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Magnetic nanoparticles (MNP) are widely investigated for biomedical applications in diagnostics (e.g. imaging), therapeutics (e.g. hyperthermia) and general biosensing. For all these applications, the MNPs’ unique magnetic relaxation mechanism in an alternating magnetic field (AFM) is stimulated to induce desired effects. Whereas magnetic fluid hyperthermia (MFH) and magnetic particle imaging (MPI) are the most prominent examples for biomedical application, we investigate the relatively new biosensing application of frequency mixing magnetic detection (FMMD) from a fundamental perspective. Generally, we ask how specific MNP parameters (core size, magnetic anisotropy) influence the signal, specifically we predict the most effective MNP core size for signal generation. In FMMD, simultaneously two AFM are applied: a low-frequency magnetic driving field, driving MNP close to saturation, and a high-frequency excitation field that probes MNP susceptibility: . Resulting from the nonlinear magnetization of the MNP, harmonics of both individual incident frequencies as well as intermodulation products of these frequencies are generated. In this work, we present numerical Monte-Carlo(MC)-based simulations of the MNP relaxation process, solving the Landau-Lifshitz-Gilbert (LLG) equation to predict FMMD signals: As Figure 1 shows for the first four intermodulation signals , with , we can clearly see that larger core sizes generally increase the signal intensity. Same trend is predicted by a simple Langevin-function based thermal equilibrium model. Both predictions include a lognormal size distribution. The effect of core size distribution presumably dominates the effect of magnetic anisotropy. The findings are supported by comparison with experimental data and help to identify which MNP are best suited for magnetic biosensing applications using FMMD.

Dual frequency magnetic excitation of magnetic nanoparticles (MNP) enables enhanced biosensing applications. This was studied from an experimental and theoretical perspective: nonlinear sum-frequency components of MNP exposed to dual-frequency magnetic excitation were measured as a function of static magnetic offset field. The Langevin model in thermodynamic equilibrium was fitted to the experimental data to derive parameters of the lognormal core size distribution. These parameters were subsequently used as inputs for micromagnetic Monte-Carlo (MC)-simulations. From the hysteresis loops obtained from MC-simulations, sum-frequency components were numerically demodulated and compared with both experiment and Langevin model predictions. From the latter, we derived that approximately 90% of the frequency mixing magnetic response signal is generated by the largest 10% of MNP. We therefore suggest that small particles do not contribute to the frequency mixing signal, which is supported by MC-simulation results. Both theoretical approaches describe the experimental signal shapes well, but with notable differences between experiment and micromagnetic simulations. These deviations could result from Brownian relaxations which are, albeit experimentally inhibited, included in MC-simulation, or (yet unconsidered) cluster-effects of MNP, or inaccurately derived input for MC-simulations, because the largest particles dominate the experimental signal but concurrently do not fulfill the precondition of thermodynamic equilibrium required by Langevin theory.

Biomedical applications of magnetic nanoparticles (MNP) fundamentally rely on the particles’ magnetic relaxation as a response to an alternating magnetic field. The magnetic relaxation complexly depends on the interplay of MNP magnetic and physical properties with the applied field parameters. It is commonly accepted that particle core size is a major contributor to signal generation in all the above applications, however, most MNP samples comprise broad distribution spanning nm and more. Therefore, precise knowledge of the exact contribution of individual core sizes to signal generation is desired for optimal MNP design generally for each application. Specifically, we present a magnetic relaxation simulation-driven analysis of experimental frequency mixing magnetic detection (FMMD) for biosensing to quantify the contributions of individual core size fractions towards signal generation. Applying our method to two different experimental MNP systems, we found the most dominant contributions from approx. 20 nm sized particles in the two independent MNP systems. Additional comparison between freely suspended and immobilized MNP also reveals insight in the MNP microstructure, allowing to use FMMD for MNP characterization, as well as to further fine-tune its applicability in biosensing.

Frequency mixing magnetic detection (FMMD) is a sensitive and selective technique to detect magnetic nanoparticles (MNPs) serving as probes for binding biological targets. Its principle relies on the nonlinear magnetic relaxation dynamics of a particle ensemble interacting with a dual frequency external magnetic field. In order to increase its sensitivity, lower its limit of detection and overall improve its applicability in biosensing, matching combinations of external field parameters and internal particle properties are being sought to advance FMMD. In this study, we systematically probe the aforementioned interaction with coupled Néel–Brownian dynamic relaxation simulations to examine how key MNP properties as well as applied field parameters affect the frequency mixing signal generation. It is found that the core size of MNPs dominates their nonlinear magnetic response, with the strongest contributions from the largest particles. The drive field amplitude dominates the shape of the field-dependent response, whereas effective anisotropy and hydrodynamic size of the particles only weakly influence the signal generation in FMMD. For tailoring the MNP properties and parameters of the setup towards optimal FMMD signal generation, our findings suggest choosing large particles of core sizes dc > 25 nm nm with narrow size distributions (σ < 0.1) to minimize the required drive field amplitude. This allows potential improvements of FMMD as a stand-alone application, as well as advances in magnetic particle imaging, hyperthermia and magnetic immunoassays.