Recovering both the wave speed and the source function in a time-domain wave equation by injecting contrasting droplets

Soumen Senapati; Mourad Sini; Haibing Wang

doi:10.3934/dcds.2023151

Article Contents

2024, Volume 44, Issue 5: 1446-1474. Doi: 10.3934/dcds.2023151

This issue Previous Article Concentration compactness for the energy critical Zakharov system Next Article Failure of Khintchine-type results along the polynomial image of IP₀ sets

Recovering both the wave speed and the source function in a time-domain wave equation by injecting contrasting droplets

1.
RICAM, Austrian Academy of Sciences, A-4040, Linz, Austria
2.
School of Mathematics, Southeast University, Nanjing 210096, China
3.
Nanjing Center for Applied Mathematics, Nanjing 211135, China

^*Corresponding author: Soumen Senapati
^*Corresponding author: Soumen Senapati

Received: April 2023

Revised: October 2023

Early access: December 21, 2023

Published online: December 21, 2023

This work is supported by Natural Science Foundation of China (Nos. 12071072, 11971104) and the Austrian Science Fund (FWF): P 30756-NBL.

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Abstract

Dealing with the inverse source problem for the scalar wave equation, we have shown recently that we can reconstruct the space-time dependent source function from the measurement of the wave, collected at a single point $ x $ for a large enough interval of time, generated by a small scaled droplets, enjoying large contrasts of its bulk modulus, injected inside the domain to image. Here, we extend this result to reconstruct not only the source function but also the variable wave speed. Indeed, from the measured waves, we first localize the internal values of the travel-time function by looking at the behavior of this collected wave in terms of time. Then from the Eikonal equation, we recover the wave speed. Second, we recover the internal values of the wave generated only by the background (in the absence of the small droplets) from the same measured data by inverting a Volterra integral operator of the second kind. From this reconstructed wave, we recover the source function at the expense of a numerical differentiation.

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Mathematics Subject Classification: 35R30, 45Q05, 65N21.

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References

[1]	M. Agranovsky, P. Kuchment and L. Kunyansky, On reconstruction formulas and algorithms for the thermoacoustic tomography, in Photoacoustic Imaging and Spectroscopy, CRC Press, (2009), 89-101.
[2]	C. R. Anderson, X. Hu, H. Zhang, J. Tlaxca, A. E. Decl'eves, R. Houghtaling, K. Sharma, M. Lawrence, K. W. Ferrara and J. J. Rychak, Ultrasound molecular imaging of tumor angiogenesis with an integrin targeted microdroplet contrast agent, Investigative Radiology, 46 (2011), 215-224. doi: 10.1097/RLI.0b013e3182034fed.
[3]	M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Appl. Anal., 83 (2004), 983-1014. doi: 10.1080/0003681042000221678.
[4]	M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation, J. Math. Pures Appl., 85 (2006), 193-224. doi: 10.1016/j.matpur.2005.02.004.
[5]	M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics. Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.
[6]	A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR., 260 (1981), 269-272.
[7]	R. E. Caflisch, M. J. Miksis, G. C. Papanicolaou and L. Ting, Effective equations for wave propagation in a bubbly liquid, J. Fluid Mec., 153 (1985), 259-273. doi: 10.1017/S0022112085001252.
[8]	R. E. Caflisch, M. J. Miksis, G. C. Papanicolaou and L. Ting, Wave propagation in bubbly liquids at finite volume fraction, J. Fluid Mec., 160 (1985), 1-14. doi: 10.1017/S0022112085003354.
[9]	D. P. Challa, A. P. Choudhury and M. Sini, Mathematical imaging using electric or magnetic droplets as contrast agents, Inverse Probl. Imaging, 12 (2018), 573-605. doi: 10.3934/ipi.2018025.
[10]	D. P. Challa, A. Mantile and M. Sini, Characterization of the acoustic fields scattered by a cluster of small holes, Asymptot. Anal., 118 (2020), 235-268. doi: 10.3233/ASY-191560.
[11]	A. Dabrowski, A. Ghandriche and M. Sini, Mathematical analysis of the acoustic imaging modality using droplets as contrast agents at nearly resonating frequencies, Inverse Probl. Imaging, 15 (2021), 555-597. doi: 10.3934/ipi.2021005.
[12]	E. C. Fear, P. M. Meaney and M. A. Stuchly, Microwaves for breast cancer?, IEEE Potentials, 22 (2003), 12-18. doi: 10.1109/MP.2003.1180933.
[13]	D. Finch and Rakesh, Recovering a function from its spherical mean values in two and three dimensions, in Photoacoustic Imaging and Spectroscopy, CRC Press, (2009), 12 pp.
[14]	A. Ghandriche and M. Sini, Photo-acoustic inversion using plasmonic contrast agents: The full Maxwell model, J. Differential Equations, 341 (2022), 1-78. doi: 10.1016/j.jde.2022.09.008.
[15]	A. Ghandriche and M. Sini, Simultaneous reconstruction of optical and acoustical properties in photo-acoustic imaging using plasmonics, SIAM J. Appl. Math., 83 (2023), 1738-1765. doi: 10.1137/22M1534730.
[16]	J. Hou, Z. C. Zheng and J. S. Allen, Time-domain immersed-boundary simulation of acoustic propagation between two spherical gas bubbles, JASA Express Lett., 3 (2023), 094002. doi: 10.1121/10.0020811.
[17]	T. Ilovitsh, A. Ilovitsh, J. Foiret, et al., Enhanced microdroplet contrast agent oscillation following 250 kHz insonation, Sci Rep, 8 (2018), 16347. doi: 10.1038/s41598-018-34494-5.
[18]	O. Y. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations, Comm. Partial Differential Equations, 26 (2001), 1409-1425. doi: 10.1081/PDE-100106139.
[19]	O. Y. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems, 17 (2001), 717-728. doi: 10.1088/0266-5611/17/4/310.
[20]	T. Sh. Kalmenov and D. Suragan, A boundary condition and spectral problems for the Newtonian potential, Oper. Theory Adv. Appl., 216 (2011), 187-210. doi: 10.1007/978-3-0348-0069-3_11.
[21]	Y. Kian and G. Uhlmann, Determination of the sound speed and an initial source in photoacoustic tomography, preprint, 2023, arXiv: 2302.03457.
[22]	C. Knox and A. Moradifam, Determining both the source of a wave and its speed in a medium from boundary measurements Inverse Problems, 36 (2020), 025002, 15 pp. doi: 10.1088/1361-6420/ab53fc.
[23]	R. A. Kruger, W. L. Kiser, D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography using a conventional linear transducer array, Med. Phys., 30 (2003), 856-60. doi: 10.1118/1.1565340.
[24]	H. Liu and G. Uhlmann Determining both sound speed and internal source in thermo- and photo-acoustic tomography Inverse Problems, 31 (2015), 105005, 10 pp. doi: 10.1088/0266-5611/31/10/105005.
[25]	S. Qin, C. F. Caskey and K. W. Ferrara, Ultrasound contrast micro-droplets in imaging and therapy: Physical principles and engineering, Phys Med Biol., 54 (2009), R27. doi: 10.1088/0031-9155/54/6/R01.
[26]	E. Quaia, Microdroplet ultrasound contrast agents: An update, European Radiology, 17 (2007), 1995-2008. doi: 10.1007/s00330-007-0623-0.
[27]	V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press BV, Utrecht, 1987.
[28]	F.-J. Sayas, Retarded Potentials and Time Domain Boundary Integral Equations. A Road Map, Springer Series in Computational Mathematics, 50. Springer, [Cham], 2016. doi: 10.1007/978-3-319-26645-9.
[29]	P. S. Sheeran and P. A. Dayton, Phase-change contrast agents for imaging and therapy, Curr Pharm Des., 18 (2012), 2152-2165. doi: 10.2174/138161212800099883.
[30]	M. Sini and H. Wang, The inverse source problem for the wave equation revisited: A new approach, SIAM J. Math. Anal., 54 (2022), 5160-5181. doi: 10.1137/21M1463689.
[31]	P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16 pp. doi: 10.1088/0266-5611/25/7/075011.
[32]	P. Stefanov and G. Uhlmann, Instability of the linearized problem in multiwave tomography of recovery both the source and the speed, Inverse Probl. Imaging, 7 (2013), 1367-1377. doi: 10.3934/ipi.2013.7.1367.
[33]	P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications, Trans. Am. Math. Soc., 365 (2013), 5737-5758. doi: 10.1090/S0002-9947-2013-05703-0.
[34]	M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 65-98. doi: 10.1016/S0021-7824(99)80010-5.
[35]	P. Zhang, L. Li, L. Lin, J. Shi and L. V. Wang, In vivo superresolution photoacoustic computed tomography by localization of single dyed droplets, Light: Science and Applications, 8 (2019), Article number: 36. doi: 10.1038/s41377-019-0147-9.