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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:
[1]

G. S. Alberti, On multiple frequency power density measurements, Inverse Problems, 29 (2013), 115007, 25pp.

[2]

G. Alessandrini, Global stability for a coupled physics inverse problem, Inverse Problems, 30 (2014), 075008, 10pp.

[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.

[4]

H. AmmariE. BonnetierY. CapdeboscqM. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573. doi: 10.1137/070686408.

[5]

H. AmmariY. CapdeboscqF. deGournayA. Rozanova-Pierrat and F. Triki, Microwave imaging by elastic deformation, SIAM J. Appl. Math., 71 (2011), 2112-2130. doi: 10.1137/110828241.

[6]

H. Ammari, J. Garnier and W. Jing, Resolution and stability analysis in acousto-electric imaging, Inverse Problems, 28 (2012), 084005, 20pp.

[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.

[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.

[9]

G. Belizzi and O. M. Bucci, Microwave cancer imaging exploiting magnetic nanaparticles as contrast agent, IEEE Transactions on Biomedical Engineering, 58 (2011).

[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.

[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.

[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).

[13]

E. C. FearP. M. Meaney and M. A. Stuchly, Microwaves for breast cancer, IEEE Potentials, 22 (2003), 12-18.

[14]

N. Honda, J. McLaughlin and G. Nakamura, Conditional stability for a single interior measurement, Inverse Problems, 30 (2014), 055001, 19pp.

[15]

A.I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2), 128 (1988), 531-576. doi: 10.2307/1971435.

[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.

[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.

[18]

A. G. Ramm, Wave Scattering by Small Bodies of Arbitrary Shapes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.

[19]

J. D. SheaP. KosmasB. 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.

[20]

F. Triki, Uniqueness and stability for the inverse medium problem with internal data, Inverse Problems, 26 (2010), 074009, 22 pp.

[21]

Open the link http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/magprop.html, then click on 'Tables' then 'Magnetic properties'.

[22]

https://en.wikipedia.org/wiki/Relative_permittivity

show all references

References:
[1]

G. S. Alberti, On multiple frequency power density measurements, Inverse Problems, 29 (2013), 115007, 25pp.

[2]

G. Alessandrini, Global stability for a coupled physics inverse problem, Inverse Problems, 30 (2014), 075008, 10pp.

[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.

[4]

H. AmmariE. BonnetierY. CapdeboscqM. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573. doi: 10.1137/070686408.

[5]

H. AmmariY. CapdeboscqF. deGournayA. Rozanova-Pierrat and F. Triki, Microwave imaging by elastic deformation, SIAM J. Appl. Math., 71 (2011), 2112-2130. doi: 10.1137/110828241.

[6]

H. Ammari, J. Garnier and W. Jing, Resolution and stability analysis in acousto-electric imaging, Inverse Problems, 28 (2012), 084005, 20pp.

[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.

[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.

[9]

G. Belizzi and O. M. Bucci, Microwave cancer imaging exploiting magnetic nanaparticles as contrast agent, IEEE Transactions on Biomedical Engineering, 58 (2011).

[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.

[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.

[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).

[13]

E. C. FearP. M. Meaney and M. A. Stuchly, Microwaves for breast cancer, IEEE Potentials, 22 (2003), 12-18.

[14]

N. Honda, J. McLaughlin and G. Nakamura, Conditional stability for a single interior measurement, Inverse Problems, 30 (2014), 055001, 19pp.

[15]

A.I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2), 128 (1988), 531-576. doi: 10.2307/1971435.

[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.

[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.

[18]

A. G. Ramm, Wave Scattering by Small Bodies of Arbitrary Shapes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.

[19]

J. D. SheaP. KosmasB. 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.

[20]

F. Triki, Uniqueness and stability for the inverse medium problem with internal data, Inverse Problems, 26 (2010), 074009, 22 pp.

[21]

Open the link http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/magprop.html, then click on 'Tables' then 'Magnetic properties'.

[22]

https://en.wikipedia.org/wiki/Relative_permittivity

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