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June  2021, 15(3): 555-597. doi: 10.3934/ipi.2021005

Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies

Alexander Dabrowski ±,, , Ahcene Ghandriche ±, and  Mourad Sini ‡,

RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria

* Corresponding author: Ahcene Ghandriche

± This author is supported by the Austrian Science Fund (FWF): P 30756-NBL.
‡ This author is partially supported by the Austrian Science Fund (FWF): P 30756-NBL

Received  June 2020 Revised  October 2020 Published  December 2020

We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves).

We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixed single backscattering direction. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region.

1.We show that, for each fixed used bubble, the contrasted far-fields reach their maximum value at, incident, frequencies close to the Minnaert resonance (or the body-wave resonances depending on the types of bubbles we use). Hence, we can reconstruct this resonance from our data. The explicit dependence of these resonances in terms of the background mass density of the background allows us to recover it, i.e. the mass density, in a straightforward way.

2.In addition, this measured contrasted far-fields allow us to recover the total field at the location points of the bubbles (i.e. the total field in the absence of the bubbles). A numerical differentiation argument, for instance, allows us to recover the bulk modulus of the targeted region as well.

Keywords: Acoustic imaging, bubbly media, Minnaert resonance, surface-wave resonances, Newtonian potential, body-wave resonances.
Mathematics Subject Classification: Primary: 35R30, 35C20; Secondary: 00A69.
Citation: Alexander Dabrowski, Ahcene Ghandriche, Mourad Sini. Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies. Inverse Problems & Imaging, 2021, 15 (3) : 555-597. doi: 10.3934/ipi.2021005
References:
[1]

A. Alsaedi, B. Ahmad, D. P. Challa, M. Kirane and M. Sini, A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index, Math. Methods Appl. Sci., 39 (2016), 3607-3622.  doi: 10.1002/mma.3805.  Google Scholar

[2]

H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, Springer, Berlin, 2008. doi: 10.1007/978-3-540-79553-7.  Google Scholar

[3]

H. Ammari, D. P. Challa, A. P. Choudhury and M. Sini, The point-interaction approximation for the fields generated by contrasted bubbles at arbitrary fixed frequencies, Journal of Differential Equations, 267 (2019), 2104–2191. doi: 10.1016/j.jde.2019.03.010.  Google Scholar

[4]

H. Ammari, D. P. Challa, A. P. Choudhury and M. Sini, The equivalent media generated by bubbles of high contrasts: Volumetric metamaterials and metasurfaces, Multiscale Model. Simul., 18 (2020), 240-293.  doi: 10.1137/19M1237259.  Google Scholar

[5]

H. Ammari, A. Dabrowski, B. Fitzpatrick, P. Millien and M. Sini, Subwavelength resonant dielectric nanoparticules with high refractive indices, Mathematical Methods in the Applied Sciences, 42 (2019), 6567-6579.  doi: 10.1002/mma.5760.  Google Scholar

[6]

H. Ammari, B. Fitzpatrick, D. Gontier, H. Lee and H. Zhang, Minnaert resonances for acoustic waves in bubbly media, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1975–1998. doi: 10.1016/j.anihpc.2018.03.007.  Google Scholar

[7]

H. Ammari, B. Fitzpatrick, H. Lee, S. Yu and H. Zhang, Double-negative acoustic metamaterials., Quarterly of Applied Mathematics, 77 (2019), 767-791.  doi: 10.1090/qam/1543.  Google Scholar

[8]

H. Ammari, B. Fitzpatrick, D. Gontier, H. Lee and H. Zhang, Sub-wavelength focusing of acoustic waves in bubbly media, Proceedings of the Royal Society A., 473 (2017), 20170469, 17pp. doi: 10.1098/rspa.2017.0469.  Google Scholar

[9]

H. Ammari and H. Zhang, Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency, SIAM J. Math. Anal., 49 (2017), 3252-3276.  doi: 10.1137/16M1078574.  Google Scholar

[10]

C. R. Anderson, X. Hu, H. Zhang, J. Tlaxca, A. E. Declèves, R. Houghtaling, K. Sharma, M. Lawrence, K. W. Ferrara and J. J. Rychak, Ultrasound molecular imaging of tumor angiogenesis with an integrin targeted microbubble contrast agent, Investigative Radiology, 46 (2011), 215-224.  doi: 10.1097/RLI.0b013e3182034fed.  Google Scholar

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R. Caflisch, M. Miksis, G. Papanicolaou and L. Ting, Effective equations for wave propagation in a bubbly liquid, J. Fluid Mec., 153 (1985), 259-273.   Google Scholar

[12]

R. Caflisch, M. Miksis, G. Papanicolaou and L. Ting, Wave propagation in bubbly liquids at finite volume fraction, J. Fluid Mec., 160 (1986), 1-14.   Google Scholar

[13]

D. P. Challa and M. Sini, Multiscale analysis of the acoustic scattering by many scatterers of impedance type, Z. Angew. Math. Phys., 67 (2016), Art. 58, 31pp. doi: 10.1007/s00033-016-0652-0.  Google Scholar

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-030-30351-8.  Google Scholar

[15]

G. Dassios and R. Kleinman, Low Frequency Scattering, Oxford Mathematical Monographs, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar

[16]

A. Ghandriche and M. Sini, Mathematical Analysis of the Photo-acoustic imaging modality using resonating dielectric nanoparticles: The 2D TM-model, arXiv: 2003.03162 Google Scholar

[17]

T. Ilovitsh, A. Ilovitsh, J. Foiret, et al., Enhanced microbubble contrast agent oscillation following 250 kHz insonation, Sci Rep, 8 (2018), 16347. doi: 10.1038/s41598-018-34494-5.  Google Scholar

[18]

G. C. Papanicolaou, Diffusion in Random Media, Surveys in Applied Mathematics, volume 1, Edited by J P. Keller, D W. McLaughlin and G C. Papanicolaou, Plenum Press, NewYork, 1995. Google Scholar

[19]

S. Qin, C. F. Caskey and K. W. Ferrara, Ultrasound contrast microbubbles in imaging and therapy: Physical principles and engineering, Phys Med Biol., 54 (2009), R27. doi: 10.1088/0031-9155/54/6/R01.  Google Scholar

[20]

E. Quaia, Microbubble ultrasound contrast agents: An update, European Radiology, 17 (2007), 1995-2008.  doi: 10.1007/s00330-007-0623-0.  Google Scholar

[21]

T. Meklachi, S. Moskow and J. C. Schotland, Asymptotic analysis of resonances of small volume high contrast linear and nonlinear scatterers, J. Math. Phys., 59 (2018), 083502, 20pp. doi: 10.1063/1.5031032.  Google Scholar

[22]

A. Song, J. Li, C. Shen, X. Peng, X. Zhu, T. Chen1 and S. A. Cummer, Broadband high-index prism for asymmetric acoustic transmission, Appl. Phys. Lett., 114 (2019), 121902. Google Scholar

[23]

F. Zangeneh-Nejad and R. Fleury, Acoustic analogues of high-index optical waveguide devices, Sci Rep, 8 (2018), 10401. doi: 10.1038/s41598-018-28679-1.  Google Scholar

show all references

References:
[1]

A. Alsaedi, B. Ahmad, D. P. Challa, M. Kirane and M. Sini, A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index, Math. Methods Appl. Sci., 39 (2016), 3607-3622.  doi: 10.1002/mma.3805.  Google Scholar

[2]

H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, Springer, Berlin, 2008. doi: 10.1007/978-3-540-79553-7.  Google Scholar

[3]

H. Ammari, D. P. Challa, A. P. Choudhury and M. Sini, The point-interaction approximation for the fields generated by contrasted bubbles at arbitrary fixed frequencies, Journal of Differential Equations, 267 (2019), 2104–2191. doi: 10.1016/j.jde.2019.03.010.  Google Scholar

[4]

H. Ammari, D. P. Challa, A. P. Choudhury and M. Sini, The equivalent media generated by bubbles of high contrasts: Volumetric metamaterials and metasurfaces, Multiscale Model. Simul., 18 (2020), 240-293.  doi: 10.1137/19M1237259.  Google Scholar

[5]

H. Ammari, A. Dabrowski, B. Fitzpatrick, P. Millien and M. Sini, Subwavelength resonant dielectric nanoparticules with high refractive indices, Mathematical Methods in the Applied Sciences, 42 (2019), 6567-6579.  doi: 10.1002/mma.5760.  Google Scholar

[6]

H. Ammari, B. Fitzpatrick, D. Gontier, H. Lee and H. Zhang, Minnaert resonances for acoustic waves in bubbly media, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1975–1998. doi: 10.1016/j.anihpc.2018.03.007.  Google Scholar

[7]

H. Ammari, B. Fitzpatrick, H. Lee, S. Yu and H. Zhang, Double-negative acoustic metamaterials., Quarterly of Applied Mathematics, 77 (2019), 767-791.  doi: 10.1090/qam/1543.  Google Scholar

[8]

H. Ammari, B. Fitzpatrick, D. Gontier, H. Lee and H. Zhang, Sub-wavelength focusing of acoustic waves in bubbly media, Proceedings of the Royal Society A., 473 (2017), 20170469, 17pp. doi: 10.1098/rspa.2017.0469.  Google Scholar

[9]

H. Ammari and H. Zhang, Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency, SIAM J. Math. Anal., 49 (2017), 3252-3276.  doi: 10.1137/16M1078574.  Google Scholar

[10]

C. R. Anderson, X. Hu, H. Zhang, J. Tlaxca, A. E. Declèves, R. Houghtaling, K. Sharma, M. Lawrence, K. W. Ferrara and J. J. Rychak, Ultrasound molecular imaging of tumor angiogenesis with an integrin targeted microbubble contrast agent, Investigative Radiology, 46 (2011), 215-224.  doi: 10.1097/RLI.0b013e3182034fed.  Google Scholar

[11]

R. Caflisch, M. Miksis, G. Papanicolaou and L. Ting, Effective equations for wave propagation in a bubbly liquid, J. Fluid Mec., 153 (1985), 259-273.   Google Scholar

[12]

R. Caflisch, M. Miksis, G. Papanicolaou and L. Ting, Wave propagation in bubbly liquids at finite volume fraction, J. Fluid Mec., 160 (1986), 1-14.   Google Scholar

[13]

D. P. Challa and M. Sini, Multiscale analysis of the acoustic scattering by many scatterers of impedance type, Z. Angew. Math. Phys., 67 (2016), Art. 58, 31pp. doi: 10.1007/s00033-016-0652-0.  Google Scholar

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-030-30351-8.  Google Scholar

[15]

G. Dassios and R. Kleinman, Low Frequency Scattering, Oxford Mathematical Monographs, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar

[16]

A. Ghandriche and M. Sini, Mathematical Analysis of the Photo-acoustic imaging modality using resonating dielectric nanoparticles: The 2D TM-model, arXiv: 2003.03162 Google Scholar

[17]

T. Ilovitsh, A. Ilovitsh, J. Foiret, et al., Enhanced microbubble contrast agent oscillation following 250 kHz insonation, Sci Rep, 8 (2018), 16347. doi: 10.1038/s41598-018-34494-5.  Google Scholar

[18]

G. C. Papanicolaou, Diffusion in Random Media, Surveys in Applied Mathematics, volume 1, Edited by J P. Keller, D W. McLaughlin and G C. Papanicolaou, Plenum Press, NewYork, 1995. Google Scholar

[19]

S. Qin, C. F. Caskey and K. W. Ferrara, Ultrasound contrast microbubbles in imaging and therapy: Physical principles and engineering, Phys Med Biol., 54 (2009), R27. doi: 10.1088/0031-9155/54/6/R01.  Google Scholar

[20]

E. Quaia, Microbubble ultrasound contrast agents: An update, European Radiology, 17 (2007), 1995-2008.  doi: 10.1007/s00330-007-0623-0.  Google Scholar

[21]

T. Meklachi, S. Moskow and J. C. Schotland, Asymptotic analysis of resonances of small volume high contrast linear and nonlinear scatterers, J. Math. Phys., 59 (2018), 083502, 20pp. doi: 10.1063/1.5031032.  Google Scholar

[22]

A. Song, J. Li, C. Shen, X. Peng, X. Zhu, T. Chen1 and S. A. Cummer, Broadband high-index prism for asymmetric acoustic transmission, Appl. Phys. Lett., 114 (2019), 121902. Google Scholar

[23]

F. Zangeneh-Nejad and R. Fleury, Acoustic analogues of high-index optical waveguide devices, Sci Rep, 8 (2018), 10401. doi: 10.1038/s41598-018-28679-1.  Google Scholar

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