Numerical Simulation of Low-Frequency Magnetic Fields and Gradients for Magnetomechanical Applications

Abstract

This study aims to identify optimal parameters for the clinical implementation of magnetic fields in therapeutic contexts, with a particular focus on in vitro magneto-mechanical actuation in biological systems. This approach relies on the transduction of magnetic energy into mechanical stress at low frequencies (<<100 Hz). Accordingly, the investigation centers on evaluating the magnetic field gradients responsible for initiating the motion of intracellular magnetic nanoparticles and the resulting mechanical forces acting upon them. To achieve this, a novel, custom-built, and highly adaptable three-dimensional turntable system was designed, calibrated, and implemented. This apparatus allows the generation of magnetic fields with precisely tunable amplitude and frequency, enabling controlled activation of magneto-mechanical mechanisms. In vitro experiments using this device facilitated the exposure of cancer cells to well-characterized magnetic fields, thereby inducing mechanical stimulation in the presence of nanoparticles distributed within intracellular or extracellular environments. Quantitative measurements of magnetic field intensities were performed, providing estimations of the forces exerted by magnetic nanoparticles with diverse physical characteristics (phase, size, and shape) under varying magnetic field gradients.

1. Introduction

In recent years, numerous studies have initially centered on magnetic hyperthermia; however, alongside explanations based on intracellular heating, several authors have proposed that many of the observed effects could also arise from non-thermal mechanisms [1,2,3,4]. These non-thermal hypotheses are supported by physical reasoning. When magnetic nanoparticles (MNPs) are subjected to an external magnetic field, they not only produce heat—resulting from magnetic relaxation and reorientation of their magnetic moments (μ)—but also experience mechanical effects. To suppress the heating contribution and enhance the mechanical response, several research groups have adopted static magnetic fields or employed field frequencies many orders of magnitude lower than those typically used for magnetic hyperthermia. Under low-frequency alternating magnetic fields (AMF) with magnetic flux densities $\mathrm{B}\ll 1 \mathrm{T}$ and frequencies $\mathrm{f}<100 \mathrm{Hz},$ both local and bulk thermal effects on cells and tissues can be considered negligible, regardless of the MNP concentration. In these conditions, the applied magnetic field primarily induces mechanical motion of MNPs, which can in turn influence cells, subcellular structures, and biomacromolecules to which the particles are bound [5].

Other studies [6,7] have utilized rotating or precessing magnetic fields, where the rotation axis of the magnetic induction vector is perpendicular or inclined relative to the main field direction. The amplitude of these fields may remain constant or vary periodically with time. Such rotating fields—often generated by reciprocating or rotating permanent magnets with a phase shift of π/2 between them—can be regarded as a specific case of an AMF. The biological responses observed under these conditions have been rationalized through theoretical analyses that propose the conversion of magnetic field energy—whether static or non-heating AMF ($f\ll 1 \mathrm{kHz}$)—into mechanical deformation of cells or macromolecules attached to MNPs [8,9,10].

More recently, innovative strategies have emerged that exploit low-frequency magneto-mechanical (MM) effects to target cancer cells [11,12,13]. These approaches often employ anisotropic magnetic particles capable of exerting torques or forces on cellular membranes. Depending on the specific mechanism, such interactions can lead to necrosis by physically disrupting cell integrity (magnetoporation or magnetolysis) or can trigger apoptosis, the programmed cell death pathway [14,15]. In the latter case, magneto-mechanical stimulation induces intracellular ionic imbalances, particularly affecting calcium homeostasis, thereby initiating the apoptotic cascade. This is significant because many cancer cells lose the ability to undergo apoptosis due to oncogenic mutations, leading to uncontrolled proliferation, tumor growth, and metastasis. Apoptosis, characterized by cell shrinkage, membrane blebbing, and the formation of apoptotic bodies subsequently cleared by macrophages, proceeds without releasing cytotoxic content into the extracellular space. Consequently, unlike necrosis, it does not elicit inflammatory responses [16], making apoptosis the preferred pathway for cancer therapies seeking minimal side effects.

At the nanoscale, realignment of MNPs with an external magnetic field can generate localized stresses in MNP-linked cellular membranes. In a uniform magnetic field, an MNP experiences a torque $T$, whereas in a non-uniform field, it experiences a translational force $F$, leading to rotational or linear motion, respectively. The resulting cellular deformation—tensile, compressive, torsional, or shear—depends on the orientation of the MNP magnetic moments (μ_i), the magnetic flux density vector $B$, and the cellular attachment sites of the particles [17]. Therefore, accurate estimation of the magnetic field amplitude, direction, and gradient—and consequently the induced mechanical forces—is crucial. Theoretical calculations indicate that for magnetic fields with $B<1 \mathrm{T}$, forces on the order of several thousand piconewtons (pN) can be achieved, resulting in deformations of tens of nanometers—sufficient to produce biologically relevant effects. Mechanical forces of this magnitude are known to modulate cell differentiation [18]: cytoskeletal stress occurs at forces between 1 and 10 pN [19]; DNA stretching and chromosome segregation require 10–100 pN [20]; neuronal mechanosensitivity thresholds range from 5 to 70 pN [21]; and forces exceeding 500 pN can drive intracellular transport of macrophages [22], surpassing actin filament strength and inducing cellular damage [23]. Furthermore, results from Single-Molecule Force Spectroscopy demonstrate that such force magnitudes and deformations can trigger macromolecular transitions and processes fundamental to nanomedicine and drug delivery [24]. These findings highlight the potential of magneto-mechanical actuation for biomedical applications under experimentally accessible magnetic field conditions, with $B$ values ranging from a few tens to several hundred millitesla.

In the present work, we focus on the quantitative identification and analysis of magnetic field distributions generated by a previously developed magneto-mechanical actuation system. While the design and construction of this device have been reported elsewhere [25], the current study aims to characterize and compare the distinct magnetic field configurations—static, rotating, and alternating—and their corresponding mechanical effects on magnetic nanoparticle (MNP) systems. Numerical simulations were performed to determine the spatial field gradients and to estimate the forces acting on MNPs exhibiting different intracellular uptake behaviors, as observed in in vitro environments relevant to magnetic hyperthermia (MHT) and magneto-mechanical cancer therapy. By integrating MNP physical properties and their experimentally inferred intracellular concentrations, we calculated the forces experienced by individual cells under various field regimes. This systematic framework provides a quantitative basis for correlating magnetic field parameters with the resulting mechanical stresses, advancing the understanding of magneto-mechanical stimulation as a complementary or alternative strategy to magnetic hyperthermia for regulating biological processes.

2. Materials and Methods

The apparatus simulated here can be found in our previous works [25,26]. Here we present a more detailed and systematic analysis of the magnetic fields simulated and also the mechanical forces generated by the various fields’ configurations. A three-dimensional polymer turntable, fabricated via 3D printing was developed as the core rotating component of the experimental setup. The device was driven by a direct current (DC) motor powered at variable voltages ranging from 3 to 12 V, enabling adjustable rotational frequencies between 0 and 60 Hz. MNPs subjected to a low-frequency magnetic field undergo a mechanical effect. In a uniform magnetic field H with magnetic flux density B, there is a torque [27]:

τ = μ × B

(1)

while in a non-uniform magnetic field, the rotation moment is supplemented by force:

F = (μ·∇) B

(2)

which, as we can see from Equation (2), is strongly dependent on the gradient field ∇B.

The various configurations of permanent Nd-Fe-B magnets that are capable of generating strong constant magnetic field were simulated by using COMSOL Multiphysics (v.3.5a). Firstly, the CAD model of each configuration was imported in COMSOL, and then the appropriate properties and conditions were applied. The “AC/DC magneto-statics, no currents” application mode, which handles magnetic fields without currents, was used in order to calculate the resulting magnetic flux density in a 3D computational domain. When no currents are present, the problem is easier to solve using the magnetic scalar potential V_m, which is related to the permanent magnet’s magnetic field H according to the formula

H = −∇V_m

(3)

At the boundaries of the computational domain, we applied the magnetic insulation condition that sets the normal component of the magnetic flux density Β to zero. This condition is useful at these boundaries, confining a surrounding region of air, while the continuity boundary condition was applied at the magnets’ boundaries. This is the natural boundary condition that ensures continuity of B in the entire computational domain studied. This application mode solves the following Equation:

$$\nabla \mathit{{\rm B}}=0⟺-\mathsf{\nabla }\cdot \left({\mu }_0{\mu }_r\mathsf{\nabla }{V}_m-{\mathit{B}}_{\mathit{r}}\right)=0$$

(4)

where μ₀ is the permeability of vacuum, μ_r is the relative permeability of Nd-Fe-B magnet, which is equal to 1.05, and B_r is the remnant magnetic flux density, which varies from 0 to 1.2 T outside and inside the magnet, respectively. This Equation assumes the following constitutive relation between B and H:

$$\mathit{B}={\mu }_0{\mu }_r\mathit{H}+{\mathit{B}}_{\mathit{r}}$$

(5)

By implementing this method, we visualize the spatial distribution of the magnetic flux density magnitude |B| and direction, along with the magnitude of the gradient of magnetic flux density |∇(B)|.

When MNPs are exposed to magnetic fields, they experience a magnetic force acting on them. We are going to determine this magnetic force by taking into account a model for the magnetic response of a single nanoparticle, as proposed in the literature by Furlani et al. [28,29]. There, the authors used a linear magnetization model with saturation, as shown in the below saturation:

$$\mathit{M}={\chi }_p{\mathit{H}}_{in}$$

(6)

where ${\chi }_p$ and M are the susceptibility and the magnetization of the particle, respectively. Above saturation, |M| = Ms, where Ms is the saturation magnetization of the particle. In Equation (6), H_in is the field inside the particle and is given by

H_in = H_α − H_demag

(7)

H_in is different from the externally applied field H_α because the magnetization of the nanoparticle itself gives rise to a self-demagnetization field H_demag that opposes H_α. For a uniformly magnetized sphere with magnetization M in free space,

H_demag = M/3

(8)

H_demag is the field inside the sphere due to the “magnet change” ${\mathit{\sigma }}_m=M·\widehat{\mathit{n}}$ at its surface, where $\widehat{\mathit{n}}$ is the unit vector normal to the surface. According to the linear model, the magnetic field intensity required to saturate the particle is |H_in,sat| = Ms/${\chi }_p$, and by embodying Equations (7) and (8) in Equation (6), we find the (below saturation) magnetization expression:

$$\mathit{M}={\displaystyle \frac{3{\chi }_p}{3+{\chi }_p}}{\mathit{H}}_a$$

(9)

Based on this assumption, Furlani et al. analytically determined the magnetic force applied to a single spherical magnetic nanoparticle. They used an effective dipole moment approach in which the particle is replaced by an ‘‘equivalent’’ point dipole, located at its center. The force depends on the magnetic field at the location of the dipole and reads

$${\mathit{F}}_{mp}={\mu }_0 V_p{\displaystyle \frac{3{\chi }_p}{{\chi }_p+3}}\left({\mathit{H}}_a·\nabla \right){\mathit{H}}_a$$

(10)

where ${\mu }_0=4\pi \times {10}^{-7 }[\mathrm{{\rm T}}·\mathrm{m}·{\mathrm{A}}^{-1}]$ is the vacuum magnetic permeability and $V_p={\displaystyle \frac{4}{3}}\pi R_p^3$ is the volume of a single spherical magnetic nanoparticle of radius R_p.

In our simulations, we consider the motion in 3D space (x-y, x-z, y-z planes), and therefore, the magnetic field H and the magnetic force ${\mathit{F}}_{mp}$ can be expressed as

$$\mathit{H}\left(x,y,z\right)=H_X\left(x,y,z\right)\widehat{\mathit{x}}+H_y\left(x,y,z\right)\widehat{\mathit{y}}+H_z\left(x,y,z\right)\widehat{\mathit{z}}$$

(11)

$${\mathit{F}}_{mp}=F_{mpx}\widehat{\mathit{x}}+F_{mpy}\widehat{\mathit{y}}+F_{mpz}\widehat{\mathit{z}}$$

(12)

where the magnetic force can be decomposed into its components as

$$F_{mpx}={\mu }_0 V_p{\displaystyle \frac{3{\chi }_p}{{\chi }_p+3}}\left[H_X\left(x,y,z\right){\displaystyle \frac{\partial H_X\left(x,y,z\right)}{\partial x}}+H_y\left(x,y,z\right){\displaystyle \frac{\partial H_X\left(x,y,z\right)}{\partial y}}+H_z\left(x,y,z\right){\displaystyle \frac{\partial H_X\left(x,y,z\right)}{\partial z}}\right]$$

$$F_{mpy}={\mu }_0 V_p{\displaystyle \frac{3{\chi }_p}{{\chi }_p+3}}\left[H_X\left(x,y,z\right){\displaystyle \frac{\partial H_y\left(x,y,z\right)}{\partial x}}+H_y\left(x,y,z\right){\displaystyle \frac{\partial H_y\left(x,y,z\right)}{\partial y}}+H_z\left(x,y,z\right){\displaystyle \frac{\partial H_y\left(x,y,z\right)}{\partial z}}\right]$$

$$F_{mpz}={\mu }_0 V_p{\displaystyle \frac{3{\chi }_p}{{\chi }_p+3}}\left[H_X\left(x,y,z\right){\displaystyle \frac{\partial H_z\left(x,y,z\right)}{\partial x}}+H_y\left(x,y,z\right){\displaystyle \frac{\partial H_z\left(x,y,z\right)}{\partial y}}+H_z\left(x,y,z\right){\displaystyle \frac{\partial H_z\left(x,y,z\right)}{\partial z}}\right]$$

For ferromagnetic nanoparticles, ${\chi }_p$ >> 1 and thus the term ${\displaystyle \frac{3{\chi }_p}{{\chi }_p+3}}\approx {\displaystyle \frac{3{\chi }_p}{{\chi }_p}}=3$ while for superparamagnetic (SPM), nanoparticles ${\chi }_p$ can be estimated from the initial susceptibility and is given by

$${\chi }_p={\chi }_i=\left({\displaystyle \frac{\partial \mathit{M}}{\partial \mathit{H}}}\right)={\displaystyle \frac{{\mu }_0\varphi M_d^2{V}_p}{3kT}}$$

(13)

where $\varphi$ is the fraction of saturation magnetization M_S of the magnetic nanoparticles to the bulk magnetization M_d: $\varphi$ = M_S/M_d.

In the case of an ensemble of N MNPs and under the assumption that all particles exert equal magnitude forces, the magnitude of the total magnetic force ${\mathit{F}}_t$ is given by

$${{|\mathit{F}}_t|=N|\mathit{F}}_{mp}|$$

(14)

To estimate the mechanical forces exerted by magnetic nanoparticles (MNPs) on individual cells in vitro, it is essential to determine the cellular uptake, that is, the number of nanoparticles internalized per cell. This parameter is experimentally quantified in in vitro studies using inductively coupled plasma–optical emission spectroscopy (ICP–OES), which measures the intracellular iron concentration as an indicator of nanoparticle loading.

For the evaluation of the proposed device—after its complete calibration and analysis under various magnetic field configurations—the generated mechanical forces were estimated for specific MNP systems internalized in distinct cancer cell types. Accordingly, experimental uptake data from three representative in vitro studies were selected from the literature, covering different magnetic regimes: superparamagnetic (SPM), single-domain (SD), and multidomain (MD) nanoparticles. These datasets, obtained from studies employing either magnetic hyperthermia (MHT) or magneto-mechanical approaches, provided realistic benchmarks for force estimation in our simulations.

(i)

Jordan et al. [29] investigated the endocytosis of superparamagnetic magnetite nanoparticles (13 nm) and their intracellular hyperthermic effect on human mammary carcinoma BT20 cells. The reported maximum uptake was 500 pg of Fe per cell, corresponding to approximately $N=1.1\times {10}^8$ magnetite nanoparticles per cell, assuming a magnetite density of $5200 {\mathrm{kg}/\mathrm{m}}^3$. In this study, human carcinoma cells were incubated with dextran-coated magnetite nanoparticles and exposed to a high-frequency alternating magnetic field (520 kHz, 7–13 kA m⁻¹) to investigate the feasibility of intracellular magnetic particle hyperthermia. Cells were allowed to internalize nanoparticles for up to 100 h at a concentration of 0.8 mg ferrite/mL. Cells were subsequently heated either by water bath hyperthermia or by AC magnetic field exposure. Although theoretical calculations predicted sufficient intracellular heating for cell inactivation (SAR ≈ 44 mW/mL), no enhanced cytotoxic effect was observed for magnetic hyperthermia compared with conventional water-bath heating. Electron microscopy revealed lysosomal degradation of the dextran coating, leading to particle aggregation and SAR reduction. As a result, this study concluded that dextran-coated magnetite nanoparticles were not suitable for intracellular MFH-induced cancer cell killing. Thus, no enhancement of cancer cell killing beyond conventional thermal effects was observed despite significant nanoparticle uptake, demonstrating the limitations of intracellular magnetic hyperthermia under these conditions.

(ii)

Chalkidou et al. [30] examined the heating efficiency of Fe/MgO core–shell nanoparticles (with a 75 nm single-domain iron core) under an alternating magnetic field, using human breast cancer MCF7 cells. Cellular uptake and cytotoxicity assays were conducted to evaluate the biological response. The maximum iron uptake measured was 250 pg Fe per cell, equivalent to $N=1.4\times {10}^5$ nanoparticles per cell, considering an iron density of $7900 {\mathrm{kg}/\mathrm{m}}^3$. Nanoparticles were internalized through energy-dependent endocytosis. AC magnetic field exposure produced specific absorption rates (SAR) in the range of 100–500 W g⁻¹ Fe, leading to a rapid temperature rise of ~15 °C within 10 min. Selective cytotoxicity assays demonstrated that magnetic hyperthermia caused statistically significant cancer cell death, with biological response depending on cancer subtype. SAR values scaled proportionally with field amplitude and inversely with nanoparticle concentration. The authors explicitly characterize this work as a first proof-of-principle in vitro demonstration of effective intracellular magnetic hyperthermia, while emphasizing that further optimization is required before in vivo translation.

(iii)

Spyridopoulou et al. [31] studied the influence of magnetic fields, generated using a similar experimental device, on the growth of magnetite nanoparticle-treated HT29 colon cancer cells. The nanoparticles consisted of aqueous magnetite dispersions with an average hydrodynamic diameter of 100 nm and a core size of approximately 85 nm, comprising several single-domain crystallites—thus classifying them as multidomain particles. Each nanoparticle featured a magnetite core coated with a hydrophilic starch polymer to prevent aggregation. The maximum uptake was 11 pg Fe per cell, corresponding to $N=2.25\times {10}^4$ nanoparticles per cell, assuming a magnetite density of $5200 {\mathrm{kg}/\mathrm{m}}^3$. The field-mode-dependent biological responses are demonstrated in the absence of measurable temperature rise, with static and rotating fields inducing growth inhibition and alternating fields promoting frequency-dependent proliferation. This study provides quantitatively defined intracellular nanoparticle uptakes together with well-characterized magnetic field protocols and distinct biological endpoints, thereby enabling a consistent physical estimation of magneto-mechanical forces per cell and supporting the relevance of the present numerical simulations for both hyperthermic and non-thermal magneto-mechanical cancer treatment concepts.

Finally, magneto-mechanical forces acting on internalized MNPs under a rotating magnetic field were calculated using finite element simulations in COMSOL Multiphysics. A time-varying magnetic field was imposed through a rotating mesh configuration to mimic the experimental actuation conditions. Five representative spatial points within the culture well were selected, corresponding to characteristic positions previously analyzed in the device. For each point, the magnetic field gradients and resulting magnetic forces on MNPs were computed over time, assuming literature-based intracellular particle loadings. The resulting force-time signals were exported and post-processed in MATLAB (R2023a). A Fast Fourier Transform (FFT) was applied to quantify the dominant frequency components of the magneto-mechanical stimulus, enabling validation of force periodicity induced by the rotating magnetic field.

3. Results

3.1. Magnetic Field

All the magnetic configurations geometries implemented in COMSOL are illustrated in Figure 1 and in Figures S1–S10 of the Supporting Material A, along with the numerical results for |B|, B direction and |∇(B)| measured at the level coordinates, where the hypothetical cell culture Petri dish (∅: 3.5 cm) is positioned each time. In the main text, we present the configuration that generated the highest forces for all the aforementioned case studies, while all the other 2D and 3D field configurations are presented in the Supporting Materials A. To evaluate our results, we compared the |B| values to the corresponding experimental ones, which were measured with the Hall probe at the different measuring points for the precise Petri dish location shown in Figure 1a. The results are shown in

Table 1. Experimental measurements of total flux density |B_exp| in 5 selected points and their comparison with the numerically resulted |B_the| from COMSOL calculations. Values of the gradient field magnitude |∇(B)| are also presented.

Position	|Bexp| (mT)	|Bthe| (mT)	Deviation (%)	|∇(B)| (T/m)
1	85	86	1	20
2	39	38	3	0
3	75	80	7	20
4	28	25	11	5
5	31	35	13	5

, where the values of gradient field measure |∇(B)| are also calculated at the aforementioned measuring points. The mesh consists of 2.86 × 10⁵ elements, while the values of the dependent variables were solved for 3.13 × 10⁵ degrees of freedom (DOFs). Henceforth, all the magnetic field configurations, simulations and evaluations follow the same sequence of calculations and are presented in Tables S1–S10 of the Supporting Materials A section. The arrays of magnets are positioned in two different slots, as shown in Figure 1a,b. If we inverse the polarity of the magnets in one of the two slots, an alternating field is generated with a frequency value that is the half of the simple rotating mode. To quantitatively characterize the magnetic field non-uniformity, the spatial distribution of the magnetic flux density and its gradient were computed numerically over the entire Petri dish surface and volume. The resulting gradient maps allow direct estimation of the spatially varying force amplitudes experienced by MNP-loaded cells across the culture area.

In order to evaluate our results, we compared the |B| values to the corresponding experimental ones which were measured with the Hall probe at the 5 measuring points depicted in Figure 1a for the precise Petri dish location. The results are shown in Table 1 where the values of the gradient field measure |∇(B)| are also calculated at the aforementioned measuring points. The mesh consists of 1.56 × 10⁶ elements, while the values of the dependent variables were solved for 9.54 × 10⁵ degrees of freedom (DOFs).

3.2. Magnetic Force

Then, by importing Equations (10)–(14) into COMSOL, we obtained the spatial distribution of the maximum force magnitude generated by the magnetic field configurations shown in Figure 1 and in Figures S1–S10 of the Supporting Materials A. The analysis was performed for three distinct case studies, denoted as J, C, and S, corresponding, respectively, to the works of Jordan et al. [29], Chalkidou et al. [30], and Spyridopoulou et al. [31]. In case study (J), the magnetic susceptibility is given by importing data of magnetic measurements reported in [29] in Equation (13) while in (C) and (S), we assume that ${\chi }_p$ >> 1. Each force distribution is presented in Figure 2 and in Figures S11–S20 of Supporting Material B, along with the corresponding configuration geometry in order to have a clearer view of the working space. For each configuration, the maximum force is then presented, and it corresponds to the maximum force value obtained in the working space. This working space is the area occupied by a typical cell culture Petri dish and depicted in our models in Figure 1 and in Figures S1–S10 of Supporting Material A.

For each configuration, the maximum force is then presented in

Table 2. Maximum force estimation for the experimental conditions of the three case studies used.

Work	Cell Line	MNPs	Size/ Morphology	N (MNPs per Cell)	Ft,max (pN)	Biological Outcome
Jordan et al. [29]	BT20	SPM 13 nm Fe3O4 dextran-coated	~13 nm, quasi-spherical	1.1 × 108	3000	magnetic hyperthermia induced cell necrosis
Chalkidou et al. [30]	MCF7	Fe/MgO core–shell, single-domain	~75 nm, spherical	1.4 × 105	2400	magnetic hyperthermia induced cell apoptosis
Spyridopoulou et al. [31]	HT29	Fe3O4 multidomain clusters, starch-coated	~85–100 nm, anisotropic composite morphology	2.25 × 104	2100	clear MM-driven cell membrane disruption; increased mechanotransduction under rotational and alternating MF

, and it corresponds to the maximum force value obtained in the working space. The first four columns of the Table at the bottom of this Figure summarize the in vitro experimental data used for simulations, while in the last column, the maximum values of ${|\mathbf{F}}_{\mathrm{t}}|$ obtained within the area occupied by the cell culture Petri dish—that is, the region of interest—are presented for the case studies considered here.

In Figure 3a, the force-time plot demonstrates oscillatory magneto-mechanical behavior across all five characteristic positions within the device. Although force amplitudes differ due to spatial variations in the magnetic field gradient, each signal exhibits a cyclic pattern with consistent periodicity, confirming that the rotating magnetic field generates alternating mechanical inputs at the cellular level. The FFT spectrum, presented in Figure 3b reveals a strong and narrow dominant frequency peak centered around the driving field frequency (~45 Hz), with no notable harmonic distortions.

This result indicates that the applied rotational field results in a clean, well-defined alternating force stimulus on the MNPs. Such reproducible and frequency-controlled magneto-mechanical signals are crucial for establishing predictable cellular stimulation conditions. The observed force magnitudes fall within ranges reported to activate mechanotransduction pathways, suggesting that this system can be used to explore frequency-dependent cellular responses to controlled magnetic forces.

To establish a rigorous and reproducible link between the applied magnetic field conditions and the resulting magneto-mechanical forces acting on magnetic nanoparticles (MNPs), a systematic numerical characterization of all three magnetic field configurations—static, rotating, and alternating—was performed. The alternating field refers to the field rotation but this time the magnets have different polarity per slot. Rather than focusing solely on the configuration yielding the maximum mechanical force, the present analysis provides a full quantitative comparison of the magnetic flux density and its spatial gradients for each field type within the effective volume of a standard culture Petri dish. For each configuration, key physical parameters including the maximum, average, and root mean square (RMS) magnetic flux density are reported, together with the corresponding frequency content for time-varying fields. In addition, the spatial distribution of magnetic field gradients is evaluated in order to directly quantify the force-generating capability of the device. This comprehensive parameterization enables direct quantitative correlation between the applied magnetic field conditions and the resulting biological stimulation and provides a reproducible physical framework for future magneto-mechanical studies.

Table 3. Global magnetic flux density parameters for the three magnetic field configurations within the Petri dish volume.

Field Configuration	Bmax (mT)	Bavg (mT)	BRMS (mT)	FFT Peak Frequency (Hz)
Static	86	60	-	0
Rotating	86	-	43	45
Alternating	86	-	43	22.5

summarizes the global magnetic flux density characteristics of the three magnetic field configurations—static, rotating, and alternating—within the effective volume of the cell culture Petri dish. For each configuration, the maximum ($B_{\max }$), average between the five points for the static case ($B_{\mathrm{avg}}$), and root mean square ($B_{\mathrm{RMS}}$) values for the rotating and alternating cases of the magnetic flux density modes are reported, together with the corresponding dominant frequency components extracted from the FFT analysis of the time-dependent magneto-mechanical force signals for the rotating and alternating magnetic field configurations are depicted in Table 3. In the static case, the RMS value coincides with the DC field magnitude, whereas for the time-varying rotating and alternating fields, the RMS values account for the temporal modulation of the magnetic flux density. This table provides a direct quantitative comparison of the field strengths generated under each operating mode of the device and establishes a well-defined physical basis for correlating magnetic field exposure conditions with the resulting magneto-mechanical forces and subsequent biological responses.

The statistical characterization of the spatial gradients of the magnetic flux density within the effective culture volume for the three magnetic field configurations is presented in

Table 4. Statistical parameters of the magnetic flux density spatial gradient within the cell culture volume.

Field Configuration	$\mathsf{\nabla }\mathit{B}$max (T/m)	$\mathsf{\nabla }\mathit{B}$avg (T/m)	$\mathsf{\nabla }\mathit{B}$RMS (mT)
Static	20	10	-
Rotating	20	-	7
Alternating	36	-	25

. For static operating mode, the maximum and average values of the magnetic field gradient, $\nabla B$, are reported, while for rotating and alternating fields, the RMS value is also reported.

From Table 4, it can be seen that the change in polarity results to higher gradients due to the induced field inhomogeneity. Since the magneto-mechanical force acting on magnetic nanoparticles is directly proportional to the magnetic field gradient, these parameters provide the most relevant physical link between the applied magnetic field and the resulting mechanical stimulation at the cellular level.

4. Discussion

To evaluate the resulting magneto-mechanical forces in vitro, we present the following table, which displays the activation threshold forces for various cell responses. By comparing those values to the obtained force distributions, calculated above for the various configurations, we see that all studied setups, simulated and calibrated here, generate Magneto-mechanical forces that safely overcome all the effects thresholds reported in

Table 5. Thresholds for the effects of magneto-mechanical forces.

Effects	Threshold Forces (pN)	References
Diffusion of ions and biologically relevant molecules in solutions	102–103	[32]
Magnetically assisted cell migration and positioning	102–103	[33]
Change the membrane potential	103–104	[34]
Local change in membrane potential	103–104	[34]
Change the probability of channel switch on/off events	102–103	[35]
Tumor cells arrest	103–104	[34]
Magnetically assisted cell division	102–104	[34,36]
Change the differentiation pathway and gene expression	1–100	[37]
Magnetically assisted endocytosis	10–100	[38]
Cell swelling	100–200	[35]

.

It is evident that magneto-mechanical forces arising from the magnetokinetic behavior of endocytosed magnetic nanoparticles (MNPs) under externally applied magnetic fields, as presented and simulated in this study, can induce a variety of cellular responses and biological processes of significant biomedical relevance. In summary, this work introduces a custom-designed device developed to investigate magneto-mechanical effects in MNP systems. Numerical simulations were performed to quantitatively characterize multiple magnetic field configurations generated by the apparatus, and the simulated magnetic field amplitudes were validated through experimental measurements. Furthermore, based on the device geometry, the spatial gradients of the magnetic flux density were numerically computed within the volume of a standard cell culture Petri dish to estimate the resulting mechanical forces acting on cells.

By combining these calculated field configurations with literature-derived data on intracellular MNP uptake across different magnetic regimes, the study provides estimates of force magnitudes experienced by single cells under various experimental conditions. The simulations reveal a broad range of force intensities, demonstrating the tunability and versatility of the proposed setup. Overall, this easily operable and adaptable system, together with its corresponding “force-per-cell” mapping, offers a valuable reference framework for future research on magneto-mechanical phenomena in cellular environments, where MNPs subjected to controlled magnetic field variations may initiate or regulate biologically relevant processes.

It is important to note here that the magneto-mechanical response of magnetic nanoparticles is strongly dependent on their shape, magnetic regime, and surface physicochemical properties, all of which critically influence torque generation, force transmission, cellular internalization, and intracellular mechanical coupling. In the study by Jordan et al. [29], the nanoparticles were quasi-spherical, dextran-coated superparamagnetic magnetite with a small core diameter (~13 nm). This geometry and magnetic regime favor Néel relaxation over Brownian rotation, resulting in dominant thermal dissipation and very limited mechanical torque or force transmission at low frequencies, which explains the poor magneto-mechanical efficacy and the rapid intracellular aggregation observed. In contrast, the Fe/MgO core–shell nanoparticles used by Chalkidou et al. [30] consisted of large, nearly spherical single-domain iron cores (~75 nm), a size range that supports strong magnetic moments and enhanced mechanical coupling under alternating magnetic fields; however, their spherical symmetry limits directional torque efficiency, rendering their primary biological action hyperthermia-driven rather than mechanically driven. Finally, in the work of Spyridopoulou et al. [31], the magnetite nanoparticles were multidomain with a hydrodynamic diameter of ~100 nm and an irregular, non-ideal spherical morphology arising from clustered single-domain crystallites forming a composite core. This structural anisotropy, combined with their larger magnetic moment and Brownian rotational freedom, is particularly favorable for magneto-mechanical actuation under rotating and alternating low-frequency fields, enabling efficient conversion of magnetic excitation into mechanical forces at the cellular membrane and cytoskeleton. Additionally, surface coatings differ significantly among the three systems, ranging from dextran to MgO and starch, affecting colloidal stability, intracellular trafficking, and mechanical coupling to cellular structures. These differences justify the wide span of calculated force magnitudes in the present study and confirm that nanoparticle geometry, magnetic domain structure, and surface chemistry must be considered jointly when interpreting magneto-mechanical force predictions and their associated biological effects.

Recent experimental findings further support the importance of nanoparticle morphology in magneto-mechanical actuation [39]. While theoretical models often predict that spherical nanoparticles should exhibit the highest mechanical torque in alternating and rotating low-frequency magnetic fields, experimental evidence demonstrates that anisotropic—particularly rod-shaped—nanoparticles can outperform spherical ones in magneto-mechanical efficacy. As reported by Prishchepa et al. [39], elongated particles display enhanced sensitivity to field configuration and can undergo oscillatory motion resembling a propeller-like rotation in rotating low-frequency magnetic field, resulting in stronger deformation of biomolecules and more efficient mechanical energy transfer. These observations are fully consistent with the differences among the three nanoparticle systems discussed in the present work [39] and further underscore the necessity of considering nanoparticle shape when interpreting magneto-mechanical force predictions.

Beyond the in vitro force estimations and field–force mapping provided in this study, the presented framework also highlights clear pathways for transitioning magneto-mechanical actuation toward preclinical investigation. A critical next step involves the systematic characterization of MNP behavior within tissue-like environments, where viscosity, extracellular matrix constraints, and heterogeneous nanoparticle uptake may significantly modify the effective forces exerted on cells. Integrating these biological variables into physics-based models will enable the prediction of magnetomechanical force thresholds at the tissue level and guide the establishment of exposure protocols suitable for small-animal studies. Equally important is the development of magnetic field applicators capable of generating spatially localized gradients at centimeter-scale penetration depths, while retaining the tunability and non-thermal characteristics demonstrated in vitro. Coupling such field delivery systems with targeted or stimuli-responsive MNP formulations could facilitate selective mechanical stimulation of malignant tissues or other pathological microenvironments in vivo. Taken together, these considerations outline a realistic and technically achievable pathway from controlled laboratory experiments to the first preclinical demonstrations of magneto-mechanical stimulation as a non-thermal biomedical intervention.

5. Conclusions

The magneto-mechanical effect was investigated as a promising non-thermal yet spatially localized mechanism for transferring magnetic energy to cellular membranes and other biological structures in proximity to MNPs. To achieve this, a novel, custom-built, and fully tunable three-dimensional device was designed, calibrated, and implemented to generate magnetic fields with adjustable amplitudes and frequencies. Various magnetic field configurations produced by the system were quantified and analyzed, along with the corresponding mechanical forces induced on MNPs exhibiting different magnetic characteristics under distinct field gradients.

The in vitro application of this device enables the controlled exposure of cancer cells to well-defined magnetic fields, thereby inducing mechanical stress through the motion of intracellular MNPs. Cellular uptake data obtained from previous in vitro studies—covering both magnetic hyperthermia and magneto-mechanical regimes—were incorporated to estimate the mechanical forces experienced by single cancer cells. This approach provides a quantitative means to correlate magnetic field parameters (amplitude and gradient) with the resulting intracellular mechanical forces. It is emphasized that the magnetic field over the Petri dish surface is not uniform by design, since spatial magnetic field gradients are a necessary condition for the generation of translational magneto-mechanical forces acting on magnetic nanoparticles. While a spatially uniform magnetic field can only induce rotational torque on nanoparticles, the presence of magnetic field gradients enables the development of net mechanical forces responsible for cellular stimulation. For this reason, the spatial non-uniformity of the magnetic field is a functional feature of the device rather than a limitation.

Overall, the mechanical manipulation of MNPs proposed in this study offers an alternative to conventional magnetic hyperthermia for harnessing MNP–field interactions and converting magnetic energy into biologically relevant mechanical stimuli. The estimated forces, together with a comparative assessment of single-cell heating effects, demonstrate that the magneto-mechanical approach constitutes an effective and controllable pathway for actuating and probing cellular processes within living systems.

Magneto-mechanical forces generated by the response of endocytosed MNPs to externally applied magnetic fields can induce a wide range of cellular reactions and processes at the single-cell level, many of which hold significant biomedical potential. Building upon the findings presented in this study, it is proposed that by establishing controlled and reproducible experimental conditions, predictive models of cellular behavior under magnetic field exposure can be developed. Such a priori simulations are essential for elucidating, quantifying, and ultimately controlling the key parameters that govern cellular responses to magnetic fields and MNP–field interactions.