Fundamentals and Applications of Dual-Frequency Magnetic Particle Spectroscopy: Review for Biomedicine and Materials Characterization

Abstract

Superparamagnetic nanoparticles (MNP) offer exciting applications for engineering and biomedicine in imaging, diagnostics, and therapy upon magnetic excitation. Specifically, if excited at two distinct frequencies f₁ and f₂, MNP responds with magnetic intermodulation frequencies m·f₁ ± n·f₂ caused by their nonlinear magnetization. These mixing frequencies are highly specific for MNP properties, uniquely characterizing their presence. In this review, the fundamentals of frequency mixing magnetic detection (FMMD) as a special case of magnetic particle spectroscopy (MPS) are reviewed, elaborating its functional principle that enables a large dynamic range of detection of MNP. Mathematical descriptions derived from Langevin modeling and micromagnetic Monte-Carlo simulations show matching predictions. The latest applications of FMMD in nanomaterials characterization as well as diagnostic and therapeutic biomedicine are highlighted: analysis of the phase of the FMMD signal characterizes the magnetic relaxation of MNP, allowing to determine hydrodynamic size and binding state. Variation of excitation amplitudes or magnetic offset fields enables determining the size distribution of the particles' magnetic cores. This permits multiplex detection of polydisperse MNP in magnetic immunoassays, realized successfully for various biomolecular targets such as viruses, bacteria, proteins, and toxins. A portable magnetic reader enables portable immunodetection at point-of-care. Future applications toward theranostics are summarized and elaborated.

Keywords: magnetic biosensing; magnetic fluid hyperthermia (MFH); magnetic frequency mixing detection (FMMD); magnetic immunoassays; magnetic particle imaging (MPI); magnetic particle spectroscopy (MPS); micromagnetic simulation.

Figure 1

a) Illustration of a magnetic nanoparticle, consisting of a single magnetic core (e.g., magnetite) and an organic shell to yield the particle hydrophilic. In an aqueous solution, water molecules aggregate at the particle surface, additionally contributing to the overall hydrodynamic diameter. b) Multi‐core particle for which the organic shell contains several magnetic cores of different sizes.

Figure 2

Field‐dependent magnetic properties of spherical magnetite/Fe₃O₄ nanoparticles as a function of the magnetic field, for different particle sizes, at 45 °C temperature (typical temperature inside a measurement head). The saturation magnetic moment corresponding to the particle size is given in the legend. (a) Average relative magnetic moment and (b) particle susceptibility. The dashed lines at ξ = 1.37 (this value is taken from Table 1 in Section 3.1.1) indicate (a) the field at which the curvature is at its maximum and thus denoting the crossover from linear increase to the onset of saturation and (b) the value of steepest susceptibility decline.

Figure 3

The magnetic response of MNP in FMMD: (a) When a dual‐frequency magnetic field consisting of a high frequency component f ₁ and a low frequency component f ₂ is applied to superparamagnetic MNP, their nonlinear magnetization (b) leads to a distorted (“flattened”) magnetization response (c) which saturates at high(er) field amplitudes. The induced voltage (d) in the detection coil is given by the time derivative of the particles’ magnetization. In the frequency spectrum (e) of the induced voltage, not only the excitation frequency lines f ₂ and f ₁ are observed, but also their higher harmonics (e.g., 3f ₂, 5f ₂, 3f ₁, …) and frequency mixing components (e.g., f ₁–2f ₂, f ₁+2f ₂, …) are found in the Fourier‐transformed response signal.

Figure 4

Principle of field‐and‐core‐size dependent magnetic response of MNP to dual frequency excitation. (a) Superparamagnetic MNP magnetization as a function of the applied (single frequency) magnetic field for differently sized magnetite MNP for core sizes ranging d_C =  12 nm to 26 nm (cf. Figure 2). Below, the excitation field with B ₁ =  0.5 mT for varying offset fields, B ₀, are shown. The dark‐colored triangles approximate the signal intensity for the given excitation field at different static offset field values according to Equation (17). These mark the size‐dependent “corridor of influence” that a specific AMF has on monodisperse MNP. For simplicity, the drive field B ₂ is neglected here. (b) FMMD signal generated from approximating the size‐dependent magnetic response capability of the MNP, i.e., change of magnetization per field, $\frac{\mathrm{\Delta }M}{\mathrm{\Delta }B}(B_0)$, shows strong dependence on MNP core size and the specific offset‐field, i.e., the nonlinear section of the magnetization curve probed (based on the triangles shown in panel (a)). (c) Size‐dependent MNP magnetization response for dual‐frequency excitation: The drive field (shown with amplitude B ₂ =  10 mT) pushes the “corridor of influence” for the excitation field (again: B ₁ =  0.5 mT) along the size‐dependent individual particle magnetization curve in the same way as a static offset field (for the same exciation field), cf. subfigure (a). The triangles of $\frac{\mathrm{\Delta }M}{\mathrm{\Delta }B}(t)$ are equivalent to (a), if the same reference fields with B ₀ = B ₂ (t) are chosen. Dynamics are indicated by lighter color, i.e., lighter color represents a field later in time. (d) Approximated size‐dependent FMMD signal for the three exemplary points in time/field amplitudes B ₂(t) – analogous to (b).

Figure 5

Illustration of the magnetic moment detection scales for individual particles of magnetite/Fe₃O₄ ranging from 1 µ_B (= 0.5 × 10⁻²² Am²) at the bottom to 10¹⁵ µ_B (= 10⁻⁸ Am²) at the top. (dark gray) The saturation moments of a single Fe₃O₄ molecule and of particles with exemplary sizes between 12 nm and 26 nm are represented by the colored horizontal lines. (light gray) The maximum achievable nonlinear magnetic moments of these particles at the mixing components f ₁ + 2·f ₂ and f ₁ + 4·f ₂ at typical FMMD excitation fields are shown by the horizontal lines at the left, together with a proportional visualization of their sizes. Note that the nonlinear moments are 30‐ to 3000‐fold smaller than the saturation moments. The theoretical magnetic moment resolution of an FMMD device is indicated by the dashed line in black.

Figure 6

Lognormal size distributions with median d ₀ = 15 nm, for different width parameters σ on a logarithmic scale. The solid vertical lines mark the most probable values (denoted as mode), the dashes indicate the volumetric mean, i.e., the third moment of the distribution.

Figure 7

Nonlinear magnetic moment of MNP at mixing frequencies f ₁ + n · f ₂, with n  =  (1, 2, 3, 4) as a function of the static magnetic offset field B ₀ for quasi‐monodisperse (σ  =  0.05) magnetite particles of different mean core diameters d _C (color‐coded), calculated for B ₁ = 1.29 mT, f ₁ = 30.5 kHz, B ₂ = 16.4 mT and f ₂ = 63 Hz. (a)) is calculated from the demodulated Langevin function (Equation 46), while (b)) is derived from DRS. Violet semi‐transperent arrows are a guide to the eye, demonstrating peak‐shifts with changing particle sizes. Note that values in (b) are arbitrary and scaled to facilitate comparison of trends with (a)).

Figure 8

(a) Photograph of the MagiCoil portable device with a smartphone application. The overall dimensions of the device are 212 mm (L) × 84 mm (W) × 72 mm (H). (i) Device shell is 3D printed using the material. (ii) Disposable, USP type I glass vial containing the MNP sample. (iii) Sample loading port. (iv) Smartphone application. (b) Photograph of the internal structures of the MagiCoil device. (c) 3D model of the MagiCoil device with (v) top and (vi) bottom circuit boards, and (vii) three sets of copper coils for generating magnetic driving fields and collecting dynamic magnetic responses of MNPs. (d) Side view of the 3D model with length (L) and height (H) labeled. (e) Discrete time voltage signal collected from pick‐up coils during two periods of low‐frequency field. The dynamic magnetic responses of MNPs cause visible spikes as highlighted in gray regions. (f) Frequency domain MPS spectra from (e). Higher harmonics are observed. (g) Enlarged view of higher harmonics (the 3rd to the 41st harmonic) between 5 and 7 kHz. Reproduced under terms of the CC‐BY license with permission from K. Wu et al., ACS Applied Materials & Interfaces 13, 7966.^{[ 102 ]} Copyright 2021 American Chemical Society.

Figure 9

Exemplary characteristic feature extraction from FMMD signal of a lognormally distributed ensemble of magnetite MNP with mean core size d_c =  12 nm in dual‐frequency excitation with B ₁ =  0.1 mT, f ₁ =  1260 Hz, B ₂ =  0.1 mT, f ₂ =  63 Hz, derived from Langevin function demodulation modeling (section 3.2.1, Equation 47). (a) Nonlinear magnetic moments per particle of the first four mixing frequencies for varied static magnetic offset fields B ₀ and different distribution widths σ. (b) Develelopment of the FMMD characteristic features for maximum (peak) position, zero‐crossing and minimum position as a function of the distribution width σ for the first four mixing frequencies (directly extracted from (a)). Note that the dimensionless field variable ξ (Equation 7) is shown as the top axis in both figures.

Figure 10

Modeled particle response to drive field amplitude scans. (a) Numerical mapping of the normalized f ₁ + 2f ₂ intermodulation component when scanning the drive field amplitude, B ₀, for different of the core size. (b) Exemplary intermodulation components for f ₁ + nf ₂, n  =  2,  4,  6 for mean core sizes d_c =  20 nm (red) and d_c =  35 nm (green). All intermodulation components are normalized to each individually scanned B ₂‐scan. Reproduced under terms of the CC‐BY license with permission from T. Bikulov et al., Int. J. Magn. Part. Imaging 10, 2 403 014.^{[ 269 ]} Copyright 2024 Infinite Science Publishing GmbH.