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June  2018, 12(3): 573-605. doi: 10.3934/ipi.2018025

## Mathematical imaging using electric or magnetic nanoparticles as contrast agents

 1 Faculty of Mathematics, Indian Institute of Technology Tirupati, Tirupati, India 2 Technische Universität Darmstadt, Institute of Mathematics, Schlossgartenstr. 7, 64289 Darmstadt, Germany 3 RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria

* Corresponding author

Received  April 2017 Revised  January 2018 Published  March 2018

Fund Project: The first author was partially supported by the Austrian Science Fund (FWF): P28971-N32 and DST SERB MATRICS (Mathematical Research Impact Centric Support) MTR/2017/000539. The second author was supported by the DFG International Research Training Group IRTG 1529 on Mathematical Fluid Dynamics at TU Darmstadt. The third author was partially supported by the Austrian Science Fund (FWF): P28971-N32

We analyse mathematically the imaging modality using electromagnetic nanoparticles as contrast agent. This method uses the electromagnetic fields, collected before and after injecting electromagnetic nanoparticles, to reconstruct the electrical permittivity. The particularity here is that these nanoparticles have high contrast electric or magnetic properties compared to the background media. First, we introduce the concept of electric (or magnetic) nanoparticles to describe the particles, of relative diameter $δ$(relative to the size of the imaging domain), having relative electric permittivity (or relative magnetic permeability) of order $δ^{-α}$ with a certain $α>0$, as $0<δ<<1$. Examples of such material, used in the imaging community, are discussed. Second, we derive the asymptotic expansion of the electromagnetic fields due to such singular contrasts. We consider here the scalar electromagnetic model. Using these expansions, we extract the values of the total fields inside the domain of imaging from the scattered fields measured before and after injecting the nanoparticles. From these total fields, we derive the values of the electric permittivity at the expense of numerical differentiations.

Citation: Durga Prasad Challa, Anupam Pal Choudhury, Mourad Sini. Mathematical imaging using electric or magnetic nanoparticles as contrast agents. Inverse Problems & Imaging, 2018, 12 (3) : 573-605. doi: 10.3934/ipi.2018025
##### References:
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show all references

##### References:
 [1] G. S. Alberti, On multiple frequency power density measurements, Inverse Problems, 29 (2013), 115007, 25pp.  Google Scholar [2] G. Alessandrini, Global stability for a coupled physics inverse problem, Inverse Problems, 30 (2014), 075008, 10pp.  Google Scholar [3] A. Alsaedi, F. Alzahrani, D. P. Challa, M. Kirane and M. Sini, Extraction of the index of refraction by embedding multiple and close small inclusions, Inverse Problems, 32 (2016), 045004, 18pp.  Google Scholar [4] H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573.  doi: 10.1137/070686408.  Google Scholar [5] H. Ammari, Y. Capdeboscq, F. deGournay, A. Rozanova-Pierrat and F. Triki, Microwave imaging by elastic deformation, SIAM J. Appl. Math., 71 (2011), 2112-2130.  doi: 10.1137/110828241.  Google Scholar [6] H. Ammari, J. Garnier and W. Jing, Resolution and stability analysis in acousto-electric imaging, Inverse Problems, 28 (2012), 084005, 20pp.  Google Scholar [7] H. Ammari and H. Kang, Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities, J. Math. Anal. Appl., 296 (2004), 190-208.  doi: 10.1016/j.jmaa.2004.04.003.  Google Scholar [8] H. Ammari and H. Kang, Polarization and Moment Tensors, With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Springer, New York, 2007.  Google Scholar [9] G. Belizzi and O. M. Bucci, Microwave cancer imaging exploiting magnetic nanaparticles as contrast agent, IEEE Transactions on Biomedical Engineering, 58 (2011). Google Scholar [10] D. P. Challa and M. Sini, On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul., 12 (2014), 55-108.  doi: 10.1137/130919313.  Google Scholar [11] 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.  Google Scholar [12] Y. Chen, I. J. Craddock and P. Kosmas, Feasibility study of lesion classification via contrast-agent-aided UWB breast imaging, IEEE Transactions on Biomedical Engineering, 57 (2010). Google Scholar [13] E. C. Fear, P. M. Meaney and M. A. Stuchly, Microwaves for breast cancer, IEEE Potentials, 22 (2003), 12-18.   Google Scholar [14] N. Honda, J. McLaughlin and G. Nakamura, Conditional stability for a single interior measurement, Inverse Problems, 30 (2014), 055001, 19pp.  Google Scholar [15] A.I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2), 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar [16] R.G. Novikov, A multidimensional inverse spectral problem for the equation $-Δψ +(v(x)-Eu(x))ψ = 0$, Funktsional. Anal. i Prilozhen, 22 (1988), 11-22, 96.   Google Scholar [17] A.G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886.  doi: 10.1088/0266-5611/4/3/020.  Google Scholar [18] A. G. Ramm, Wave Scattering by Small Bodies of Arbitrary Shapes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.  Google Scholar [19] J. D. Shea, P. Kosmas, B. D. Van Veen and S. C. Hagness, Contrast-enhanced microwave imaging of breast tumors: A computational study using 3D realistic numerical phantoms, Inverse Problems, 26 (2010), 1-22.   Google Scholar [20] F. Triki, Uniqueness and stability for the inverse medium problem with internal data, Inverse Problems, 26 (2010), 074009, 22 pp.  Google Scholar [21] Open the link http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/magprop.html, then click on 'Tables' then 'Magnetic properties'. Google Scholar [22]
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