The effect of size range on ultrasound-induced translations in microbubble populations

Abstract

Microbubble translations driven by ultrasound-induced radiation forces can be beneficial for applications in ultrasound molecular imaging and drug delivery. Here, the effect of size range in microbubble populations on their translations is investigated experimentally and theoretically. The displacements within five distinct size-isolated microbubble populations are driven by a standard ultrasound-imaging probe at frequencies ranging from 3 to 7 MHz, and measured using the multi-gate spectral Doppler approach. Peak microbubble displacements, reaching up to 10 μm per pulse, are found to describe transient phenomena from the resonant proportion of each bubble population. The overall trend of the statistical behavior of the bubble displacements, quantified by the total number of identified displacements, reveals significant differences between the bubble populations as a function of the transmission frequency. A good agreement is found between the experiments and theory that includes a model parameter fit, which is further supported by separate measurements of individual microbubbles to characterize the viscoelasticity of their stabilizing lipid shell. These findings may help to tune the microbubble size distribution and ultrasound transmission parameters to optimize the radiation-force translations. They also demonstrate a simple technique to characterize the microbubble shell viscosity, the fitted model parameter, from freely floating microbubble populations using a standard ultrasound-imaging probe.

I. INTRODUCTION

Microbubbles undergo coupled volumetric oscillations and translations when acted upon by traveling acoustic waves due to the ensuing spatial pressure gradients. These produce a net bubble displacement in the direction of the acoustic wave propagation owing to the change in the bubble's volume during the acoustic cycle (Dayton et al., 1997). Consequently, systemically injected ultrasound contrast agent microbubbles subject to a regular medical ultrasound field are pushed in the direction away from the probe, enhancing thus their contact with the distal wall of the blood vessel. An enhanced contact with the endothelium can facilitate ligand-receptor binding for molecular imaging (Zhao et al., 2004) and local deposition of drug or other carrier materials for image-guided therapy (Lum et al., 2006).

The acoustically induced radiation force, also known as the primary Bjerknes force, pushes bubbles in a size-dependent manner, as has been shown through optical measurements of individual bubble displacements (Acconcia et al., 2018; Palanchon et al., 2005). This feature has been exploited in acoustic microbubble sorting (Segers and Versluis, 2014). Furthermore, microbubbles used in clinical applications are encapsulated typically with a lipid, polymer, or protein shell, which ensures stability against dissolution and coalescence. The viscoelasticity of the shell has a strong influence on the mechanical response of the bubble to the acoustic driving, and therefore also on their displacements. Doinikov and Dayton (2006) showed numerically that encapsulated microbubbles displace significantly more under the effect of the radiation force at high-frequency acoustic driving (>4 MHz) than free bubbles of equivalent size. This effect was explained by the encapsulating shell increasing the resonance frequency of the bubble, resulting thus in larger radiation forces at high frequencies close to its resonance. Translations of encapsulated microbubbles have also been shown to have a destabilizing effect on the bubble dynamics, inducing nonspherical shape oscillations (Liu et al., 2018).

Patil et al. (2009) proposed a technique to simultaneously translate and image microbubble populations within large blood vessels using customized dual-frequency sequences. They studied displacements within bubble populations flowing through a flow phantom by measuring the intensity of static microbubbles adhered to the phantom vessel wall due to the radiation force. They varied the excitation frequency and bubble concentration, but treated the microbubble population response with mean values, such as with approximative resonance frequency associated to the mean bubble size. Fan et al. (2014) used high-speed videomicroscopy to characterize the dynamic activities of microbubble population driven by pulsed ultrasound. They captured displacements of individual microbubbles within the population and described them qualitatively. However, no study to date has systematically investigated the effect of bubble size distribution within a population on their translations.

The translation of microbubbles was initially predicted as the result of the complex balance between the ultrasound radiation force and the fluid drag force (Tortoli et al., 2000). More detailed predictions are challenged by the complexity of modeling the microbubble viscoelastic shell. Dayton et al. (2002) compared measurements of individual microbubble displacements captured by high-speed streak imaging with a theoretical model based on a modified Rayleigh-Plesset equation. While they found a fair agreement of the general trends with the various acoustic driving parameters, the theory overestimated the displacements. The authors attributed this discrepancy to the friction between the bubble shell and the containment tube. More recently, Acconcia et al. (2018) compared measurements with theoretical translations using two different viscoelastic interfacial rheological models: a Rayleigh-Plesset-type equation with the shell model as proposed by Marmottant et al. (2005), and a modified Herring equation extended by shell terms following the work of Sarkar et al. (2005). The ability of these models to predict the displacements was strongly dependent on the driving pressures: the Marmottant model, which accounts for buckling and rupture, worked better at low pressures, while the Sarkar model, which assumes a constant surface elasticity, performed better at higher pressures.

In addition to modeling, characterizing the shell viscoelasticity of the microbubbles is rather demanding, as it generally requires sophisticated optical or acoustic measurement techniques on isolated or monodisperse microbubbles, for example through high-speed imaging (van der Meer et al., 2007), forward laser-scattering (Lum et al., 2016), attenuation (Segers et al., 2018), or deflation (Guidi et al., 2010). Vos et al. (2007) highlighted Doppler imaging on microbubble populations as a potentially convenient method to characterize the shell viscoelastic properties.

In this study, translations of freely floating microbubble populations are investigated. Microbubble translations are induced with a commercial linear array ultrasound probe and measured with the same probe through the multi-gate spectral Doppler approach (Blue et al., 2018). The main objectives are to (i) quantify differences between the measured translations (peak values and statistical behavior) within bubble populations of distinct size ranges, (ii) to determine how well they compare with theory, and (iii) to demonstrate that they can serve as a relatively simple technique to characterize the viscosity of the microbubble shell, which are validated with supporting forward laser-scattering measurements of individual microbubble oscillations.

II. THEORY

A. Radial equation of motion

The radial motion of a spherical bubble is described by a modified Herring equation, which includes pressure terms to account for a Newtonian surface elasticity and dilatational surface viscosity as proposed by Sarkar et al. (2005), and for a coupling effect from the bubble translation,

$$\begin{array}{c}{\rho }_l\left(R\ddot{R}+\frac{3}{2}{\dot{R}}^2\right)=\left[p_0+\frac{2\sigma (R_0)}{R_0}\right]{\left(\frac{R_0}{R}\right)}^{3\gamma }\left(1-\frac{3\gamma }{c}\dot{R}\right)\hfill \\ -\frac{4\mu \dot{R}}{R}-\frac{4{\kappa }_S\dot{R}}{R^2}{\left(\frac{R_0}{R}\right)}^2\hfill \\ -\frac{2\chi }{R}{\left(\frac{R_0}{R}\right)}^2\left(1-\frac{R_0}{R}\right)\hfill \\ +\frac{{\rho }_l{\mathit{u}}^2}{4}-\left[p_0+p_a(t)\right].\hfill \end{array}$$ (1)

The left-hand side of Eq. (1) comprises the inertial terms describing the bubble oscillations, where ρ_l is the liquid density, R is the bubble radius, and the over-dots indicate time derivatives. The right-hand side comprises the pressure terms, which account for the acoustic driving, the gas inside the bubble, liquid compressibility, the shell, and the bubble translation. Here, R₀ is the bubble's radius at equilibrium, p₀ is the atmospheric pressure, $\sigma (R_0)$ is the initial surface tension of the shell (set to zero to comply with bubble stability), γ is the polytropic exponent of the gas core, c is the speed of sound in the liquid, μ is the liquid viscosity, κ_S is the dilatational surface viscosity, χ is the surface elasticity, and u is the bubble's translational velocity. The acoustic driving pressure, p_a, is expressed as

$$p_a(t)=p_{\text{pk}}\hspace{0.17em}\text{sin}\hspace{0.17em}\left(\frac{2\pi f_0}{c}z-2\pi f_{0}t\right),$$ (2)

where $p_{\text{pk}}$ is the peak negative pressure, f₀ is the driving frequency, and z is the bubble translation.

Here, the Sarkar shell model is chosen over the Marmottant shell model (Marmottant et al., 2005) because of the use of high driving pressure ($p_{\text{pk}}$ = 200 kPa) in this study. It has been shown by Acconcia et al. (2018) that at high driving pressure, the Sarkar model provides a better prediction of the microbubble translations [referred to as the “Hoff” model by Acconcia et al. (2018)]. The authors suggested that for such high-amplitude oscillations, the Marmottant shell model causes the bubble to remain in the buckling or ruptured regime most of the time with little effect of the elasticity. Instead the Sarkar shell model assumes that the bubble always remains in the elastic regime, with a constant surface elasticity. Furthermore, the modified Herring model is preferred over Rayleigh-Plesset, as the weak liquid compressibility, which facilitates the dispersive radiation of energy, becomes non-negligible for high-amplitude oscillations. The second-last term in Eq. (1) represents the pressure term arising due to the bubble translation, which can be derived from potential flow theory. However, this term is expected to have little contribution because the driving pressures in this work are significantly lower than the values causing important coupling effects between the bubble translation and the radial motion (Doinikov, 2002).

B. Translational equation of motion

To determine the translational displacement of the bubble, we compute the balance of forces acting on it (Dayton et al., 2002; Igualada-Villodre et al., 2018; Liu et al., 2018; Reddy and Szeri, 2002; Rensen et al., 2001; Tortoli et al., 2000; Vos et al., 2007) as

$${\rho }_b{V}_0\frac{\partial \mathit{u}}{\partial t}=\Sigma \mathit{F}={\mathit{F}}_{US}+{\mathit{F}}_D+{\mathit{F}}_{AM},$$ (3)

where $V_0=(4\pi /3)R_0^3$ is the initial bubble volume, u is the translational velocity of the bubble, ${\mathit{F}}_{US}$ is the ultrasound radiation force, ${\mathit{F}}_D$ is the drag force, and ${\mathit{F}}_{AM}$ is the “added mass” force. Secondary radiation forces, which can yield attractions between bubbles, and secondary scattered ultrasound fields are neglected for simplicity: in the experiment, the bubble concentration was kept low to reduce these effects (estimated smallest mean distance between bubbles was approximately 50 times the bubble radius). The buoyancy force is orders of magnitude smaller compared to the other forces on the microbubble and can therefore also be neglected. The individual force components are computed as follows.

The primary radiation force results from a pressure wave acting on the bubble's volume. It reads

$${\mathit{F}}_{US}=-V_b(t)\mathbf{\nabla }{p}_a(z,t)=V_b(t)\frac{1}{c}\frac{\partial p_a}{\partial t}\mathit{e},$$ (4)

where $V_b(t)=(4\pi /3)R{(t)}^3$ is the bubble volume and $\mathbf{\nabla }{p}_a(z,t)=\partial p_a/\partial z$ is the local pressure gradient. The latter can be approximated using the time gradient $-(1/c)(\partial p/\partial t)\mathit{e}$, where e represents the unit vector in the direction of the acoustic wave propagation, because the bubble size is small compared to the wavelength of the transmitted sound.

Viscous drag induces a force expressed as

$${\mathit{F}}_D=-\frac{1}{4}\pi C_D{\text{Re}}_{t}R\mu \mathit{u},$$ (5)

where

$${\text{Re}}_t=\frac{2R{\rho }_l\left|\mathit{u}\right|}{\mu }$$ (6)

and

$$C_D=\frac{24}{{\text{Re}}_t}+\frac{6}{1+\sqrt{{\text{Re}}_t}}+\mathrm{0.4.}$$ (7)

Here, C_D is the drag coefficient of a sphere, Re_t is the translational Reynolds number, and u is the translational velocity of the bubble. The drag coefficient is derived from a parametric fit to extensive experimental data on flow around smooth solid spheres (the microbubble shell enforces a no-slip boundary condition similar to solid spheres), validated for ${\text{Re}}_t<2\times {10}^5$ (White and Corfield, 1991). The drag is assumed quasisteady and only dependent on the instantaneous bubble radius and velocity.

The force due to added mass represents the inertia added to the system as the accelerating and decelerating microbubble moves some volume of the surrounding liquid. For microbubbles, it represents the added mass for a sphere, which is commonly known as

$${\mathit{F}}_{\text{AM}}=-\frac{1}{2}{\rho }_l\left(V_b\frac{\partial \mathit{u}}{\partial t}+\mathit{u}\frac{d{V}_b}{dt}\right),$$ (8)

where $d{V}_b/dt=4\pi R^2\dot{R}$.

For simplicity and to reduce the computational cost, the Basset force, or the history term, is omitted. It has been shown that for large translational and radial Reynolds numbers, i.e., ${\text{Re}}_t=2R{\rho }_l\left|\mathit{u}\right|/\mu \gg 1$ and ${\text{Re}}_r=\left|\dot{R}/\mathit{u}\right|{\text{Re}}_t\gg 1$, it is generally negligible (Magnaudet and Legendre, 1998). However, it has been demonstrated that history effects do affect bubble displacements for low-amplitude driving and off-resonance conditions, where the Reynolds number can remain below unity (Acconcia et al., 2018; Garbin et al., 2009; Igualada-Villodre et al., 2018). They showed that omitting the history term would yield an overestimation of the displacement by slowing down the convergence to a steady translational velocity. However, as the current study mostly focuses on high-amplitude and on-resonance driving, and as both the translational and radial Reynolds number in these conditions generally stay above 5 where history effects have little effect on the bubble displacements (Acconcia et al., 2018), the history term is neglected. However, the possibility that this assumption may cause a slight overestimation of the peak displacement is not excluded.

Finally, combining Eqs. (3), (4), (5), and (8) yields the bubble's translational acceleration,

$$\begin{array}{c}\frac{\partial \mathit{u}}{\partial t}=\frac{1}{\frac{1}{2}{\rho }_l{V}_b-{\rho }_b{V}_0}\hfill \\ \times \left[V_b\frac{1}{c}\frac{\partial p}{\partial t}\mathit{e}-\frac{1}{4}\pi C_D{\text{Re}}_{t}R\mu \mathit{u}-\frac{1}{2}{\rho }_l\mathit{u}\frac{d{V}_b}{dt}\right].\hfill \end{array}$$ (9)

Equations (1) and (9) are recursively solved using a variable-step, fourth-order Runge-Kutta method, finally providing the translational velocity u and thereby the displacement $\Delta z$ within one driving pulse,

$$\Delta z={\int }_0^{\Delta t}\mathit{u}dt,$$ (10)

where $\Delta t$ stands for the pulse duration.

Table I summarizes the parameters used in the numerical computation. The choices of the empirically determined surface elasticity and viscosity are explained hereafter.

TABLE I.

List of tested parameters.

Atmospheric pressurea	p0	$8\times {10}^4$ Pa
Liquid density	ρl	1000 kg/m3
Sound speed in liquid	c	1500 m/s
Liquid viscosity	μ	$8.9\times {10}^{-4}$ Pas
Driving frequency	f0	3–7 MHz
Peak negative pressure	$p_{\text{pk}}$	200 kPa
Pulse repetition frequency	$f_{\text{pr}}$	4 kHz
Pulse duration	$\Delta t$	10 μs
Equilibrium radius	R0	0.5–4 μm
Shell elasticity	χ	0.61 N/m
Shell viscosity	κS	0.1–$6\times {10}^{-8}$ kg/s
Density of gas core	ρb	9.7 kg/m3
Polytropic exponent for gas core	γ	1.07
Initial shell surface tension	$\sigma (R_0)$	0 N/m

^aBoulder CO is located at 1600 m above sea level.

III. EXPERIMENTS

A. Microbubble synthesis

As shown in Fig. 1, we used several microbubble populations with distinct size ranges, and one population with polydisperse sizes. They were either synthesized in our lab (for the 1.5–2 μm range and polydisperse population) or purchased from Advanced Microbubbles Laboratories LLC (Boulder, USA) (0.5–1, 2–2.5, and 2.5–4 μm). The constituents of microbubble shell and the protocol to synthesize microbubbles in both cases were similar. The main phase lipid, 1,2-distearoyl-sn-glycero-3-phosphocholine (DSPC), and the emulsifier, 1,2-distearoyl-sn-glycero-3-phospho-ethanolamine-N-[amino(polyethylene glycol)-2000] (DSPE-PEG2000) (Avanti Polar Lipids, Alabaster, AL, USA), were mixed at a molar ratio of 9:1 into phosphate buffer saline to a concentration of 2 mg/mL, after which the solution was stirred, heated to 60 °C, and further mixed using a probe sonicator at low amplitude (30%). Microbubbles were formed by sonicating the surface of the lipid solution at high power (100% amplitude) while simultaneously flowing perfluorobutane gas over it, after which the microbubble suspension was cooled down to room temperature and washed using centrifugation. Differential centrifugation was used to size-isolate the microbubbles into specific size distributions (Feshitan et al., 2009), which were measured using a Multisizer III (Beckman Coulter Life Sciences, Indianapolis, IN, USA). Further details on the microbubble synthesis may be found in Guidi et al. (2019).

FIG. 1.

(Color online) Measured number size distributions for five different bubble populations. PMB stands for polydisperse microbubbles. The error bars show the standard deviation of three measurements.

B. Microbubble displacement measurements

Microbubble populations of five distinct size ranges were tested, their size-isolated distributions peaking at radii 0.5–1, 1.5–2, 2–2.5, and 2.5–4 μm, and one with a polydisperse size distribution. Their number-weighted size distributions are displayed in Fig. 1. For each experiment, the microbubbles were diluted with purified water to a concentration corresponding to a void fraction of approximately ${10}^{-6}$ within an 800-mL suspension contained in a glass beaker. This concentration was selected as a compromise to avoid strong attenuation and simultaneously guarantee sufficient signal. The attenuation was assessed by verifying that the intensity levels were similar across a range of depths. Sufficient signal was assessed to be at least 20 dB above the root-mean-squared noise level. It should be noted that the number concentrations for the different bubble populations were different ($3.3\times {10}^5$ mL⁻¹ for 0.5–1 μm, $5\times {10}^4$ mL^− 1 for 1.5–2 μm, $2.3\times {10}^4$ mL⁻¹ for 2–2.5 μm, and $1.4\times {10}^4$ mL⁻¹ for both 2.5–4 μm and polydisperse populations), because the volume concentration was kept constant instead to prevent significant changes in attenuation. A tilted rubber absorber was placed at the bottom of the beaker to minimize measurement artifacts from acoustic reflections. Figure 2 shows a schematic of the experimental setup.

FIG. 2.

(Color online) Schematic of the experimental setup (left), and a typical multigate spectral Doppler profile in plane wave ultrasound transmission mode (right). The colorbar shows the signal amplitude in arbitrary units. $f_0=4$ MHz.

The microbubble solution was insonified using the Esaote LA332 linear array probe (Esaote SpA, Florence, Italy), which has 144 elements covered by a silicone lens with an elevational focus at 23 mm. The probe elements have a 0.245-mm pitch, a 4.6-MHz center frequency and 100% (−6 dB) bandwidth. The probe was driven by the 64-channel ULtrasound Advanced Open Platform (ULA-OP) (Tortoli et al., 2009). The ultrasound transmission was performed with 240 000 pulses of 10 μs length in plane-wave mode (Blue et al., 2018) at a pulse repetition frequency of 4 kHz and with a peak negative pressure of 200 kPa located approximately at 2.7 cm from the probe. The peak negative pressure was sufficient to result in considerable microbubble displacements without destroying them. Bubble destruction was only observed beyond approximately 250 kPa, which resulted in a drop in bubble concentration monitored with a particle sizer between every experiment. Here, the pulses were longer than those used in typical clinical ultrasound imaging in order to produce large microbubble displacements. However, the bubble displacements are scalable as they vary linearly with the burst length (Guidi et al., 2019). The driving frequency was varied between 3 and 7 MHz. The peak negative pressure at each frequency was calibrated using a hydrophone.

The microbubble displacements were measured using the multi-gate spectral Doppler approach (Vos et al., 2007). The echo signals were acquired along single axial line from the center of the probe. The signals corresponding to each depth were grouped into blocks of size 512, weighed by a Hanning window, and transformed into the frequency domain through a 512-point fast Fourier transform (FFT) moving through time with sliding steps of 32. This constructs a multi-gate spectral Doppler (MSD) profile for a single axial line, an example of which is shown in Fig. 2. It is possible to identify individual bubbles from the MSD profile given that (i) their signal amplitude is above a fixed amplitude threshold $A_{\text{TH}}$ and (ii) they displace sufficiently to appear far from the zero-frequency centerline, which is typically crowded by numerous still and slowly-moving bubbles. $A_{\text{TH}}$ was heuristically chosen to be 20 dB superior to the root-mean-squared noise level, estimated by reference measurements in water in the absence of bubbles while being low enough to allow for sufficient spectral contributions produced by microbubble scattering. The displacement for each identified bubble at depth z was computed from the measured frequency shift, $\Delta f(z)$, through the Doppler equation

$$\Delta z(z)=\frac{\Delta f(z)c}{2f_0{f}_{\text{pr}}},$$ (11)

where $f_{\text{pr}}$ is the pulse repetition frequency. In Fig. 2 it can be observed that the bubbles have an apparent length in the axial dimension on the MSD display owing to the burst length and the bubble displacement occurring during the time contained within the FFT window. For the statistical analysis of bubble displacements, individual bubbles that produced displacements exceeding the threshold $\Delta z_{\text{TH}}=$ 0.8 μm/pulse, were counted at a single depth (2.7 cm) across the entire acquisition. Since individual bubbles can be counted multiple times due to the apparent length of the bubbles in the MSD display (see Fig. 2), this effect being more prominent for slowly moving bubbles compared to the faster ones, the counts are weighted to avoid the resulting skewing of the distributions with the following weight function (Guidi et al., 2019):

$$w(\Delta z)=\frac{32\Delta z}{0.6\times 512\Delta z+c\Delta t/2},$$ (12)

where the factors 512 and 32 correspond to the FFT window and sliding size, respectively, and 0.6 corrects for the Hanning window applied prior to FFT. Further details on the MSD acquisition may be found in Guidi et al. (2019).

C. Characterization of acoustic streaming

The measured microbubble displacements are affected by acoustic streaming, where the bulk fluid is driven into motion. This effect may be particularly significant in plane-wave ultrasound transmission, where a large volume of fluid is insonified. The micro-streaming produced by oscillating microbubbles was assumed negligible, as the bubble concentrations were kept relatively low. To distinguish the effects of the primary acoustic streaming and the radiation force on the displacements, the streaming velocities were measured through the peak displacements of 3-μm latex bead tracking particles driven by the tested acoustic driving conditions across a range of frequencies. The latex beads provide a reasonable approximation of the fluid streaming velocity and a relevant baseline for bubble displacements, as the contribution of the radiation force on the bead displacement is assumed insignificant. Their concentration was kept low ($5\times {10}^4$/mL) to minimize their influence on the attenuation of ultrasound in water that drives the streaming. The resulting velocities, shown in Fig. 3, were subtracted from the measured microbubble displacements.

FIG. 3.

Peak displacements of latex beads driven by primary acoustic streaming for a range of driving frequencies, used to compensate for this streaming in the microbubble experiments. The error bars show the standard deviation of at least three measurements.

D. Characterization of the shell viscoelasticity

To reliably determine the viscoelasticity of the bubble shells as a reference to the radiation force measurements, individual microbubble responses to acoustic driving were recorded using the same photoacoustic technique as detailed by Lum et al. (2018). The size of the microbubbles, contained between a glass slide and a cover slip, was determined from a microscope image. The bubble oscillations were driven by low-amplitude acoustic waves ($p_{\text{pk}}\sim 3$–5 Pa), which were generated by focusing an amplitude-modulated continuous wave laser driven sinusoidally at ultrasonic frequencies, approximately 100 μm away from the bubble. The water was heated locally by the laser and its subsequent thermal expansion results in soundwave emission through the photoacoustic effect. The relative amplitude of the bubble oscillation was measured through forward light scattering by focusing another continuous wave laser at the location of the bubble and measuring the laser light intensity with a photodetector placed at the distal side of the microbubble. By scanning through a range of driving frequencies, such measurements provide resonance curves and thereby the resonance frequency and the full-width-at-half-maximum associated with each bubble radius, which, respectively, provide the elasticity and viscosity of the bubble's shell (Lum et al., 2018).

Theoretically, the radial excursions of the bubble driven by such weak pressures are computed via a modified Rayleigh-Plesset equation, similar to Eq. (1), but better suited to weak bubble oscillations. The presence of the glass substrate, against which the microbubbles lay due to buoyancy, is known to decrease their resonance frequency and thereby affect the measured shell elasticity (Strasberg, 1953; van der Meer et al., 2007). This was accounted for in the computations through the last term on the right-hand side of Eq. (16). The boundary is modeled as an image bubble, which induces, according to the continuity equation, a radial flow with velocity

$$v=\frac{R^2\dot{R}}{r^2},$$ (13)

where r is the radial distance from the center of the image bubble. The pressure field is computed from the momentum equation for liquid motion,

$${\rho }_l\frac{\partial v}{\partial t}+\frac{\partial p}{\partial r}=0,$$ (14)

where the nonlinear convective term $\rho v(\partial v/\partial r)$ is omitted as it is significantly smaller (on the order of $r^{-5}$) compared to the first term (Mettin et al., 1997). Combining Eqs. (13) and (14) yields an expression for the radial pressure gradient,

$$\frac{\partial p}{\partial r}=-\frac{{\rho }_l}{r^2}\frac{\partial }{\partial t}\left[R^2\dot{R}\right],$$ (15)

the integration of which at a distance $r=2R$ provides the expression for the pressure exerted on the bubble by the boundary $p_b=({\rho }_l/2R)[2R{\dot{R}}^2+R^2\ddot{R}]$. The modified Rayleigh-Plesset equation with boundary term thus reads

$$\begin{array}{c}{\rho }_l\left(R\ddot{R}+\frac{3}{2}{\dot{R}}^2\right)=p_0{\left(\frac{R_0}{R}\right)}^{3\gamma }-\frac{4\mu \dot{R}}{R}-\frac{4{\kappa }_S\dot{R}}{R^2}\hfill \\ -\frac{2\chi }{R}\left(R^2/R_0^2-1\right)-\left[p_0+p_a(t)\right]\hfill \\ -\frac{{\rho }_l}{2R}\left[2R{\dot{R}}^2+R^2\ddot{R}\right].\hfill \end{array}$$ (16)

IV. RESULTS

A. Microbubble resonance curves

The resonance curves of bubbles of three different sizes, measured using the method described in Sec. III D, are shown in Fig. 4(a) and compared with theory. The computations use the surface elasticity and viscosity yielding the best fit to the measurements, and both the measurements and the computed curves are normalized to the peak amplitude. The maximum oscillation amplitude is associated with a resonance frequency for each bubble radius.

FIG. 4.

(Color online) (a) Measured and theoretical relative microbubble oscillation amplitudes, normalised to peak amplitude, as a function of frequency for $R_0=2,$ 2.5, and 3 μm. The solid lines show the theoretical curves. (b) Damped resonance frequency as a function of bubble radius for DSPC-lipid coated microbubbles measured using the photoacoustic technique and compared with theory with and without accounting for the effect of the wall. The computations used the following parameters: $p_{\text{pk}}$ = 3 Pa, $\hspace{0.17em}{\text{log}}_{10}({\kappa }_s)=-8.8+0.4R_0$, and $\chi =0.61$ N/m.

These frequencies are displayed in Fig. 4(b) across a range of bubble radii and compared with the computations with and without accounting for the presence of the wall. The presence of a wall results in a decrease of the theoretical resonance frequency by an average 16% for bubbles of radii in the range $R_0=1$–4 μm, which is consistent with the values reported in the literature (Overvelde et al., 2011; Strasberg, 1953).

The range of the measured values for both surface elasticity and viscosity were overall in good agreement with those reported in the literature (Segers et al., 2018; van der Meer et al., 2007; van Rooij et al., 2015). The elasticity was found to be approximately constant at $\chi =0.61\hspace{0.17em}\pm \hspace{0.17em}0.10$ N/m (mean ± standard deviation). The slightly higher elasticity here compared to the $\chi \approx 0.5$ N/m typically reported for DPPC and DPPE-coated microbubbles (Segers et al., 2018; van der Meer et al., 2007) is explained by the DSPC-phospholipids, which have a longer chain length, resulting in a stiffer shell (Lum et al., 2016; van Rooij et al., 2015).

Figure 5 shows the dilatational surface viscosity yielding the best fits to the photoacoustic measurements in Fig. 4 for a range of bubble radii (blue dots, $R_0>1.5$ μm). The viscosities measured using the photoacoustic technique vary significantly with the bubble radius. This is explained through the rheological thinning behavior of the phospholipid monolayer shell, decreasing its viscosity with increasing dilatation rate, which in turn is higher for smaller bubbles (van der Meer et al., 2007). For convenience in the computations, the surface viscosity is expressed here as a function of the bubble radius, although it would be more correct to do so as a function of the dilatation rate.

FIG. 5.

(Color online) Shell viscosity as a function of the bubble radius, as fitted with the measurements of the photoacoustic technique (PA) and the peak microbubble displacements (PMD) of a 1.5–2-μm bubble population. The error bars result from the standard deviation of the PMDs [see Fig. 8(a)]. The dashed line shows the logarithmic fit to PA and PMD data combined: $\hspace{0.17em}{\text{log}}_{10}({\kappa }_s)=-8.8+0.4R_0$, where R₀ has units of μm.

B. Individual bubble displacements

Figure 6 shows the sensitivity of the theoretical microbubble displacement on the shell parameters, namely the surface elasticity, viscosity, and initial surface tension. The displacements driven by a single 10-μs pulse are shown as a function of radius in Figs. 6(a), 6(c), and 6(e). In all radius-displacement curves, a prominent peak was observed, in most cases at a radius around 1.2 μm, which roughly corresponds to the resonating bubble radius at 4-MHz excitation frequency [see Fig. 4(b)]. Some of these curves, mostly corresponding to weakly viscous or highly elastic shells, also showed nonlinear harmonic behavior at approximately 2.2-μm-radius, for which 4 MHz is the second harmonic frequency. The variation of the displacement for a bubble of radius $R_0=1.2$ μm (close to resonant bubble size at 4-MHz frequency) is shown as a direct function of the shell parameters in Figs. 6(b), 6(d), and 6(f). This is compared with the peak displacement across the full range of bubble radii present in the bubble population. The surface elasticity does not significantly affect the theoretical peak displacements within the examined range (all measurements in Sec. IV A remained within the range $\chi =0.5$–0.9 N/m, with a mean value of $\chi =0.61\hspace{0.17em}\pm \hspace{0.17em}0.1$ N/m), yielding an uncertainty lower than 1 μm, and was therefore kept constant for the computations hereafter. The displacements and the radius producing the peak displacement are, however, highly sensitive to the surface viscosity. This highlights the need for a careful determination of this value when computing microbubble displacements, and could be a source of discrepancies between experiments and theory reported in the literature. The initial surface tension of the phospholipid shell has little effect on the bubble displacements, and therefore it is kept at a constant value $\sigma (R_0)=0$ N/m to comply with the microbubble stability.

FIG. 6.

(Color online) Theoretical microbubble displacement for varying (a),(b) surface elasticity, (c),(d) surface viscosity, and (e),(f) initial surface tension. The top and bottom figures show the displacements as a function of bubble radius and shell parameters, respectively. The displacement for $R_0=$ 1.2 μm (solid line) is compared with the peak displacement within the full range of bubble radii (dashed line). Here, $f_0=4$ MHz. Default values are $\chi =0.61$ N/m, ${\kappa }_s=0.4\times {10}^{-8}$ kg/s and $\sigma (R_0)=0$ N/m.

The size and frequency dependencies of the microbubble displacements are displayed in Fig. 7. The displacements followed the expected behavior (Vos et al., 2007): larger bubbles (lower resonance frequencies) undergo larger displacements than smaller bubbles (higher resonance frequencies). Therefore, within the microbubble population comprising a range of sizes, the peak microbubble displacements are expected to decrease with increasing frequency.

FIG. 7.

(Color online) Microbubble displacements as a function of (a) bubble radius and (b) driving frequency. Examples of peak microbubble displacements (PMD) are indicated with arrows. The surface viscosity was defined as $\hspace{0.17em}{\text{log}}_{10}({\kappa }_s)=-8.8+0.4R_0$.

The peak microbubble displacement (PMD) is defined as the largest displacement observed in the multi-gate spectral Doppler profile, such as the one in Fig. 2(b), across the entire acquisition. Figure 8 displays the measured PMDs achieved within the 1.5–2 μm microbubble population (see Fig. 1 for size distribution) as a function of the driving frequency, after subtracting the streaming velocities shown in Fig. 3. The PMD is expected to result from one or more resonant (or close-to-resonant) bubbles, which in principle should always be present within the bubble population regardless of its relative proportion. The length of the acquisition as well as bubble concentration were determined to be sufficient to justify the assumption that we always capture such a resonant bubble at the peak elevational pressure. In agreement with theory, which in Fig. 8 accounts for peak displacements across all bubble sizes present in the population, the experimental results also show decreasing a PMD with increasing frequency.

FIG. 8.

(Color online) Measured and theoretical peak microbubble displacements of 1.5–2 μm microbubble population as a function of driving frequency. The dashed line shows the theoretical displacements for a single radius of 1.75 μm. The error bars show the standard deviation of at least three measurements. Here, $f_0=4$ MHz and $\hspace{0.17em}{\text{log}}_{10}({\kappa }_s)=-8.8+0.4R_0$.

The selection of the surface viscosity was done by finding the best fit to the peak microbubble displacements (PMD) measured at a range of acoustic driving frequencies. For each tested frequency, the surface viscosity producing the closest theoretical peak displacement to the measured PMD was determined, as well as the associated bubble radius. For instance, Fig. 6(b) shows displacements driven at a 4-MHz frequency, which produces a $\sim 7$ μm displacement per pulse according to Fig. 8. The surface viscosity producing the closest peak displacement in Fig. 6(b) is ${\kappa }_s=0.4\times {10}^{-8}$ kg/s, and the corresponding radius is $R_0=1.2$ μm. The combinations of κ_s and R₀ determined this way across all tested frequencies were added to Fig. 5 (gray dots, $R_0<1.5$ μm). The history term, as mentioned in Sec. II, should not affect the determined κ_s and R₀ combinations since they correspond to resonant bubbles (Acconcia et al., 2018). Within the error and the scatter, the surface viscosities followed the same trend as those obtained through the photoacoustic technique. They also conveniently reached smaller values of bubble radii, for which surface viscosities are harder to measure through the photoacoustic technique due to a decreased signal-to-noise ratio. For the remaining computations, the surface viscosity was determined by a logarithmic fit to the PMD and photoacoustic data as a function of radius. Figure 8 shows the theoretical peak displacements computed with this way of determining the surface viscosity as a function of frequency, and they result in a good agreement with the experiments.

Finally, it is important to note that Fig. 8 highlights the importance of accounting for the size range present in the microbubble population: the agreement between the measured and theoretical PMDs is significantly better when accounting for the whole size range (solid line) compared to the single radius of 1.75 μm (dashed line).

C. Displacements in microbubble populations

The measured and theoretical peak microbubble displacements for all the tested different-sized microbubble populations are displayed in Fig. 9 as a function of the driving frequency. The theoretical peak displacements for each size-isolated population only consider the bubble sizes with more than 2% (chosen heuristically) relative to the peak of a Gaussian fit to the respective microbubble number distribution as displayed in Fig. 1. The highest of the displacements computed for these bubble sizes is defined as the theoretical PMD. For the polydisperse microbubble population, all bubble sizes are included. As expected, the values obtained for all populations practically overlap because each population contains at least one resonant microbubble within the plane-wave transmission zone over the acquisition time. Only the population containing the largest bubbles, 2.5–4.0 μm, yields smaller displacements beyond $f_0=4$ MHz, as this size range is too far from the resonant size at these frequencies. The corresponding measurements are in reasonable agreement with the theoretical predictions. There are some fluctuations in the measurements, in contrast to the smooth monotonic decrease in the theoretical PMD. Such fluctuations probably result from uncertainty in the probe calibration, since the streaming velocities (Fig. 3) showed similar behavior.

FIG. 9.

(Color online) Measured and theoretical peak microbubble displacements of different size-isolated microbubble populations as a function of frequency: (a) 0.5–1 μm, (b) 1.5–2 μm, (c) 2–2.5 μm, (d) 2.5–4 μm, and (e) polydisperse microbubbles (PMB). The theoretical displacements for PMB serve as a reference (dashed line). The error bars show the standard deviation of at least three measurements.

In order to gain some insight into the statistical behavior of the translating bubbles, the number distributions of all identified displacement events were obtained at a single depth (elevation focus) displacing more than the displacement threshold $\Delta z_{\text{TH}}=0.8$ μm/pulse (before subtracting the streaming velocities). The histograms in Fig. 10 confirm that the peak microbubble displacements are not representative of the bubble population from a statistical point of view. For example, for 0.5–1.0 μm bubbles, which have a high resonance frequency, show a lot more displacement events at 7 MHz compared to 4 MHz, yet the rare large displacements at 3 MHz are larger than those at 7 MHz. Polydisperse bubbles combine the behaviors of the small and large bubbles, as expected.

FIG. 10.

(Color online) Measured histograms of displacements at f₀ = 3 and 7 MHz for (a) 0.5–1 μm, (b) 1.5–2 μm, (c) 2–2.5 μm, (d) 2.5–4 μm, and (e) polydisperse microbubbles. Each histogram represents a single measurement.

A statistical quantity of interest is the total number of displacements counted during the entire acquisition. This can be useful especially considering small bubbles that displace very little even at their favorable frequency compared to bigger bubbles (see Fig. 7), while there may be a large number of bubbles being pushed. Figure 11 shows the measured counts for the five distinct bubble populations as a function of frequency. The strong scatter in the data is likely to be partly due to the limited measurement resolution for distinguishing individual bubble displacements, and partly due to uncertainty in the pressure calibration of the probe. Differences can be observed between the different bubble populations: at low frequencies, the largest bubbles show significantly larger median displacements compared to the smaller bubbles, while at high frequencies the smaller bubbles show more displacements compared to larger bubbles. Polydisperse bubbles, again, combine the behaviors of the small and large bubbles.

FIG. 11.

(Color online) Measured counts of different size-isolated microbubble populations as a function of frequency: (a) 0.5–1 μm, (b) 1.5–2 μm, (c) 2–2.5 μm, (d) 2.5–4 μm, and (e) polydisperse microbubbles (PMB). The error bars show the standard deviation of at least three measurements.

V. DISCUSSION

The peak microbubble displacements are well predicted by the theory and are generally the same for all populations, as they can result from rare, transient events from the resonant proportion of the population. This makes PMD a less reliable indicator of the performance of a microbubble population. However, the PMD is valuable for characterizing the shell properties, especially the surface viscosity, for a microbubble formulation. The use of multi-gate spectral Doppler to characterize microbubble shells was first demonstrated by Vos et al. (2007). Here the approach is extended to quantify the size dependence of the surface viscosity. The surface viscosity values presented in Fig. 5 are in good agreement with values reported in the literature [e.g., both van der Meer et al. (2007) and Segers et al. (2018) reported similar logarithmic trends with R₀, with viscosities estimated between ${\kappa }_s\approx 5\times {10}^{-9}$ kg/s and ${10}^{-8}$ kg/s for a $R_0=1.5$–2 μm range], and confirmed by extrapolation of the measurement results obtained by the photoacoustic technique on individual bubbles. The use of the multi-gate spectral Doppler approach to characterize these properties has the advantage of involving a relatively simple experiment. It requires no challenging alignments of components typical to those needed in forward laser-scattering or high-speed imaging experiments. Furthermore, there is no need to isolate single bubbles to perform the experiment: one can obtain the surface viscosity by measuring easily reproducible displacements of entire bubble populations floating freely in a beaker. The technique benefits from polydisperse microbubble populations and is capable of characterizing smaller microbubbles that are more difficult to measure optically. The next steps for refining this technique include transmitting at a wider range of frequencies to expand the covered range of bubble radii in Fig. 5, and carefully verifying the assumption of the neglected history force, which is a computationally costly addition to the theoretical model.

Our theoretical analysis showed that surface viscosity has a strong influence on the radiation force-induced displacement. Additionally, we showed experimentally that smaller microbubbles, which have a higher resonance frequency and therefore larger dilatational strain rate, yield a lower apparent surface viscosity. The mechanism for this rheological thinning behavior is still unknown. However, this result is useful in that it suggests that small microbubbles pushed at high (resonant) frequencies provide more efficient radiation force displacement and less energy dissipation by the shell.

The overall findings of this study are a step toward optimizing the pulse sequence and the instrumentation with the bubble formulation to achieve the largest fraction and magnitude of radiation force-induced bubble translations. Statistical quantities such as the number of counted events provide valuable benchmarks for system performance. Here, the statistical analysis verified the size-dependent behavior of bubble translations: transmission at 3 MHz selectively translates more 2.5–4.0 μm bubbles, whereas transmission at 7 MHz selectively translates more 0.5–1.0 μm bubbles. This result is relevant to in vivo applications, where it is generally preferable to increase the fraction of microbubbles that are pushed against the endothelium. This fraction can be maximized by using a narrow size distribution and matching the ultrasound transmission to the resonance frequency of the median size.

VI. CONCLUSION

This study presented an investigation of the primary radiation force effects on microbubble populations. Displacements within freely floating microbubble populations were measured using a standard ultrasound-imaging probe and multi-gate spectral Doppler approach. The theoretical displacements computed by combining a modified Herring model and a balance of translational forces were generally in good agreement with the measurements. As expected, large bubbles translate more at low frequencies and small bubbles at high frequencies. Peak microbubble displacements were almost identical between the distinct-sized bubble populations, but when combined with theory the PMD was useful for characterizing the microbubble shell properties, especially the strain rate-dependent dilatational surface viscosity. Statistical analysis, however, revealed a strong dependence between the number of displaced microbubbles and the size distribution. This suggests that narrow size distributions may be useful to improve in vivo control over the displacements and thus improve the efficacy of targeting or therapy.