Magnetic nanoparticle density mapping from the magnetically induced displacement data: a simulation study

Modeling and Measurement of the Nonlinear Force on Nanoparticles in Magnetomotive Techniques

Magnetomotive (MM) ultrasound (US) imaging is the identification of tissue in which magnetic nanoparticle tracers are present by detecting a magnetically induced motion. Although the nanoparticles have a magnetization that depends nonlinearly on the external magnetic field, this has often been neglected, and the presence of resulting higher harmonics in the detected motion has not been reported yet. Here, the magnetization of nanoparticles in gelatin was modeled according to the Langevin theory of superparamagnetism. This nonlinear relationship has a fundamental effect on the resulting force and motion. However, the magnetic field must contain regions with a strong magnetic gradient and a low absolute magnetic field to allow the significant generation of higher harmonics in the force. To validate the model, an MM setup that has a constant magnetic gradient on one axis superimposed by a homogeneous time-varying magnetic field was used. After the magnetic characterization of the nanoparticles and calculations of the expected displacement in the setup, experiments were conducted. A laser Doppler vibrometer was used to quantify the small displacements at higher harmonics. The experimental results followed theoretical predictions. Deviations between model and experiment were attributed to a simplified mechanical modeling and temperature rise during measurements. It is concluded that in MM techniques, the nonlinear magnetization of nanoparticles must generally be considered to reconstruct quantitative parameters, to achieve optimum matching of fields and particles, or to exploit nanoparticle magnetization for tissue characterization. In addition, with the presented experimental setup, the magnetization properties of nanoparticles can be determined by MM techniques alone.

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.