Mathematical imaging using electric or magnetic nanoparticles as contrast agents

Durga Prasad Challa; Anupam Pal Choudhury; Mourad Sini

doi:10.3934/ipi.2018025

Article Contents

2018, Volume 12, Issue 3: 573-605. Doi: 10.3934/ipi.2018025

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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
* Corresponding author

Received: March 31, 2017

Revised: December 31, 2017

Published: March 2018

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.

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Abstract

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.

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Mathematics Subject Classification: 35R30, 35C20.

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