A non-linear three-dimensional model for quantifying microbubble dynamics

Abstract

A three-dimensional non-linear model for simulating microbubble response to acoustic insonation is presented. A 1 μm radius microbubble stimulated using positive and inverted 2.4 MHz pulses produced radius-time curves that matched (error <10%) with the experimental observation. A bound 2.3 μm radius microbubble insonated using 2.25 MHz 6 cycle pulse was observed to oscillate with max∕min oscillations 45% lower than that of the free microbubble, this correlated (∼10% error) with the observations of Garbin et al. [Appl. Phys. Lett. 90, 114103 (2007)]. The adherent microbubble oscillated asymmetrically in the plan view and symmetrically in the elevation view, consistent with the previous experimental results.

Introduction

Ultrasound contrast agents are gas filled microspheres encapsulated by lipid, protein, or polymer shells. The linear resonance frequencies of the microbubbles are within the diagnostic ultrasound limit, thus making them valuable in medical imaging. In the past decade, the therapeutic applications of microbubbles have emerged, primarily motivated by the need of a suitable drug delivery∕gene delivery agent.1

One-dimensional (1D) radially symmetric models have been widely reported in the literature for quantifying the microbubble’s behavior in response to acoustic insonation. However, most 1D (Ref. 2) models are incapable of predicting phenomena such as higher order nonspherical microbubble oscillations and a consequent shell rupture. Also, phenomena such as the microbubble-microbubble interaction and the microbubble-cell interactions are poorly understood. A thorough understanding of these phenomena can aid in selectively guiding, imaging, and destroying adherent microbubbles, thus enhancing local drug delivery. Criteria such as the Blake threshold,3 the peak negative velocity,4 and the surface kinetic energy of a collapsing microbubble and its cutoff shell stress∕strain5 have been suggested in the literature as thresholds for predicting the onset of an inertial microbubble cavitation and a successive fragmentation. Although some of the above formulations have been validated using high speed camera experiments, ad hoc modifications in these 1D models have been suggested to fully or partially explain them. Here, we propose a full non-linear three-dimensional (3D) finite element analysis (FEA) model for quantifying the microbubble dynamics. This model estimates the coupled 3D oscillatory-translational motion, the shell stress∕strain, and the backscattered acoustic pressure.

Methods

PZFlex (Weidlinger Associates Inc., Mountain View, CA), a 3D FEA software was used in this work.6 In PZFlex, the coupled acoustomechanical problems can be simulated by precisely accounting for the non-linear propagation of the acoustic waves. Free (1 and 2.3 μm radii) and adherent (2.3 μm radius) microbubbles were simulated in PZFlex. A free microbubble [Fig. 1a] is a microbubble that is surrounded by saline or blood and is not in contact with any other entity (vessel wall or other microbubble). An adherent microbubble is a microbubble that is attached to an arterial vessel wall [Fig. 1b]. Each microbubble was simulated in PZFlex by meshing a sphere. The microbubble was encompassed by a cube that simulated the surrounding media (e.g., saline or blood). The meshed cube was ten times the diameter of the meshed microbubble. Eight-node skewed hexahedral elements7 were used to mesh the model. A target element size of approximately 1∕100 of the length of the cube was chosen for meshing the model. The mesh was refined and the model was executed until the output was mesh density invariant and convergent. A final node density of approximately 2×10⁶ nodes per model was used in the simulations reported in this study. In the case of an adherent microbubble, the lipid8 wall properties were assigned to a column of elements to simulate a lumen-vessel wall interface [Fig. 1b]. The adhered microbubble was simulated to be in contact with the arterial wall such that a zero relative motion condition was imposed on the contact shell elements (2×2 or 4, element size ∼140×140×1 nm³) between the wall and the microbubble shell [Fig. 1b].

Figure 1.

(a) Cross-section view of a free microbubble enclosed by a meshed cube representing saline. The microbubble shell is 1 nm thick. The microbubble shell encloses perfluorobutane gas. (b) Cross-section view of an adherent microbubble attached to a lipid wall. The external excitation is applied to XY plane near the origin. The acoustic wave propagates in positive Z direction. The radii of the free and the adherent microbubbles are 1 and 2.3 μm, respectively.

Table 1 lists the values corresponding to various parameters used in the simulations. The local material properties of the media and the microbubble gas (continuum elements) were defined using the material’s bulk modulus, shear modulus, Poisson’s ratio, and density. External acoustic stimulation was applied to a side of the model (along XY plane at Z=0, Fig. 1). Boundary conditions on all the other sides were defined such that they absorbed the acoustic energy to prevent reverberations and undesired reflection. The coupled differential equations that govern the transient behavior of the model are as follows:

$$M_{\mu \mu }\frac{d^2\mu }{d{t}^2}+C_{\mu \mu }\frac{d\mu }{dt}+C_{\mu \psi }\frac{d\psi }{dt}+K_{\mu \mu }\mu =F,$$ (1)

$$M_{\psi \psi }\frac{d^2\psi }{d{t}^2}-C_{\mu \psi }^T\frac{d\mu }{dt}+C_{\psi \psi }\frac{d\psi }{dt}+K_{\psi \psi }\psi =0.$$ (2)

Equation 1 defines the mechanical behavior of the system and Eq. 2 defines the acoustic behavior of the system. The global unknowns, μ and Ψ, are the elastic displacement vector and the velocity potential vector, respectively. F is the external excitation function applied to a side of the model. M, C, and K are mass, damping, and stiffness matrices, respectively. The above equations are applicable to all the material phases’ continuum and shells of the FEA model. The equations are integrated step by step using an explicit time-domain approach.6

Table 1.

List of all parameters used in the 3D simulations.
	Saline	C4F10	Lipid wall	Bubble shell
Density ρ	1000 kg∕m3	11.2 kg∕m3	1000 kg∕m3	1000 kg∕m3
Poisson’s ratio υ	0.499	⋯	0.499	0.499
Shear modulus G	0	0	E∕(2(1+υ))	E∕(2(1+υ))
Elastic modulus E	⋯	⋯	1 MPa	χ∕ε
Sound speed c	1480 m∕s	108 m∕s	⋯	⋯
Dilation modulus χ	⋯	⋯	⋯	0 N∕m, 0.3 N∕m
Bulk modulus K	ρc2	ρc2	E∕(3(1−2υ))	E∕(3(1−2υ))
B∕A	5	0.006	⋯	⋯

In non-linear wave propagation, the wave speed of the propagating wave is inherently pressure dependent by the virtue of the pressure and density relationship,

$$c=\sqrt{\frac{dp}{d\rho }},p=p_o+A\left[\theta +\frac{B}{A}\frac{{\theta }^2}{2}\right],\theta =\frac{\rho }{{\rho }_o}-1,$$ (3a)

where θ is dilatation or expansion and B∕A (Ref. 9) is the parameter of nonlinearity in the truncated Taylor series, c is the velocity of sound through a medium, ρ is the material density, and p is the wave pressure. The incrementally linear, second order accurate,10 explicit algorithm used in PZFlex for simulating the non-linear propagation is based on the velocity and force update equations for a computational element and is stated as

$$v^n=v^{n-1}+\frac{\Delta t{F}^{n-1∕2}}{m}\Rightarrow {\mu }^{n+1∕2}+v^n\Delta t\Rightarrow {\theta }^{n+1∕2}=div.u^{n+1∕2},$$ (3b)

$$p^{n+1∕2}=p_o+A\left[{\theta }^{n+1∕2}+\frac{B}{A}\frac{{\left({\theta }^{n+1∕2}\right)}^2}{2}\right]\Rightarrow F^{n+1∕2},$$ (3c)

where νⁿ, μ^n−1∕2, and F^n−1∕2 are the nodal velocity, displacement, and force, respectively, m is the lumped mass, Δt is the time step, and superscripts refer to the time level. This non-linear propagation model is referred to as “tisu” model in PZFlex. The non-linear pressure-volume relationship in the gas domain of a microbubble can be modeled in PZFlex using the tisu material model and the B∕A parameter. The B∕A values for a given gas can be calculated as follows,

$$\frac{B}{A}=\frac{\gamma -1}{\rho },$$ (4)

where γ is the polytropic expansion constant of the gas and ρ is the density. The B∕A values of air, perflurobutane (C₄F₁₀), and water are 0.4, 0.006, and 5.0, respectively. The gas core of the microbubble was modeled as perflurobutane gas (C₄F₁₀) with density11 11.2 kg∕m³ and dilatational wave speed of approximately 108 m∕s. The shell parameters (shell elements) such as shell elastic (E), bulk (K), and shear moduli (G) can be estimated from the area dilatational modulus or shell stiffness modulus. The area dilatational modulus is defined as, χ=Eε, where E and ε are Young’s modulus and the thickness of the microbubble shell. For all simulations reported in this work, the microbubble’s shell was assumed to be of 1 nm thickness.12 The Poisson’s ratio for biomaterials such as lipid and protein is in the range5 0.49<υ<0.5. For all simulations reported in this work, the Poisson’s ratio was assumed to be 0.499. The surrounding water media was modeled as a viscous Newtonian fluid. The density and the viscosity of water were assumed to be 1000 kg∕m³ and 0.001 Pa s, respectively.

Results

Free and adherent 2.3 μm radius microbubbles were simulated in the 3D FEA software. The microbubbles were stimulated using a windowed 6 cycle 2.25 MHz sine pulse with a peak negative pressure of 240 kPa [Fig. 2d]. Similar experimental parameters were used in works published by Garbin et al.,13 Zhao et al.,14 and Vos et al.15 In the adherent microbubble model, the vessel wall was simulated using the mechanical properties of a stiff arterial wall8 with the stiffness of (E=1 MPa) and the density of 1000 kg∕m³. The arterial wall was assumed to be incompressible. Figure 2a illustrates the superimposed radial oscillations of the free and adherent microbubbles. The adherent microbubble exhibits reduced radial excursion when compared with the free microbubble. The maximum positive and negative excursions of the adherent microbubble are approximately 0.8 and 0.6 μm, respectively [Fig. 2a], whereas the maximum excursions of the free microbubble are 1.5 and 1.3 μm, respectively. The radial excursions of the adherent microbubble are approximately 45% smaller than that of the free microbubble. This observation correlates with the results published by Garbin et al.,13 who reported a 50% reduction. As illustrated in Fig. 2b, the center of the adherent microbubble follows its radial oscillation, but with reduced amplitude, implying that the microbubble is continuously attached to the simulated wall. The center of the free microbubble follows a uniform oscillatory-translational motion in the direction away from the excitation source [Fig. 2b]. Similar experimental results have been demonstrated by Vos et al.15 The spectrum of the echo received from the adherent microbubble exhibits reduced power in the higher order harmonics and an increased ratio in the power of the fundamental to harmonics (20% fundamental second and 50% fundamental third) when compared with the spectrum of the free microbubble [Fig. 2c]. Experimental results reporting the same were first published by Zhao et al.16 (30% fundamental second and 70% fundamental third at 2 MHz∕210 kPa). It should be noted that the experimental results were reported for a cloud of microbubble rather than a single microbubble.

Figure 2.

A 2.3 μm radius microbubble (0.3 N∕m area dilation modulus) stimulated using a 2.25 MHz, 6 cycle sine pulse with a peak negative pressure of 240 kPa. (a) Superimposed radius-time curves of free and adherent microbubble, (b) superimposed motion of the center of mass (translational motion) of free and adherent microbubble, (c) superimposed spectra of the received echoes from free and adherent microbubble, and (d) excitation pulse.

Figure 3 illustrates the 3D oscillatory motion of an adherent microbubble in two different orthogonal planes, azimuth-range (XZ plane) and azimuth-elevation (XY plane). Here after, these planes are referred to as plan view and elevation view, respectively. The adherent microbubble oscillates asymmetrically (second order mode) in the plan view and symmetrically in the elevation view. Thus, our 3D simulations validate the experimental results independently reported by Vos et al.15 and Zhao et al.14 3D FEA simulations may provide insight in designing pulse codes capable of uniquely enhancing signals from adherent microbubbles and suppressing signals from free microbubbles, 1D models are incapable of providing such valuable information.

Figure 3.

Time snapshots of the 2.3 μm adherent microbubble. The microbubble was stimulated using a 2.25 MHz 6 cycle sine pulse (240 kPa) along the plane parallel to azimuth-elevation plane. Plan view (azimuth-range plane) of the adherent microbubble (first row). Elevation view (azimuth elevation) of the adherent microbubble (second row). The last two rows list the corresponding maximum and minimum shell stresses at various time instants in the excitation cycle. The elastic modulus (E) of the microbubble shell was simulated to be 300 MPa. The peak negative normal shell stress or compressive shell stress (at t=1.6 μs) was approximately 147 MPa (<0.5 E).

Figure 4a illustrates the R-T (radius-time) curves of a free 1 μm radius microbubble stimulated using a 2.4 MHz Gaussian modulated sine wave with 50% fractional bandwidth (BW) and a peak negative pressure of 360 kPa. The 3D FEA R-T curves obtained using the above parameters and the viscoelastic model published by Church et al.17 are superimposed on the optical R-T curves obtained from Morgan et al.12 [Fig. 4c in Ref. 12]. The rms error between the 3D FEA simulations and the optical R-T curves is 9%. Figure 4b illustrates the superimposed spectra of the echo response obtained by insonating a 1 μm microbubble with 0° and 180° phase pulses. A 0° phase pulse is a Gaussian modulated sine wave and a 180° pulse is an inverted Gaussian modulated sine wave. The shift between the centroids of the spectra is approximately 0.5 MHz and is within the range of the frequency shifts reported by Morgan et al.12

Figure 4.

A 1 μm radius microbubble was stimulated using a 2.4 MHz 50% fractional BW Gaussian modulated sine wave with 0° phase pulse at 360 kPa. (a) Superimposed PZFlex simulations, optical R-T curves (Ref. 12), and 1D simulations obtained using viscoelastic model published in Ref. 17. (b) Spectrum of the echoes received from 0° and 180° phase pulses. R(t) is the instantaneous microbubble radius, whereas R_o is the resting microbubble radius.

Conclusion

In this letter, we introduced and validated a 3D non-linear microbubble simulation framework. This model has great potential in addressing complex problems such as analyzing 3D asymmetric mode of vibrations and providing complementary information in the form of shell stress and strain that is not available from the conventional 1D models. The complementary information provided by this model may further aid in deciphering the microbubble-cell interaction, formulating microbubble cavitation limits, and analyzing multiple microbubble-microbubble interactions that may aid in selectively imaging microbubble and guiding local drug delivery.