A challenging issue in photoacoustic biomedical imaging is to take into account the presence of dispersive acoustic media, since these are prone to induce amplitude attenuation and scattering of the photoacoustic frequency components. These perturbations are largely the cause for which the photoacoustic tomographic image reconstruction from projections lacks a plane-wave transport formalism. Attending this problem, we further develop an analytic formalism of the transport and its numerical implementation accounting for dispersive acoustic media. We differentiate three variations of an acoustically perturbing media. Our object of interest is a numerical description of the light absorption map of a coronal human breast image. Then, we analyze conditions for which the propagation of photoacoustic perturbations can obey the generalized Heaviside telegraph equation. In addition, we provide a study of the causality consistency of the wave propagation models. We observe transport implications due to the presence of dispersive acoustic media and derive model adjustments that include attenuation and diffusion approximations within the two-dimensional forward problem. Next, we restore the inverse problem description with the deduced perturbation components. Finally, we solve the nonlinear inverse problem with a numerical strategy for a filtered backprojection reconstruction. At a stage prior to the image reconstruction, we compensate for the effect of acoustic attenuation and diffusion to calculate the inversions of the wave perturbations located within the projections. In this way, we manage to significantly reduce reconstruction artifacts. In consequence, we prevent the use of some additional image processing of noise reduction. We demonstrate a feasible strategy on how to solve the stated nonlinear inverse problem of photoacoustic tomography accounting for dispersive acoustic media. In particular, we emphasize efforts to achieve an analytical description, and thus an algorithm is placed, for imaged sound perturbations to be cleaned from acoustic scattering in a simplified manner.
We have applied a homogenization theory<sup>1 </sup>, which is based on the Fourier formalism, to calculate the effective parameters of phononic crystals having liquid inclusions embedded in a solid host matrix. The theory provides explicit formulas for determining all the components of the effective mass density and stiffness tensors, which are valid in the long wavelength limit for arbitrary Bravais lattice and any form of the inclusions inside the unit cell. In the previous work<sup>1</sup>, it was shown that rectangular two-dimensional lattices of water-filled holes in an elastic host matrix exhibit solid-like behavior with strongly anisotropic mass density in the low-frequency limit. Such metamaterials were called metasolids. In the present work, we analyze the metasolid behavior of liquid-solid three-dimensional phononic crystals. In particular we have analyzed the effect of the type of Bravais lattice and form of the liquid inclusions on the anisotropy of the effective mass density. In the analysis we have considered different solid host materials (Al, Si, and ribbon) with isolated inclusions of water. We have established that the anisotropy of the effective mass density is considerably strong when the homogenized phononic crystals do not possess inversion symmetry because of the inclusion shape. Our results could be useful for designing metamaterials with predetermined elastic properties.