Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies

Abstract

We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves).We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixed single backscattering direction. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region.

On the Origin of Minnaert Resonances

“…An integration by parts allows to transform this integro-differential equation into a solely integral equation (see detailed computations in [12,Sec. 3])…”

“…This implies an enhancement of the scattered and far-fields, and motivates the definition of resonant frequency, which became widely used in a somehow generic sense. The limit behaviour of the scale-dependent scattering problem (1.6) has been described in [7], [12], where the asymptotic analysis is developed using layer potential techniques. The expansions provided in these works are carried out under specific constraints allowing to consider the scattering problem when the frequency is only close to the resonant value, but does not coincide with.…”

The inverse source problem for the wave equation revisited: A new approach

“…The idea can also be used to acoustic imaging modalities using bubbles as contrast agents. The mathematical analysis of such imaging modalities are given in [5,7,9,10]. The main idea there is that contrasting the fields measured before and after injecting the contrasting particles, one can reconstruct the total fields on the location of those particles.…”

“…The main idea there is that contrasting the fields measured before and after injecting the contrasting particles, one can reconstruct the total fields on the location of those particles. In [7,9,10], this is justified in the time harmonic regimes (both for the electromagnetic and acoustic models) when the used incident waves are sent at frequencies close to the resonating frequencies of the used particles. These resonances exist for certain scales of the contrasts of the particles.…”

“…In this time-domain regime, the same scales of the contrasts allow us to extract the Fourier modes, in time, of the internal field using the eigenvalues of the Newtonian operator; see (1.8)-(1.9). Observe that for these scales the resonances, called the dielectric resonances, used in [7,9] are related, and actually due, to the eigenvalues of the Newtonian operator as well.…”