Inversion of displacement fields to quantify the magnetic particle distribution in homogeneous elastic media from magnetomotive ultrasound.

Magnetomotive ultrasound (MMUS) contrasts superparamagnetic iron-oxide nanoparticles (SPIOs) that undergo submicrometer-scale displacements in response to a magnetic gradient force applied to an imaging sample. Typically, MMUS signals are defined in a way that is proportional to the medium displacement, rendering an indirect measure of the density distribution of SPIOs embedded within. Displacement-based MMUS, however, suffers from 'halo effects' that extend into regions without SPIOs due to their inherent mechanical coupling with the medium. To reduce such effects and to provide a more accurate representation of the SPIO density distribution, we propose a model-based inversion of MMUS displacement fields by reconstructing the body force distribution. Displacement fields are modelled using the static Navier-Cauchy equation for linear, homogeneous, and isotropic media, and the body force fields are, in turn, reconstructed by minimizing a regularized least-squares error functional between the modelled and the measured displacement fields. This reconstruction, when performed on displacement fields of two tissue-mimicking phantoms with cuboidal SPIO-laden inclusions, improved the range of errors in measured heights and widths of the inclusions from 54%-282% pre-inversion to-15%-20%. Likewise, the post-inversion contrast to noise ratios (CNRs) of the images were significantly larger than displacement-derived CNRs alone (p   =  0.0078, Wilcoxon signed rank test). Qualitatively, it was found that inversion ameliorates halo effects and increases overall detectability of the inclusion. These findings highlight the utility of model-based inversion as a tool for both signal processing and accurate characterization of the number density distribution of SPIOs in magnetomotive imaging.