The breakup of intravascular microbubbles and its impact on the endothelium

Abstract

Encapsulated microbubbles (MBs) serve as endovascular agents in a wide range of medical ultrasound applications. The oscillatory response of these agents to ultrasonic excitation is determined by MB size, gas content, viscoelastic shell properties and geometrical constraints. The viscoelastic parameters of the MB capsule vary during an oscillation cycle and change irreversibly upon shell rupture. The latter results in marked stress changes on the endothelium of capillary blood vessels due to altered MB dynamics. Mechanical effects on microvessels are crucial for safety and efficacy in applications such as focused ultrasound-mediated blood–brain barrier (BBB) opening. Since direct in vivo quantification of vascular stresses is currently not achievable, computational modelling has established itself as an alternative. We have developed a novel computational framework combining fluid–structure coupling and interface tracking to model the nonlinear dynamics of an encapsulated MB in constrained environments. This framework is used to investigate the mechanical stresses at the endothelium resulting from MB shell rupture in three microvessel setups of increasing levels of geometric detail. All configurations predict substantial elevation of up to 150 % for peak wall shear stress upon MB breakup, whereas global peak transmural pressure levels remain unaltered. The presence of red blood cells causes confinement of pressure and shear gradients to the proximity of the MB, and the introduction of endothelial texture creates local modulations of shear stress levels. With regard to safety assessments, the mechanical impact of MB breakup is shown to be more important than taking into account individual red blood cells and endothelial texture. The latter two may prove to be relevant to the actual, complex process of BBB opening induced by MB oscillations.

Appendix: Validation of the IFT algorithm

An IFT algorithm with a modified pressure boundary condition according to Eq. (7) is employed in this study to model the dynamics of a non-spherically oscillating encapsulated MB immersed in liquid. It allows for variations of $\sigma _{s}$ and $\mu _{s}$ as a function of MB size, including abrupt changes at shell breakup or during buckling. This algorithm is validated with respect to analytical predictions from Eq. (5).

Fig. 9

Comparison of radius response curves from theoretical predictions and numerical results for a configuration with $f_{\mathrm{ex}} = 2.9$ MHz, $P_{\mathrm{ex}} = 130$ kPa, $\kappa _{s} = 15\, \hbox {nN}\, \hbox {s}\, \hbox {m}^{-1},\, \chi = 1\, \hbox {N}\, \hbox {m}^{-1}$ and $\tilde{a} = \hbox {a}_{0} = 0.975\, \upmu \hbox {m}$. The numerically determined a(t) shows excellent agreement with the analytical prediction, having a relative deviation of less than 0.45 %

The validation setup comprises a single MB of equilibrium radius $a_{0} = 0.975\, \upmu \hbox {m}$ with variable settings for $\kappa _{s}$ and $\chi$. This MB is enclosed in a fluid sphere of radius $R_{l} = 100\, \upmu \hbox {m}$, mimicking an infinitely large liquid envelope. The liquid is considered to be water with $\rho _{l} = 1000\, \hbox {kg}\, \hbox {m}^{-3}$ and $\mu _{l} = 10^{-3}\, \hbox {kg}\, \hbox {m}^{-1}\, \hbox {s}^{-1}$. This setup is exposed to a transient pressure field at various excitation frequencies $f_{\mathrm{ex}}$ and either constant or linearly growing excitation amplitude $P_{\mathrm{ex}}$.

In a first step, the conformity with predictions in the elastic regime and for buckling is established. The maximum tension that the shell can sustain is set to $\sigma _{c} = 1\, \hbox {N}\, \hbox {m}^{-1}$, which is deliberately high to preclude shell rupture at this stage of the validation process. Figure 9 shows a comparison between the transient radius plots acquired with IFT and predicted by the modified RPE in Eq. (5) for the oscillation of a MB driven by an US field of $f_{ex} = 2.9$ MHz and $P_{\mathrm{ex}} = 130$ kPa. The shell properties were chosen to fit a SonoVue$^{{\circledR }}$ UCA with $\kappa _{s }= 15\, \hbox {nN}\, \hbox {s}\, \hbox {m}^{-1}$, $\chi = 1\, \hbox {N}\, \hbox {m}^{-1}$ and $\tilde{a} = a_{0} = 0.975\, \upmu \hbox {m}$ (Marmottant et al. 2005). We quantify the conformity between the numerical and the analytical results using the maximum of the transient relative difference

$$\begin{aligned} \varepsilon (t)=\left| {\frac{a_{a} (t)-a_{\mathrm{n}} (t)}{a_{a,\max } -a_{a,\min } }} \right| , \end{aligned}$$

(11)

where $a_{{a}}(t)$ represents the transient analytically determined MB radius and $a_{n}(t)$ represents the transient numerically determined MB radius, while $a_{a,\max }$ and $a_{a,\min }$ denote the overall maximum and minimum of $a_{a}(t)$. The analytically predicted and the numerically modelled comportments show excellent agreement with $\varepsilon _{\mathrm{max}} = 0.45$ % in a simulation interval of 10  $\upmu \hbox {s}$. The numerically acquired data also closely correspond to the results based on simulations using the same parameters published by Marmottant et al. (2005). As expected, the radius pattern exhibits compression preference (de Jong et al. 2007; Versluis 2010) and a change in slope during contraction.

Fig. 10

Comparison between numerical results and analytical prediction for shell break up of two MBs with viscoelastic encapsulation of $a_{0} = 0.975\, \upmu \hbox {m}$ and $\tilde{a} = 0.95\, \upmu \hbox {m}$. An encapsulated MB of $\kappa _{s} = 7.2\, \hbox {nN}\, \hbox {s}\, \hbox {m}^{-1}$, $\chi = 0.55\, \hbox {N}\, \hbox {m}^{-1}$, $\sigma _{c} = 0.0615\, \hbox {N}\, \hbox {m}^{-1}$ and $\mu _{r} = 0.4\, \mu _{s}$ (top) exposed to $f_{\mathrm{ex}} = 1.5$ MHz, $P_{\mathrm{ex},i} = 100$ kPa and $P_{\mathrm{ex},f} = 130$ kPa with $\Delta t_{r} = 5\, \upmu \hbox {s}$ shows breakup at 3.48  $\upmu \hbox {s}$. Another MB of $\kappa _{s }= 15\, \hbox {nN}\, \hbox {s}\, \hbox {m}^{-1}$, $\chi = 1\, \hbox {N}\, \hbox {m}^{-1}$, $\sigma _{c} = 0.148\, \hbox {N}\, \hbox {m}^{-1}$ and $\mu _{r} = 0.4\, \mu _{s}$ breaks up at 4.92 $\upmu \hbox {s}$ when excited with $f_{\mathrm{ex}} = 2.9$ MHz, $P_{\mathrm{ex},i} = 150$ kPa and $P_{\mathrm{ex},f} = 210$ kPa with $\Delta t_{r} = 5\, \upmu \hbox {s}$. Both cases show close agreement between analytical prediction and numerical results

After checking the predictive capability of the numerical model for the elastic regime and the buckling state, we examine its ability to model shell rupture. To achieve a transition from oscillation with encapsulation to oscillation without encapsulation, $P_{\mathrm{ex}}$ is ramped up from an initial $P_{\mathrm{ex},i}$ to a final value $P_{\mathrm{ex},f}$ over a time interval $\Delta t_{r}$ and stays constant at $P_{\mathrm{ex},f}$ for $t > \Delta t_{r}$. Figure 10 displays a portion of the radius–time curves for two configurations with the instants of breakup indicated by arrows. Both encapsulated MBs have a buckling radius of $\tilde{a} = 0.95\, \upmu \hbox {m}$.

The top panel of Fig. 10 shows results obtained for $f_{\mathrm{ex}} = 1.5$ MHz, pressure amplitude buildup from $P_{\mathrm{ex},i} = 100$ kPa to $P_{\mathrm{ex},f} = 130$ kPa over $\Delta t_{r} = 5\, \upmu \hbox {s}$ and shell properties of $\kappa _{s} = 7.2\, \hbox {nN}\, \hbox {s}\, \hbox {m}^{-1},\, \chi = 0.55\, \hbox {N}\, \hbox {m}^{-1}$ (Gorce et al. 2000), $\sigma _{c} = 0.0615\, \hbox {N}\, \hbox {m}^{-1}$ and $\mu _{r} = 0.4\, \mu _{s}$. The shell’s breakup tension is surpassed at $P_{\mathrm{ex}} = 121$ kPa and $t_{b} = 3.48\, \upmu \hbox {s}$, after which the oscillation amplitude still keeps growing with increasing $P_{\mathrm{ex}}$. $\varepsilon _{\mathrm{max}}$ in the elastic regime does not surpass 0.6 % and peaks at 1.9 % after shell breakup.

Figure  10 (bottom) displays findings for the oscillation of a MB with $f_{ex} = 2.9\, \hbox {MHz},$ $P_{\mathrm{ex},i} = 150\, \hbox {kPa},\, P_{\mathrm{ex},f} = 210\, \hbox {kPa},\, \Delta t_{r} = 5\, \upmu \hbox {s}$, $\kappa _{s }= 15 \hbox {nN}\, \hbox {s}\, \hbox {m}^{-1}$, $\chi = 1\, \hbox {N}\, \hbox {m}^{-1}$, $\sigma _{c} = 0.148\, \hbox {N}\, \hbox {m}^{-1}$ and $\mu _{r} = 0.4\,\mu _{s}$. Shell breakup occurs at 4.92 $\upmu \hbox {s}$ and $P_{\mathrm{ex}} = 209$ kPa. Subsequently, a(t) quickly stabilizes to a new pattern with constant amplitude. $\varepsilon _{\mathrm{max}}$ in the elastic regime does not surpass 0.4 % and increases to 2.4 % after shell breakup.

All three presented setups show an agreement between numerical results and analytical predictions with a difference below 1 % before shell rupture and lower than 2.5 % after rupture.