# Mathematical imaging using electric or magnetic nanoparticles as contrast agents

## TL;DR

This paper provides a mathematical analysis of an electromagnetic imaging method that uses high-contrast electric or magnetic nanoparticles as contrast agents to reconstruct electrical permittivity.

## Contribution

It introduces a new model for nanoparticles with contrast properties and derives asymptotic expansions of electromagnetic fields for improved imaging analysis.

## Key findings

- Derived asymptotic expansions for electromagnetic fields with high-contrast nanoparticles.
- Proposed a method to extract permittivity values from measured scattered fields.
- Analyzed the scalar electromagnetic model for nanoparticle-based imaging.

## 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 $\delta$ (relative to the size of the imaging domain), having relative electric permittivity (or relative magnetic permeability) of order $\delta^{-\alpha}$ with a certain $\alpha>0$, as $0<\delta<<1$. 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.