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.