerical predictions from all models show good agreement with the experimentally measured fundamental response, but not with the experimental second-harmonic response. The characteristic features of sub-harmonic responses and the steady response beyond the threshold are matched well by model predictions. However, prediction of the threshold value depends on estimated properties and size distribution. The variation in size distribution from sample to sample leads to variation in estimates of encapsulation properties: the lowest estimated value for surface dilational viscosity better predicts the sub-harmonic threshold.

Role of effective surface tension on the frequency dependent subharmonic threshold for contrast microbubbles

We numerically investigate the predictions from several contrast microbubble models to determine the excitation threshold for subharmonic generation. In contrast to the classical perturbative result, the minimum threshold for subharmonic generation is not always obtained near twice the resonance frequency; instead it can occur over a range of frequency from resonance to twice the resonance frequency. The quantitative variation of the threshold with frequency depends on the model, bubble radius, and encapsulation properties. All models are transformed into a common interfacial rheological form, where encapsulation is represented by two radius dependent surface properties—effective surface tension and surface dilatational viscosity. Variation of the effective surface tension with radius, specifically having an upper limit (resulting from strain softening or rupture of the encapsulation during expansion), plays a critical role. It destroys a sharp minimum at twice the resonance frequency. Without the upper limit on effective interfacial tension, the threshold is extremely large especially near the resonance frequency.

High-resolution bubble sizing through detection of the subharmonic response with a two-frequency excitation technique

Sizing bubbles in fluid using a two-frequency excitation technique is not prone to the same drawbacks of some other sizing methods—it has a global maximum at the bubble resonance frequency and allows good spatial resolution. The bubble is insonated with a high fixed imaging signal and a variable pumping signal tuned to the resonant frequency of the bubble, which are coupled at resonance by the high-amplitude oscillation of the bubble wall, with the formation of sum-and-difference terms. This paper examines both the resonance and off-resonance behavior of such combination frequency signals. A coupling of the subharmonic bubble response with the imaging frequency is shown to be a much more accurate and unambiguous detector of the bubble resonance than couplings involving the fundamental resonance. The characteristics of this subharmonic signal are examined using an automated sizing method, and the dependence of the response on the pumping signal amplitude and the frequency step size between two successive pumping frequencies is examined. The location of a definite subharmonic threshold is reported and quantified both for single bubbles held on a wire and for free rising bubbles moving through the focus of the transducers. This amplitude is found to be orders of magnitude lower than that predicted by traditional volumetric pulsation models, but agrees very closely with the theoretical onset of surface waves.

Application of the subharmonic threshold to the measurement of the damping of oscillating gas bubbles

It is suggested that the damping affecting the forced radial oscillations of gas bubbles can be determined very accurately through the measurement of the subharmonic threshold for the oscillations. The advantage of the method is that, while the subharmonic oscillations have a large amplitude and can be easily detected, the theory can be accurately based on a perturbation approach. The theoretical basis for the subharmonic generation is also reviewed and its results illustrated for the case of an air bubble in water.