Mathematical Analysis of Imaging Modalities Using Bubbles or Nano-particles as Contrast Agents
Ahcene Ghandriche
Inverse Problems and Mathematical Imaging
It is a common certainty that the traditional inverse problems of recovering objects from remote measurements
are, mostly, highly unstable. To overcome this instability it is advised, in the engineering literature,
to create the missing contrasts in the targets to image, Ω, by injecting micro-bubbles or nano-particles.
In this thesis, we follow this direction and propose an approach how to analyse mathematically the effect
of the injected agents, which are small-sized particles modelled with materials that enjoy high contrast as
compared to the ones of the background, Ω, or desired sign for the electromagnetic properties. These
enhanced fields can be measured on the accessible boundary ∂Ω. The goal is then to extract the values
of needed coefficients from the measured enhanced fields. We state and provide detailed analysis for
two classes of such imaging modalities.
1. Acoustic imaging modality. Here, the contrast agents are Micro-Bubbles modelled by the mass
density and bulk modulus enjoying high contrasting scales. These contrasting scales allow them to
resonate at certain incident frequencies as the Minnaert resonance. The goal is then to reconstruct
the acoustic coefficients given by the mass density and bulk modulus, in the target Ω, from the
remotely measured ultrasound that is enhanced by the presence of the bubbles resonance and
excited by incident frequencies close to Minnaert resonance.
2. Photo-acoustic imaging modality. This technique consists of exciting the heterogeneous (i.e.
damaged) tissue, Ω, with an electromagnetic wave at a given incident frequency, which in turn
creates a pressure wave that we can measure at the accessible part ∂Ω. The goal is then to
extract information about the optical properties (i.e. the permittivity and conductivity) of this tissue,
Ω, from these measurements. As a first step, we consider the case of 2D TM-model where we use
dielectric nano-particles (enjoying high contrasts of their electric permittivity). As a second step,
we deal with the full Maxwell system where we use plasmonic nano-particles (having permittivity
of negative sign).
We show that the curves, or the surfaces, given by the measured fields in terms of the used bands
of frequencies have peaks only at the related resonant frequencies (i.e. Minnaert for acoustic fields,
Dielectric or plasmonic for electromagnetic fields). In particular, from the denominators of these fields,
we recover these resonant frequencies and, from their numerators we derive, the fields generated before
injecting the agents. This recovered information allows us to reconstruct the needed coefficients.