Mathematical Analysis of Imaging Modalities using Nanoparticles as Contrast Agents

Content. Conventional medical imaging techniques, as microwave imaging, are known to be potentially capable of extracting features in breast cancer, for instance, in case of high contrast of the permittivity between healthy tissues and malignant ones. However, in case of benign tissue, the variation of the permittivity is quite low so that such these imaging modalities are limited to be used for early detection of such diseases. In such cases, creating such missing contrast is highly desirable. One way to do it is to use electromagnetic nanoparticles as contrast agents. The object of this proposal is to analyze mathematically such imaging modalities and estimate with high precision the inner values of the electric permitivitty encoded in the remotely measured fields. Hypotheses. In this proposal, we are concerned with the cases where the used nanoparticles exhibit very high contrasts compared to the background. Several of such used nanaparticles are reported in the literature. We study two types of such imaging modalities: (1) Injecting electric nanoparticles in the targeted region will enhance locally the electric (and hence the magnetic) field. The idea is that, from the remotely measured electric field, one can recover this enhanced local field and then the permitivitty at the location of the nanaparticles. (2) Exciting magnetic (as Gold) nanoparticles, injected in the targeted region, at certain frequencies creats heat in their surroundings which in turn creats a propagating pressure wave. The idea is to measure remotely this pressure wave and then recover the permitivitty at the location of the nanaparticles. Methods. In mathematical terms, we need to analyze the (acoustic and electromagnetic) fields generated by very highly contrasted transmission conditions. Our approach is based on the derivation of the point-interaction approximation of these fields in the presence of finite but very close and smal particles with high contrats media. The analysis is base on the method of integral equations and asymptotic expansions. Originality of the project. The advantage of our approach, compared to the known techniques in literature, is that we can characterize clearly the dominant field generated by the interaction of the small particles between each other and also with the background medium. The originality here is that we can estimate the fields due to multiple interactions (at least the second interactions) of the nanoparticles. The derived formulas encode the values of the Green`s functions, at the centers of the close nanoparticles, in a precise and useful way. The values of the unknown electric permittivity can be recovered from the singularities of these Green`s functions. The price to pay to derive such formulas is to use frequencies, of incidence, close to resonances.

Motivation. The use of contrast agents (as gas bubbles, fluid droplets or electromagnetic nanoparticles) was proposed by the engineering community in the last two decades and recently extensively developed as a means of improving the quality of traditional imaging modalities, drugs delivery ways and therapy modalities. Our goal in this proposal was originally to mathematically model, analyze and quantify to what extent such techniques can indeed go beyond what the traditional imaging techniques can offer. In short, the traditional methods are known to be potentially capable of extracting features in case of high contrasts between the damaged tissues and the healthy ones. However, in case of low contrasts, as for anomalies at the early stage, such detections are not possible. To remedy to this missing contrasts, it was advised to use, whenever possible, injected contrast agents. Our approach. The imaging modalities we are considering are described in the framework of wave propagations. Typical imaging modalities we have studied are those related to Ultrasound imaging, Optical imaging or hybrid ones as Photo-Acoustic imaging. The related contrast agents are small scaled objects that enjoy high contrasting properties as compared to the normal or healthy tissues. It happens that under critical ratio between their sizes and proper contrasts, these contrast agents vibrate at specific frequencies (called local scattering resonances). The good news is that we can characterize and quantify those resonant frequencies. Therefore, taking the difference between the waves generated before and after injecting the contrast agents, we 'see' local spots around the injected contrast. These local spots encode all the information that one can be able to see. The outcome. We analyzed with great details three imaging modalities, namely the ultrasound imaging using bubbles and the optic as well as the Photo-Acoustic imaging using nanoparticles. The key features in our analysis are the following: 1. In the time-harmonic regimes, we could reconstruct the dispersion function and hence the created resonances by the contrast agents. From these dispersive functions, we could extract the acoustic and the optic properties of the object to image. 2. In the time-domain regimes, we could reconstruct the internal values of the travel time function. This function models the time needed for a wave to propagate between any two location points. With such travel time function we derive the related wave's speed and then reconstruct the acoustic properties (and the optical properties in case of optic or Photo-Acoustic imaging) of the object to image. Our findings give a solid mathematical background to these imaging modalities. We provide with quatitative results that go far beyond the known results that were obtained based on the traditional imaging modalities. Our approach is flexible enough to be applied to more challenging situations.