Abstract
Contrast agent microbubbles, which are encapsulated gas bubbles, are widely used to enhance ultrasound imaging. There are also several new promising applications of the contrast agents such as targeted drug delivery and noninvasive therapy. Here we study three models of the microbubble dynamics: a nonencapsulated bubble oscillating close to an elastic wall, a simple coated bubble and a coated bubble near an elastic wall.We demonstrate that complex dynamics can occur in these models. We are particularly interested in the multistability phenomenon of bubble dynamics. We show that coexisting attractors appear in all of these models, but for higher acoustic pressures for the models of an encapsulated bubble.We demonstrate how several tools can be used to localize the coexisting attractors. We provide some considerations why the multistability can be undesirable for applications.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Szabo, Th. L., Diagnostic Ultrasound Imaging: Inside Out, 2nd ed., Cambridge: Academic Press, 2013.
Goldberg, B.B., Raichlen, J. S., and Forsberg, F., Ultrasound Contrast Agents: Basic Principles and Clinical Applications, London: Dunitz, 2001.
Hoff, L., Acoustic Characterization of Contrast Agents for Medical Ultrasound Imaging, Dordrecht: Springer, 2001.
Klibanov, A. L., Microbubble Contrast Agents: Targeted Ultrasound Imaging and Ultrasound-Assisted Drug-Delivery Applications, Invest Radiol., 2006, vol. 41, no. 3, pp. 354–362.
Coussios, C.C. and Roy, R.A., Applications of Acoustics and Cavitation to Noninvasive Therapy and Drug Delivery, Annu. Rev. Fluid Mech., 2008, vol. 40, pp. 395–420.
Strutt, J.W. (3rd Baron Rayleigh), On the Pressure Developed in a Liquid during the Collapse of a Spherical Cavity, Philos. Mag. (6), 1917, vol. 34, pp. 94–98.
Plesset, M. S., The Dynamics of Cavitation Bubbles, J. Appl. Mech., 1949, vol. 16, pp. 277–282.
Faez, T., Emmer, M., Kooiman, K., Versluis, M., van der Steen, A., and de Jong, N., 20 Years of Ultrasound Contrast Agent Modeling, IEEE Trans. Ultrason. Ferroelectr. Freq. Control., 2013, vol. 60, no. 1, pp. 7–20.
Doinikov, A.A. and Bouakaz, A., Review of Shell Models for Contrast Agent Microbubbles, IEEE Trans. Ultrason. Ferroelectr. Freq. Control., 2011, vol. 58, no. 5, pp. 981–993.
Parlitz, U., Englisch, V., Scheffczyk, C., and Lauterborn, W., Bifurcation Structure of Bubble Oscillators, J. Acoust. Soc. Amer., 1990, vol. 88, no. 2, pp. 1061–1077.
Behnia, S., Jafari, A., Soltanpoor, W., and Jahanbakhsh, O., Nonlinear Transitions of a Spherical Cavitation Bubble, Chaos Solitons Fractals, 2009, vol. 41, no. 2, pp. 818–828.
Macdonald, C. A. and Gomatam, J., Chaotic Dynamics of Microbubbles in Ultrasonic Fields, Proc. Inst. Mech. Eng. C, 2006, vol. 220, no. 3, pp. 333–343.
Carroll, J.M., Calvisi, M. L., and Lauderbaugh, L. K., Dynamical Analysis of the Nonlinear Response of Ultrasound Contrast Agent Microbubbles, J. Acoust. Soc. Am., 2013, vol. 133, no. 5, pp. 2641–2649.
Doinikov, A.A., Aired, L., and Bouakaz, A., Acoustic Scattering from a Contrast Agent Microbubble near an Elastic Wall of Finite Thickness, Phys. Med. Biol., 2011, vol. 56, no. 21, pp. 6951–6967.
Kudryashov, N.A. and Sinelshchikov, D. I., Analytical Solutions of the Rayleigh Equation for Empty and Gas-Filled Bubble, J. Phys. A, 2014, vol. 47, no. 40, 405202, 10 pp.
Kudryashov, N.A. and Sinelshchikov, D. I., Analytical Solutions for Problems of Bubble Dynamics, Phys. Lett. A, 2015, vol. 379, no. 8, pp. 798–802.
Kudryashov, N.A. and Sinelshchikov, D. I., On the Connection of the Quadratic Liénard Equation with an Equation for the Elliptic Functions, Regul. Chaotic Dyn., 2015, vol. 20, no. 4, pp. 486–496.
de Jong, N., Hoff, L., Skotland, T., and Bom, N., Absorption and Scatter of Encapsulated Gas Filled Microspheres: Theoretical Considerations and Some Measurements, Ultrasonics, 1992, vol. 30, no. 2, pp. 95–103.
Marmottant, P., van der Meer, S., Emmer, M., Versluis, M., de Jong, N., and Hilgenfeldt, S., A Model for Large Amplitude Oscillations of Coated Bubbles Accounting for Buckling and Rupture, J. Acoust. Soc. Am., 2005, vol. 18, no. 6, pp. 3499–3505.
Tu, J., Guan, J., Qiu, Y., and Matula, T. J., Estimating the Shell Parameters of SonoVue Microbubbles Using Light Scattering, J. Acoust. Soc. Am., 2009, vol. 126, no. 6, pp. 2954–2962.
Keller, J. B. and Miksis, M., Bubble Oscillations of Large Amplitude, J. Acoust. Soc. Am., 1980, vol. 68, no. 2, pp. 628–633.
Doinikov, A.A. and Bouakaz, A., Modeling of the Dynamics of Microbubble Contrast Agents in Ultrasonic Medicine: Survey, J. Appl. Mech. Tech. Phys., 2013, vol. 54, no. 6, pp. 867–876; see also: Prikl. Mekh. Tekhn. Fiz., 2013, vol. 54, no. 6, pp. 5–16.
Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., and Prasad, A., Hidden Attractors in Dynamical Systems, Phys. Rep., 2016, vol. 637, pp. 1–50.
Dudkowski, D., Prasad, A., and Kapitaniak, T., Perpetual Points: New Tool for Localization of Coexisting Attractors in Dynamical Systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2017, vol. 27, no. 4, 1750063, 11 pp.
Leonov, G.A., Kuznetsov, N. V., and Mokaev, T.N., Homoclinic Orbits, and Self-Excited and Hidden Attractors in a Lorenz-Like System Describing Convective Fluid Motion, Eur. Phys. J. Special Topics, 2015, vol. 224, no. 8, pp. 1421–1458.
Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Dynamics of Three Vortex Sources, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 694–701.
Borisov, A.V., Kazakov, A.O., and Pivovarova, E.N., Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top, Regul. Chaotic Dyn., 2016, vol. 21, nos. 7–8, pp. 885–901.
Borisov, A.V., Kazakov, A.O., and Sataev, I.R., Spiral Chaos in the NonholonomicModel of a Chaplygin Top, Regul. Chaotic Dyn., 2016, vol. 21, nos. 7–8, pp. 939–954.
Cash, J.R. and Karp, A.H., A Variable Order Runge–Kutta Method for Initial Value Problems with Rapidly Varying Right-Hand Sides, ACM Trans. Math. Software, 1990, vol. 16, no. 3, pp. 201–222.
Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 1. Theory, Meccanica, 1980, vol. 15, pp. 9–20.
Ramasubramanian, K. and Sriram, M. S., A Comparative Study of Computation of Lyapunov Spectra with Different Algorithms, Phys. D, 2000, vol. 139, nos. 1–2, pp. 72–86.
Garashchuk, I.R., Kudryashov, N.A., and Sinelshchikov, D. I., Hidden Attractors in a Model of a Bubble Contrast Agent Oscillating near an Elastic Wall, EPJ Web Conf., 2018, vol. 173, 06006, 4 pp.
Acknowledgments
This work was supported by the Russian Science Foundation, grant number 17-71-10241.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Garashchuk, I.R., Sinelshchikov, D.I. & Kudryashov, N.A. Nonlinear Dynamics of a Bubble Contrast Agent Oscillating near an Elastic Wall. Regul. Chaot. Dyn. 23, 257–272 (2018). https://doi.org/10.1134/S1560354718030036
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354718030036