Mathematical analysis of the photo-acoustic imaging modality using resonating dielectric nano-particles: The 2D TM-model

Abstract

We deal with the photoacoustic imaging modality using dielectric nano-particles as contrast agents. Exciting the heterogeneous tissue, localized in a bounded domain Ω, with an electromagnetic wave, at a given incident frequency, creates heat in its surrounding which in turn generates an acoustic pressure wave (or fluctuations). The acoustic pressure can be measured in the accessible region ∂Ω surrounding the tissue of interest. The goal is then to extract information about the optical properties (i.e. the permittivity and conductivity) of this tissue from these measurements. We describe two scenarios. In the first one, we inject single nano-particles while in the second one we inject couples of closely spaced nano-particles (i.e. dimers). From the acoustic pressure measured, before and after injecting the nano-particles (for each scenario), at two single points x1 and x2 of ∂Ω and two single times t1≠t2 such that t1,t2>diam(Ω),(1)we localize the center point z of the single nano-particle and reconstruct the phaseless total field |u0| on that point z (where u0 is the total field in the absence of the nano-particles). Hence, we transform the photoacoustic problem into the inversion of phaseless internal electric fields.(2)we localize the centers z1 and z2 of the injected dimers and reconstruct both the permittivity and the conductivity of the tissue on those points. This is done using dielectric nano-particles enjoying high contrasts of their electric permittivity. These results are possible using frequencies of incidence close to the resonances of the used dielectric nano-particles. These particular frequencies are computable. The error of approximations are given in terms of the scales and the contrasts of the dielectric nano-particles. The results are justified in the 2D TM-model.