An introduction to the mathematics of the imaging modalities using small-scaled contrast agents

In the recent years, we witness a great interest in imaging, in a wide sense, using contrast agents. One of the reasons is that many imaging modalities, as the ones related to medical sciences, suffer from several shortcomings. The most serious one is the issue of instability. Indeed, it is, nowadays, a common certainty that classical inverse problems of recovering objects from remote measurements are, mostly, highly unstable. To recover the stability, it is advised to create, whenever possible, the missing contrasts in the targets to image. In this survey paper, we follow this direction and propose an approach how to analyze mathematically the effect of the injected agents on the different fields under consideration. These contrast agents are small-sized particles modeled with materials that enjoy high contrasts as compared to the ones of the background. These two properties allow them, under critical scales of size/contrast, to create local spots when excited from far. These local spots can be remotely recovered in stable ways. The accessible information on the target are encoded in theses spots. After stating a class of such imaging modalities that enter into this framework, as the acoustic imaging, photo-acoustic imaging, optical imaging and more, we provide detailed analysis for first two modalities where the contrast agents are micro-bubbles and nano-particles respectively. In these cases, we provide a clear and useful correspondence between the critical size/contrast scales and the main resonances, and hence the local spots, they are able to create while excited with appropriate incident frequencies. To estimate the remote dominant field generated by the background in the presence of such particles, we derive the point-interaction approximation of these fields. This dominant field that we call the Foldy–Lax field, as it is reminiscent to the Foldy–Lax field generated by Dirac-like potentials with prescribed multiplicative (scattering) coefficients, encodes the fields after the multiple scattering between the background and the different particles. This Foldy–Lax field contains the accessible information on the target to image. Using resonating incident frequencies enhances this field and makes it readable from remote measurements.

Keywords

mathematical imaging, contrast agents, micro-bubbles, nano-particles, Neumann–Poincaré operator, Newtonian operator, Minnaert resonance, plasmonic resonances, dielectric resonances

2010 Mathematics Subject Classification

35C20, 35R30

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Ahcene Ghandriche is supported by the Austrian Science Fund (FWF): P 30756-NBL.

Mourard Sini is partially supported by the Austrian Science Fund (FWF): P 30756-NBL.

Published 16 August 2022