Physico-mathematical models for interacting microbubble clouds during histotripsy

Abstract

This work proposes physico-mathematical models for cavitation of microbubble clouds under the influence of bubble–bubble interaction during histotripsy. The mathematical models are formulated to non-interacting and interparticle interacting microbubble clouds on histotripsy under considering the effect of variable surface tension. The governing equations of Keller–Miksis (KM) based on Neo-Hookean (NH) and the Quadratic Law Kelvin–Voigt (QLKV) models are transformed into ordinary differential equations using the non-dimension variables methodology, which are then solved analytically by the modified Plesset–Zwick method. The generalized case of variable surface tension is derived and evaluated for both cases of non-interacting and interacting microbubbles during histotripsy. The effects of the viscoelastic medium on the dynamics of a single microbubble dynamic and interactions between microbubbles through the histotripsy are investigated. From the analysis of the results, the behavior of single bubble growth is bigger than in the case of interaction of multi-bubbles under considering the effect of viscoelastic tissue of Young modulus, viscosity, and stiffening factor on histotripsy. Moreover, the study reveals that, when an increase in the number of cavitation microbubbles occurs, a decrease of the behavior of cavitation microbubbles occurs, on the contrary, increasing of the distance between microbubbles leads to increasing in the growth process; these processes for growth are playing a significant role during the process of histotripsy of cancerous tissues.

Appendices

Appendix A

$$Z_{1} = S_{0}^{3} \left( {\frac{{{\Omega }^{5} }}{3}} \right)^{\frac{1}{2}} \frac{2}{{\pi D_{13} }},$$

$$\begin{gathered} Z_{2} = bc^{n} + \frac{1}{2}G\left( {3\alpha - 1} \right)\left( {5 - 4{\Phi }_{0}^{\frac{1}{3}} - 4{\Phi }_{0}^{\frac{4}{3}} } \right) \hfill \\ \quad\quad+ 2G\alpha \left( {\frac{27}{{40}} + \frac{1}{8}{\Phi }_{0}^{\frac{8}{3}} + \frac{1}{5}{\Phi }_{0}^{\frac{5}{3}} + {\Phi }_{0}^{\frac{2}{3}} - 2{\Phi }_{0}^{{\frac{ - 1}{3}}} } \right) \hfill \\\quad\quad - \frac{4}{3}\eta {{\Omega \Phi }}_{0}^{{\frac{ - 1}{3}}} {\Psi }_{{\text{m}}}^{{\prime}} , \hfill \\ \end{gathered}$$

$$Z_{3} = \frac{{3{\Phi }_{0}^{{\frac{ - 1}{3}}} }}{{{\Omega }S_{0} \left( {3c + {\Omega }S_{0} {\Phi }_{0}^{{\frac{ - 2}{3}}} {\Psi }_{m}^{{\prime}} } \right)}},$$

$$\begin{gathered} Z_{4} = - \kappa P_{g0} {\Phi }_{0}^{{\frac{1}{3}\left( {3\kappa - 1} \right)}} {\Psi }_{m}^{{\prime}} \hfill \\\quad\quad - \frac{4}{3}\eta {{\Omega \Phi }}_{0}^{{\frac{ - 2}{3}}} \left( {{\Phi }_{0}^{ - 1} {\Psi }_{m}^{{\prime\prime}} + \frac{1}{3} \mathop { {\Psi }}\limits_{m}^{2} } \right) \hfill \\\quad\quad + \frac{1}{2}G\left( {3\alpha - 1} \right)\left( {\frac{4}{3}{\Psi }_{m}^{{\prime}} + 4{\Phi }_{0} {\Psi }_{m}^{{\prime}} } \right) \hfill \\ \quad\quad- 2G\alpha \left( {\frac{1}{3}{\Phi }_{0}^{\frac{7}{3}} {\Psi }_{m}^{{\prime}} + \frac{1}{3}{\Phi }_{0}^{\frac{4}{3}} {\Psi }_{m}^{{\prime}} + \frac{2}{3}{\Phi }_{0}^{\frac{1}{3}} {\Psi }_{m}^{{\prime}} + \frac{2}{3}{\Phi }_{0}^{\frac{2}{3}} {\Psi }_{m}^{{\prime}} } \right), \hfill \\ \end{gathered}$$

$$Z_{5} = \frac{{\rho_{l} {\Omega }S_{0} {\Phi }_{0}^{ - 2} }}{{9C\left( {1 + \frac{1}{3C}{\Omega }S_{0} {\Phi }_{0}^{{\frac{ - 2}{3}}} {\Psi }_{m}^{{\prime}} } \right)}},$$

$$Z_{6} = Z_{5} \left( {{\Phi }_{0}^{ - 1} \left( { {\Phi }_{0}^{ - 1} {\Psi }_{m}^{{\prime\prime}} + \frac{2}{3} { {\acute{\Psi} }}_{m}^{2} } \right) + \frac{1}{6}{{\Psi^{\prime}}}_{m}^{3} } \right).$$

Appendix B

$$\begin{aligned} S\left( t \right)& = K_{1} \left[K_{2} + \frac{1}{{\rho_{L} \Gamma^{2} R_{0}^{2} }}\left( {1 + \frac{{\Gamma R_{0} \varphi_{0}^{{\frac{ - 2}{3}}} {{\mathop \beta \limits^\prime }_m}}}{3c}} \right)\left( {B_{1} + bc^{n} } \right) \hfill \right.\\ &\quad - K_{3} + \frac{{G\left( {3\alpha - 1} \right)}}{{2\rho_{L} \Gamma^{2} R_{0}^{2} }}\left( 5\left( {1 - \frac{{\Gamma R_{0} }}{3c}\varphi_{0}^{{\frac{ - 2}{3}}} {{\mathop \beta \limits^\prime }_m} } \right)\right. \hfill \\ &\quad \left.- 4\varphi_{0}^{\frac{1}{3}} - \varphi_{0}^{\frac{4}{3}} \left( {1 - \frac{{\Gamma R_{0} }}{c}\varphi_{0}^{{\frac{ - 2}{3}}} {{\mathop \beta \limits^\prime }_m} } \right) \right) \hfill \\ &\quad + \frac{2G\alpha }{{\rho_{L} }}\left( \frac{27}{{40}}\left( {1 + \frac{{\Gamma R_{0} }}{3c}\varphi_{0}^{{\frac{ - 2}{3}}} {{\mathop \beta \limits^\prime }_m} } \right) \right.\hfill \\ &\quad + \frac{1}{8}\varphi_{0}^{\frac{8}{3}} \left( {1 - \frac{{7\Gamma R_{0} }}{3c}\varphi_{0}^{{\frac{ - 2}{3}}} {{\mathop \beta \limits^\prime }_m} } \right) \hfill \\ &\quad + \frac{1}{5}\varphi_{0}^{\frac{5}{3}} \left( {1 - \frac{{4\Gamma R_{0} }}{3c}\varphi_{0}^{{\frac{ - 2}{3}}} {{\mathop \beta \limits^\prime }_m} } \right) \hfill \\ &\quad + \beta_{m}^{{\frac{ - 2}{3}}} \left( {1 - \frac{{\Gamma R_{0} }}{3c}\varphi_{0}^{{\frac{ - 2}{3}}} {{\mathop \beta \limits^\prime }_m} } \right) \hfill \\ &\quad \left.\left.- 2\varphi_{0}^{{\frac{ - 1}{3}}} \left( {1 + \frac{{2\Gamma R_{0} }}{3c}\varphi_{0}^{{\frac{ - 2}{3}}} {{\mathop \beta \limits^\prime }_m} } \right) \hfill \right) \vphantom{ \left[K_{2} + \frac{1}{{\rho_{L} \Gamma^{2} R_{0}^{2} }}\left( {1 + \frac{{\Gamma R_{0} \varphi_{0}^{{\frac{ - 2}{3}}} {{\mathop \beta \limits^\prime }_m}}}{3c}} \right)\left( {B_{1} + bc^{n} } \right)\right.}\right]J_{a} \sqrt {\frac{24}{\pi }} \sqrt {a_{L} t} . \end{aligned}$$

Here ${K}_{1}, {K}_{2}, {K}_{3}$ and all parameters are introduced in [37].