Magnetic Nanoparticles Samples Research Articles

A multi-frequency magnetic particle spectroscopy system for systematic characterization and highly sensitive detection of magnetic nanoparticles

Systematic characterization and highly sensitive detection of magnetic nanoparticles (MNPs) is of great importance to investigate the magnetic properties of the MNPs for biomedical applications. In this study, a multi-frequency magnetic particle spectroscopy (MPS) system is developed to measure the magnetic response of the MNPs in different ac magnetic fields. A model for the description of the output signal of the MPS system is deduced to analyze the signal-to-noise ratio of the MNP harmonics. The noise source, including the Gaussian noise and harmonic distortion from the excitation magnetic field, is quantitatively measured and comprehensively analyzed to investigate their influences on the limit-of-detection (LOD) of the MPS system. The MPS signal and dynamic magnetization curve of the MNPs are measured in different ac magnetic fields with the developed MPS system. In addition, a series of MNP samples with different iron concentrations are prepared for experiments to determine the LOD, indicating an LOD of 2.3 ng in terms of iron. The dependence of the LOD on the Gaussian noise and the harmonic distortion is discussed. The present work indicates that one needs to comprehensively consider the influence factors of the Gaussian noise and harmonic distortion to develop a sensitive MPS system with multi-frequency excitation.

Magnetic nanoparticles (MNPs) based additively manufactured memory devices

Magnetic nanoparticles (MNPs) in a suspension have been shown to change resistance by an order of magnitude based on an applied field. We have prepared the MNPs samples with matrices of poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PDOT:PSS) or H2O, and Co MNPs and carried out the magnetoresistive measurements. A switch-based model is used to understand the mechanism for the change in resistance. We further propose devices for memory based on MNPs. Building blocks for these devices are then fabricated using additive manufacturing techniques and measurements of change in resistance under the influence of a magnetic field are conducted. Niche applications of additive manufacturing techniques to the fabrication of these devices are proposed. The device uses MNPs suspended in a soft matter matrix. The application of a magnetic field is used to move the MNPs to or away from electrical contacts. Depending on the change of position of the MNPs, a connection is either made or broken, which can act as a 1 or a 0. The measured change in resistance observed in such devices is more than an order of magnitude depending on the matrix solution. The proposed device and its manufacturing process could be feasible for magnetic memory and in-memory computing devices on any flexible substrates.

An efficient magnetothermal actuation setup for fast heating/cooling cycles or long-term induction heating of different magnetic nanoparticle classes

Alternating magnetic fields (AMFs) in the ∼100 kHz frequency regime cause magnetic nanoparticles (MNPs) to dissipate heat to their nanoscale environment. This mechanism is beneficial for a variety of applications in biomedicine and nanotechnology, such as localized heating of cancer tissue, actuation of drug release, or inducing conformational changes of molecules. However, engineering electromagnetic resonant circuits which generate fields to efficiently heat MNPs over long time scales, remains a challenge. In addition, many applications require fast heating/cooling cycles over ΔT= 5 °C–10 °C to switch the sample between different states. Here, we present a home-built magnetothermal actuation setup maximized in its efficiency to deliver stable AMFs as well as to enable fast heating/cooling cycles of MNP samples. The setup satisfies various demands, such as an elaborate cooling system to control heating of the circuit components as well as of the sample due to inductive losses. Fast cycles of remote sample heating/cooling (up to ±15 °C min−1) as well as long-term induction heating were monitored via contact-free thermal image recording at sub-mm resolution. Next to characterizing the improved hyperthermia setup, we demonstrate its applicability to heat different types of MNPs: ‘nanoflower’-shaped multicore iron oxide nanoparticles, core shell magnetite MNPs, as well as magnetosomes from magnetotactic bacteria (Magnetospirillum gryphiswaldense). MNPs are directly compared in their structure, surface charge, magnetic properties as well as heating response. Our work provides practical guidelines for AMF engineering and the monitoring of MNP heating for biomedical or nano-/biotechnological applications.

Temperature dependent magnetorelaxometry of magnetic nanoparticle ensembles.

Magnetorelaxometry imaging (MRXI) is a non-invasive, quantitative, imaging technique for magnetic nanoparticles (MNPs). The image resolution of this technique significantly depends on the relaxation amplitude (ΔB). For this work, we measured the room temperature (299 K) relaxation signals of eight commercial MNP sample systems with different magnetic properties, in both fluid and immobilized states, in order to select the most suitable sample for a particular MRXI setting. Additionally, the effect of elevated temperatures (up to hyperthermia temperature, 335 K) on the relaxation signals of four different MNP systems (Synomag, Perimag, BNF and Nanomag), in both states were investigated. The ΔB values of fluid samples significantly decreased with increasing temperature, and the behaviour for immobilized samples depended on their blocking temperature (TB). For samples with TB < 299 K, ΔB also decreased with increasing temperature. Whereas for samples with TB > 299 K, the opposite behaviour was observed. These results are beneficial to improving the image resolution in MRXI and show, among the investigated systems, and for our setup, Synomag is the best candidate for future in vitro and in vivo studies. This is due to its consistently high ΔB between 299 K and 335 K in both states. Our findings demonstrate the feasibility of temperature imaging by MRXI.

Magneto-structural characterization of different kinds of magnetic nanoparticles

Using well-established measurement techniques like transmission electron microscopy (TEM), dynamic light scattering (DLS), small and wide angle X-ray scattering (SAXS, WAXS), susceptometry, and magnetorelaxometry, the distribution of the physical and magnetic size (magnetic moments) and magnetic anisotropy of a variety of structurally different magnetic nanoparticle samples (MNPs) is analyzed and compared. A term which accounts for the presence of weak magnetic areas (WMAs) within the MNPs was introduced to the widespread analysis model for M(H) data, enabling a consistent interpretation of the data in most of the systems. A comparison of the size distributions as obtained for the physical and the magnetic diameter suggests a multidomain structure for three single core systems under investigation, in all probability evoked by the presence of a wustite phase, as identified by WAXS.Analyzing the relationship d < dm < dc between the average single core diameter d, the effective magnetic (domain) size dm and the cluster diameter dc quantitatively, two qualitatively different magnetic structures in multicore MNP (MCMNP) systems were identified: (i) The magnetic moments of single cores within the MCMNP of fluidMAG tend to build flux closure structures, driven by dipole–dipole interaction. (ii) The magnetic behavior of Resovist® was attributed to the presence of domain sizes of about 12 nm within MCMNP, exceeding the single core diameters of 5 nm. Thereby, WAXS revealed a bimodal crystallite size distribution suggesting a crystallite merging process within the MCMNP. The value of the effective magnetic moment of these MCMNP could be explained within the presented “random moment cluster model” (RMCM).We conclude that the combination of physical and magnetic structure parameters obtained from complementary measurement methods allows a reliable assessment of the magnetic structure of single and multicore MNPs.

Human-sized quantitative imaging of magnetic nanoparticles with nonlinear magnetorelaxometry

Objective. Magnetorelaxomety imaging (MRXI) is a noninvasive imaging technique for quantitative detection of magnetic nanoparticles (MNPs). The qualitative and quantitative knowledge of the MNP distribution inside the body is a prerequisite for a number of arising biomedical applications, such as magnetic drug targeting and magnetic hyperthermia therapy. It was shown throughout numerous studies that MRXI is able to successfully localize and quantify MNP ensembles in volumes up to the size of a human head. However, deeper regions that lie far from the excitation coils and the magnetic sensors are harder to reconstruct due to the weaker signals from the MNPs in these areas. On the one hand, stronger magnetic fields need to be applied to produce measurable signals from such MNP distributions to further upscale MRXI, on the other hand, this invalidates the assumption of a linear relation between applied magnetic field and particle magnetization in the current MRXI forward model which is required for the imaging procedure. Approach. We tackle this problem by introducing a nonlinear MRXI forward model that is also valid for strong magnetic excitation fields. Main results. We demonstrate in our experimental feasibility study that scaling up the imaging region to the size of a human torso using nonlinear MRXI is possible. Despite the extreme simplicity of the imaging setup applied in this study, an immobilized MNP sample with 6.3 cm3 and 12 mg Fe could be localized and quantified with an acceptable quality. Significance. A well-engineered MRXI setup could provide much better imaging qualities in shorter data acquisition times, making nonlinear MRXI a viable option for the supervision of MNP related therapies in all regions of the human body, specifically magnetic hyperthermia.

Harmonic signals of magnetic nanoparticle in suspension for biosensing application: Comparison of properties between Brownian- and Néel-dominant regions

We have studied harmonic signals caused by the nonlinear AC magnetization (M-H) curve of magnetic nanoparticles (MNPs) suspended in solution. First, numerical simulations were performed for wide ranges of the MNP parameters and the excitation fields that are commonly used in biosensing applications. As an example, the third harmonic signal was mainly studied. Because the AC M-H curve for the suspended MNPs is affected by both Brownian and Néel relaxations, we demonstrated the condition that determines the dominant relaxation. We emphasize that the MNP’s behavior changes from Brownian-dominant to Néel-dominant relaxation when the amplitude of the excitation field is increased, even when the MNP parameters are fixed. The properties of the harmonic signals in the Brownian- and Néel-dominant cases were compared, and the differences between the two cases were clarified. The relationships among the higher harmonic signals were also presented. Next, an analytical expression for the third harmonic signal was obtained for both the Brownian- and Néel-dominant cases. In the Néel-dominant case, the alignment of the easy axes caused by the AC field was taken into account. We demonstrated good agreement between the simulation and the analysis results. Finally, harmonic signals were measured using a commercial MNPs sample. Reasonable agreement was obtained between experiment and analysis. The results obtained will be useful for biosensing applications.

Influence of Ligands on Physicochemical Characteristics of Magnetic Nanoparticles

Magnetic-bead separation or purification serves as a technique for effective isolation of biomolecules. In presented work we prepared and characterized core-shell magnetic nanoparticle samples consisted of Fe 3 O 4 core coated with SiO 2 shell. Samples were subsequently coated with ligands MPTMS (3-(mercaptopropyl)trimethoxysilane), CPTMS (3-(chloropropyl)trimethoxysilane) and MMSP (3-(trimethoxysilyl)propyl methacrylate) with aim to increase the number of active centers for specific binding with RNA. Such samples were further investigated for their magnetic properties, size, and morphology. Magnetic properties were studied in DC field up to 5 T in temperature range 5 – 300 K. Size and morphology were determined from SEM micrographs and elemental compositions of the samples were investigated using EDX analysis. Modification of nanoparticle surface with different ligands leads to modification of active centers on the SiO 2 surface on which the DNA and RNA molecules can be bounded. It also causes the change in magnetic and structural properties of nanoparticles.

Generation of Circular Dichroism from Superposed Magnetically Oriented Magnetic Nanoparticles

Circular dichroism (CD) spectra were observed for the doubly superposed film–film and solution–solution samples, which all contained magnetically oriented achiral iron oxide magnetic nanoparticles (MNPs). The observed CD spectra critically depended on the angle difference, Δθ, between the orientation axes of MNPs in the superposed samples, giving the maximum CD value at the angle of Δθ = 45°. In a single solution cell sample with doubly superposed magnetic fields also, similar CD spectra depending on the angle difference between the magnetic fields were observed. These observed CD spectra were successfully reconstructed from the observed linear dichroism (LD) and linear birefringence (LB) spectra of each sample by the use of a superposed Mueller matrix method. Furthermore, triply and quadruply superposed solution samples exhibited the maximum CD spectra at the angle differences of Δθ = 30 and 22.5°, respectively, as suggested by the superposed Mueller matrix calculation. Thus, the origin of all CD spectra observed for the superposed achiral MNP samples has been quantitatively assigned to the helical combinations of LD and LB spectra of the magnetic field-induced optically oriented MNPs.

A new method to measure magnetic nanoparticle heating efficiency in non-adiabatic systems using transient pulse analysis.

Heating magnetic nanoparticles (MNPs) with alternating magnetic fields (AMFs) have applications in biomedical research and cancer therapy. Accurate measurement of the heating efficiency or specific loss power (SLP) generated by the MNPs is essential to assess response(s) in biological systems. Efforts to develop standardized equipment and to harmonize results obtained from various MNP samples and AMF systems have met with little success. Without a standardized magnetic nanoparticle or calorimeter device, objective comparisons of estimated thermal output among laboratories remain a challenge. In addition, the most widely used adiabatic initial slope model fails to account for thermal losses, which are unavoidable. We propose a non-adiabatic method to analyze MNP heating efficiency derived from the Box-Lucas equation, wherein the sample is subjected to several short duration heating pulses. SLP is then estimated from an arithmetic average of the Box-Lucas fitted coefficients obtained from each pulse. Heating experiments were conducted with two identical samples that were placed within vessels having different thermal insulation using the same AMF parameters. Though the samples generated different temperature curves, the pulsed Box-Lucas method produced nearly equivalent SLP estimates. Further, the pulsed test enabled analysis of the heat transfer coefficient providing quantitative measures of sample heat loss throughout the test, with robust statistical confidence. We anticipate this new methodology will aid efforts to standardize measurements of MNP heating efficiency, enabling direct comparison among varied systems.

TranSMS: Transformers for Super-Resolution Calibration in Magnetic Particle Imaging.

Magnetic particle imaging (MPI) offers exceptional contrast for magnetic nanoparticles (MNP) at high spatio-temporal resolution. A common procedure in MPI starts with a calibration scan to measure the system matrix (SM), which is then used to set up an inverse problem to reconstruct images of the MNP distribution during subsequent scans. This calibration enables the reconstruction to sensitively account for various system imperfections. Yet time-consuming SM measurements have to be repeated under notable changes in system properties. Here, we introduce a novel deep learning approach for accelerated MPI calibration based on Transformers for SM super-resolution (TranSMS). Low-resolution SM measurements are performed using large MNP samples for improved signal-to-noise ratio efficiency, and the high-resolution SM is super-resolved via model-based deep learning. TranSMS leverages a vision transformer module to capture contextual relationships in low-resolution input images, a dense convolutional module for localizing high-resolution image features, and a data-consistency module to ensure measurement fidelity. Demonstrations on simulated and experimental data indicate that TranSMS significantly improves SM recovery and MPI reconstruction for up to 64-fold acceleration in two-dimensional imaging.

Design and construction of a lab-made calorimetric system for specific loss power (SLP) measurements in suspensions of magnetic nanoparticles

The specific loss power (SLP) is a figure of merit in nanomagnetism, which allows to classify systems of magnetic nanoparticles (MNPs) according to their capability to dissipate heating power in response to the energy absorbed from an alternating magnetic field (AMF) at radiofrequencies. SLP magnitude is relevant in biological applications of MNPs, such as nanomedicine, where the thermal energy released by these nanostructures can be used to produce apoptosis in cancer cells, becoming a promising alternative for the treatment of malignant tumors. In this work we present details of the design and construction of a lab-made electromagnetic system to perform SLP measurements in systems of MNPs. The system is composed by an actuation stage, which generates the AMF to excite the sample, a thermal insulation stage between the sample and its surroundings, and two sensing stages: the first one for measuring the magnetic field applied to the sample and the second one for measuring the changes of temperature of the sample. The system developed allows to reach AMFs around 7.7 kA m−1 in the center of the actuation coil, operating at 260 kHz and with a power consumption less than 150 W. The architecture of the system allows the automatization and control of SLP measurements from remote places through the IEEE 488 bus available in the power source and a graphic interface developed in LabVIEW software, which allows to visualize temperature versus time measurements, control the magnetic field amplitude and store the data in a public folder available in the cloud. Some results are presented, where the system is employed to measure the SLP of a colloidal suspension of MNPs of maghemite (γ-Fe2O3) stabilized with sodium polyacrylate. The results obtained show a satisfactory performance of the device to measure the SLP of samples of MNPs for different purposes.

Frequency Mixing Magnetic Detection Setup Employing Permanent Ring Magnets as a Static Offset Field Source.

Frequency mixing magnetic detection (FMMD) has been explored for its applications in fields of magnetic biosensing, multiplex detection of magnetic nanoparticles (MNP) and the determination of core size distribution of MNP samples. Such applications rely on the application of a static offset magnetic field, which is generated traditionally with an electromagnet. Such a setup requires a current source, as well as passive or active cooling strategies, which directly sets a limitation based on the portability aspect that is desired for point of care (POC) monitoring applications. In this work, a measurement head is introduced that involves the utilization of two ring-shaped permanent magnets to generate a static offset magnetic field. A steel cylinder in the ring bores homogenizes the field. By variation of the distance between the ring magnets and of the thickness of the steel cylinder, the magnitude of the magnetic field at the sample position can be adjusted. Furthermore, the measurement setup is compared to the electromagnet offset module based on measured signals and temperature behavior.

Quantitative explanation of the difference in AC magnetization curves between suspended and immobilized magnetic nanoparticles for biomedical application

We study the difference in AC magnetization (M−H) curves between suspended and immobilized magnetic nanoparticles (MNPs). We use three commercial MNP samples that are often used for biomedical application. First, we measure the hysteresis area of the AC M−H curve, A, when the amplitude of the excitation field, Hac, is changed. The A vs Hac curve is compared with previously obtained analytical results by taking account of the MNP size distribution in the sample. We show that they quantitatively agree for both suspended and immobilized samples. From the comparison, we clarify the mechanism that determines the AC M−H curve of a suspended sample. For high Hac, the Néel relaxation becomes dominant in the suspended sample, and alignment of easy axes caused by the AC field increases the hysteresis loss compared to the immobilized case. The effect of Brownian relaxation in the suspended sample increases with decreasing Hac and gives additional loss at low field. The portion of Néel-relaxation- and Brownian-relaxation-dominant MNPs in the sample is quantitatively evaluated. We also clarify the difference in harmonic signals between suspended and immobilized samples. Finally, we discuss the condition for the MNP parameters and excitation field that determines the dominant relaxation mechanism in a suspended sample.

Probing particle size dependency of frequency mixing magnetic detection with dynamic relaxation simulation

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 10nm 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.

Improvement in magnetic nanoparticle tomography estimation accuracy by combining sLORETA and non-negative least squares methods

Imaging methods that can detect biofunctionalized magnetic nanoparticles (MNPs) accumulated at cancerous tumor sites are expected to be part of the in vivo cancer diagnostic techniques. An imaging technique called magnetic nanoparticle tomography (MNT), which uses a magnetic sensor array, has been proposed. High sensitivity and spatial resolution were achieved using the non-negative least-squares (NNLS) inverse solution in MNT. However, owing to the presence of measurement noise, the concentration and position of certain MNPs were estimated inaccurately, i.e., artifacts were generated. To overcome this issue, this study first applied standardized low-resolution brain electromagnetic tomography (sLORETA), a spatial filter method with no location bias, to approximate the position of MNPs. The region of analysis was restricted to where the estimated value exceeded the threshold. Subsequently, the NNLS method was applied to estimate the concentration and position of MNPs in the restricted region. In the experiment, two Resovist MNP samples (300 or 500 μgFe) were arranged at a distance of 25–502 mm, and the concentration and position estimation were performed. The estimation results demonstrated that the proposed method successfully suppresses artifacts and adequately estimates the concentration and position of MNPs within a position error of 10 mm and a concentration error of 20 %.

Dependence of hysteresis loss of immobilized magnetic nanoparticles on the easy-axis angle: Effect of easy-axis alignment

Magnetic nanoparticles (MNPs) have been widely studied for hyperthermia applications. We studied the hysteresis loss of immobilized MNPs with partially aligned easy axes. For this purpose, we investigated how the AC magnetization (M–H) curve of MNP depended on the easy-axis angle of magnetization, β, via numerical simulations of the Fokker–Planck equation. We clarified the dependences of the coercive field, Hc, and the hysteresis loss, A, on β. We, thus, obtained analytical expressions for Hc(β) and A(β) that explain simulation results for a wide range of MNP parameters and excitation conditions. The angle dependences were also discussed on the basis of magnetic moment reversal over the anisotropy energy barrier. We then examined the hysteresis loss of an MNP sample with partial alignment of the easy axes and obtained an expression for the loss by combining A(β) and a distribution function for β. We quantitatively clarified the relationship between the loss and the degree of easy-axis alignment. The loss of immobilized MNPs can be increased by a factor of 2.2 by using easy-axis alignment relative to the case of randomly oriented easy axes. Finally, we examined the alignment of easy axes induced by an AC field in suspended MNPs and showed that the loss of immobilized MNPs with partially aligned easy axes can be used to estimate the loss for suspended MNPs.

Low-temperature large-scale hydrothermal synthesis of optically active PEG-200 capped single domain MnFe2O4 nanoparticles

Herein, we report an intrinsic fluorescence in polyethylene glycol (PEG) coated manganese ferrite (MnFe2O4) magnetic nano particles (MNPs). Generally, nano ferrites (NFs) do not show fluorescence characteristics, however, imparting fluorescence in these materials can be valuable for many applications as the stipulation of external control over the properties by external field provides an opportunity to develop active-controlled devices. In the present work, PEG-200 coated MnFe2O4 MNPs have been synthesized by a large-scale, facile one-pot hydrothermal method. The optical properties of MnFe2O4 MNPs have been by photoluminescence (PL) spectroscopy suggest that the prepared sample is showing sharp emission spectra at 422 nm, corresponding to the violet band, and second at 458 nm corresponding to the blue band. The presence of oxygen vacancy created at both tetrahedral and octahedral sites and shallow hole defects are responsible for the PL behavior in the PEG-coated MnFe2O4. The time-resolved PL (TRPL) results suggest that the MnFe2O4 MNPs have two decay times τ1=1.29ns and τ2=6.85ns which are non-correlated and correspond to different defect states. The electronic bandgap of the MNPs is 3 eV obtained from the Tauc plot using UV absorption spectra. Moreover, the structural analysis has been carried out using X-ray diffraction and various optical and magnetic properties have been correlated to structural parameters and cation distribution. The ferromagnetic resonance (FMR) spectra of the MnFe2O4 MNPs sample show a broad single peak which confirms the ferromagnetic characteristics of the sample and the g-value obtained for the sample is 1.98 which suggests that the super exchange interactions (SEI) dominates over the dipolar interaction. The M-H loop confirms the single domain nature of the sample and the saturation magnetization (SM) is 68 emu/g. The obtained value of SM is higher than the previously reported literature due to an increase in the occupancy of Fe3+ and Mn2+ ions at octahedral sites (Neel’s model). Further, the FTIR spectra confirm the spinel phase formation, and the size distribution and average size are obtained using electron microscopy analysis. The intrinsic fluorescent PEG-coated MnFe2O4 are highly suitable for a diverse range of biomedical (bio-imaging, sensing, magnetic resonance imaging (MRI) contrast agent), optoelectronics, and optical applications.

Biological thermometer based on the temperature sensitivity of magnetic nanoparticle paraSHIFT

In the paper, the temperature dependence of magnetic nanoparticle (MNP) paramagnetic chemical shift (paraSHIFT) was studied by magnetic resonance (MR) spectroscopy. Based on it, iron oxide MNPs are considered as MR shifting probes for determining the temperature in liquids. With the increase in measurement temperature of the MNP reagent with MNPs, the decrease of MNP magnetization would make the peak of spectroscopy shift to the higher chemical shift area. The peak shift is related to the magnetic susceptibility of MNPs, which can be determined by MR frequency as a function of temperature and particle size. Experiments on temperature-dependent chemical shifts are performed for MNP samples with different core sizes and the estimated temperature accuracy can achieve 0.1 K. Combined with the contrast effect of magnetic nanoparticles in magnetic resonance imaging at 3 T, this technology can realize temperature imaging.

Evaluation of effective magnetic anisotropy constant of magnetic nanoparticles from coercive field of AC magnetization curve

Magnetic nanoparticles (MNPs) have been widely studied for use in biomedical application with the magnetic anisotropy constant K playing an important role in determining the performance. We estimated K near room temperature from the coercive field Hc of an AC magnetization (M–H) curve. First, we performed numerical simulation of the AC M–H curve of immobilized MNPs and clarified the dependencies of Hc on the MNP parameters and excitation conditions. Based on the simulation result, we obtained an analytical expression for Hc that was more general and included the previously obtained expression; and in addition, it could be applied to an MNP sample with a core-size distribution. Next, we measured the AC M–H curves of two commercial MNP samples and determined the dependencies of Hc on the amplitude and frequency of the excitation field. The dependencies agreed reasonably well with the analytical results. The K value was evaluated to obtain the best fit between the measured and analytical Hc, and the obtained K values were consistent with those estimated using other methods. The temperature dependence of K near room temperature was also determined. The present method will provide a useful tool to estimate the K value of MNPs.