Next Article in Journal
Deep Learning-Based Denoising for Interactive Realistic Rendering of Biomedical Volumes
Previous Article in Journal
Enhanced Sensitivity of 17-α-Ethinylestradiol (EE2) Detection Using Carbon Quantum Dots-Integrated Tapered Optical Fiber
Previous Article in Special Issue
Lean Management Framework in Healthcare: Insights and Achievements on Hazardous Medical Waste
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Applied Sciences
Volume 15
Issue 18
10.3390/app15189892
applsci-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Eddy Current Distribution in Magnetotherapy of Bones: A Qualitative and Quantitative Study

by
Przemyslaw Syrek
1,*,
Mikolaj Skowron
1 and
Piotr Kapustka
2
1
Department of Electrical and Power Engineering, Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, AGH University of Krakow, 30-059 Krakow, Poland
2
Department of Automation and Robotics, Polytechnic Faculty, University of Applied Sciences in Tarnow, 33-100 Tarnow, Poland
*
Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(18), 9892; https://doi.org/10.3390/app15189892
Submission received: 1 August 2025 / Revised: 7 September 2025 / Accepted: 8 September 2025 / Published: 9 September 2025
(This article belongs to the Special Issue Innovative Approaches and Tools for Healthcare and Medical Applications)
Browse Figures
Versions Notes

Abstract

Featured Application

The methodology and results presented in this study serve as a preliminary step toward the development and personalization of magnetotherapy parameters based on the anatomical characteristics of individual patients’ limbs. Additionally, the visualizations of eddy current distributions within the human body may be used as an educational tool to help patients better understand the principles constituting this therapeutic approach.

Abstract

Unlike pharmacological and surgical methods, electrical and especially magnetic stimulation are of interest due to their noninvasiveness and high tolerability by patients. Ensuring the repeatability of procedures and achieving the highest possible homogeneity of the eddy current distribution appears to be crucial, particularly in the context of potential clinical trials. This highlights the qualitative aspect of the therapy. Equally important, however, are the outcomes in terms of eddy current generation and their presentation in a psychological context, particularly in relation to patient communication. Many patients undergoing treatment express a desire to understand how the applicator works, what the procedure does to their body, and what sensations or effects are occurring in their limbs during therapy. On the other hand, quantitative analysis enables the rescaling of the magnetic field induced within the applicator, which in turn allows for determining the appropriate level of induced currents in the limb of a specific patient.

1. Introduction

Interest in the interaction of magnetic fields with the human body—and efforts to apply them in various forms, frequency bands, and for different medical conditions—has continued for several decades [1,2,3]. The use of a variable magnetic field to induce an electric field—even in a poorly conductive environment—results in conduction currents and is applied, for example, in transcranial brain stimulation (TMS) and magnetotherapy of wounds and bones. In the case of fractures, where it is generally necessary to concentrate the current in the immediate vicinity of the injury, rehabilitation commonly involves the use of a single applicator. However, in certain situations—such as when accelerating processes like remineralization across the largest possible area of a long bone—a group of applicators may prove more effective (according to specific criteria) than a single, strategically positioned one. The issue of current density uniformity arises in diseases, like osteoporosis, whose negative effects are particularly associated with age. Within the long bones, the process of remineralization is enhanced by the pressure and the effort imposed on them itself [4,5]. Diet and the ability to absorb nutrients are another initiators of ill bone composition. Osteoporosis is a progressive skeletal disease, manifested in a decrease in bone mass. Osteoporosis has been called a silent epidemic [6].
Contemporary research focuses on multiple aspects such as diagnostics and the detection of tissue deficits [7,8] and the influence of electric fields on bone mineral content [9]. Considered magnetic and electric components of electromagnetic fields (EMFs) are of 50 Hz, due to their wide use ingenerating devices and their proven effectiveness [10]. Magnetotherapy is applied in a variety of conditions and is not limited to bone fractures alone, being used for dysphagia [11], phrenic nerves [12], and wound healing.
The discovery and first publications on the relationship between bone healing and the electrical or magnetic component field date back to the 1950s. Examples of studies and interactions have been compiled in [13,14]. The influence of magnetic component parameters on ossification is reported in [15]. In the study [16], publications concerning the effects of magnetic fields on bone healing were compiled, both in animals and humans. Very interesting results were presented in [17], where the authors reported beneficial effects in patients (63 persons) undergoing magnetic field therapy (the parameters here were as follows: 0.5–2 mT, 5–105 Hz, sinusoidal waveform) in contrast to sham treatments.
The magnetic field distribution can be relatively easily computed using numerical methods, and the field patterns within and around the applicator coils can be represented and examined [18]. The magnetic field component can, in turn, be reconstructed from measurement data. The situation is entirely different in the case of the electric component, which is induced by a time-varying magnetic field and results in a specific distribution of eddy currents within the human body. Here, as in transcranial magnetic stimulation, calculating this distribution requires the use of appropriate field models. The complexity of these models depends on the electrical properties of the region under consideration. For instance, the presence of metallic implants necessitates the use of different modeling approaches [19] compared to those used for so-called low-conductivity regions, i.e., areas with low electrical conductivity, such as biological tissues. In most cases, the fields involved are of low frequency, which in turn allows for the omission of the so-called secondary magnetic field induced by the generated eddy currents. This is one of the fundamental assumptions. If they are not of low frequency, the analysis increases in complexity [20]. In limb magnetotherapy, solenoidal coil applicators are most commonly used, with sizes adjusted to match the dimensions of the limbs; larger coils are typically used, for example, in veterinary treatment. In the case of transcranial magnetic stimulation (TMS), various coil shapes are employed [21], including coil arrays [22]. In both cases, ongoing research aims to identify coil geometries—or specific arrangements—that can produce the desired eddy current distributions. Computational complexity and attempts to individualize therapy are already leading to efforts to apply modified U-Net architecture [23] or deep learning models [24] to predict individual TMS-induced electric fields in the brain.
The quantitative results presented in this article refer to the distributions of the magnetic field and eddy currents obtained within a selected limb model. These results are essential for demonstrating the magnitude of the induced currents under specific magnetic field conditions. Moreover, they can play an important role in patient communication—explaining what occurs in the body during therapy, which relates to the so-called psychological aspect of treatment.
The qualitative results presented in the study concern the average current values obtained in short, two-centimeter segments of bone. These results can provide general insight into the uniformity of an object’s (in this case, bone) response to magnetotherapy. Ensuring both the repeatability of treatment and the uniformity of the eddy current response is also important from the perspective of conducting future clinical studies on the effects of induced currents on bone healing, inflammation in bone-adjacent regions, and related phenomena.

2. Materials and Methods

The left arm of a 26-year-old female subject was selected for the analysis, which included information on her height and body mass. The model is based on the Virtual Family dataset, made available by the IT’IS Foundation. The available models include the following: Duke (34-years old male), Billie (11, female), and Thelonious (6, male).
The choice of the 26-year-old female model was intended as a representative adult case for illustrating the simulation methodology rather than for clinical generalization. In future work, at the stage of clinical research, we plan to create numerical models tailored for simulation-based computations for each individual patient.
The modeled region begins at the humerus, 50 mm above the elbow, and extends to the fingertips. The total arm length assumed for the calculations was 0.4 m; however, the results presented correspond to a 0.16 m segment of the full limb. The anatomical structure of the arm, along with the segmented tissues comprising it, was obtained from the mentioned database provided by [25]. Based on a spatial resolution of 2 mm, the analyzed domain consists of 153,905 elements. The relative proportions of each tissue type, along with their respective electrical conductivities, are listed in Table 1.
To prevent any physical overlap between the arm model and the applicator housing, the mesh was applied to the body using the iso2mesh open-source mesh rendering software [26]. While the magnetic field distribution can be homogenized using various approaches, one of the key aspects in the analysis of eddy current distribution is the assignment of electrical conductivity values to individual tissue types [27,28]. The tissue parameters of the limb were assigned according to [29].
Table 1. Conductivity of tissues according to [29].
Table 1. Conductivity of tissues according to [29].
TissueElectric Conductivity (S/m)Share (%)
Artery0.2610.53
Fat0.01958.65
Muscle0.23348.09
Bone0.0205.67
Skin0.000212.74
Subcutaneous fat0.019522.76
Vein (blood vessel wall)0.2610.60
Bone marrow red0.00160.95
An analysis of the limb’s anatomical structure in relation to the assigned electrical conductivities reveals significant differences between tissue types. Muscle tissue, which constitutes nearly half of the total volume, exhibits a conductivity of approximately 0.233 S/m. In contrast, bone tissue—which is critical for ensuring adequate current flow during treatment or for preventive purposes—has an electrical conductivity more than ten times lower and accounts for over 5% of the analyzed region’s volume.
The distribution of eddy currents depends on the spatial variation in the time-varying magnetic component of the external magnetic field. While the magnetic field distribution, including its amplitude and frequency, can be modeled, the resulting current distribution is influenced not only by the magnetic field itself but also by the spatial structure of the limb—namely tissue conductivities and the spatial positioning of specific points within the domain [30]. This complexity is further compounded by the fact that, even within regions of relatively high conductivity, current densities may approach zero. This occurs because, in every region, it is possible to identify curves (sets of points) where the induced current is negligible or even zero—these can be interpreted as axes around which eddy currents circulate. Moreover, this effect intensifies near the boundaries of the domain. For instance, if a limb is placed in a homogeneous magnetic field, even when the electrical conductivities of bone and subcutaneous fat are similar, the current density within the bone—especially when located near the central axis of the limb—will be significantly lower than in adjacent tissues near the surface (e.g., the skin).

3. Magnetic Flux Density Distribution

Although the magnetic vector potential is needed to obtain the eddy current distribution, a magnetic flux density is the quantity which is measured. The excitation coil of the Magnetronic applicator (Figure 1a) was used both for the measurements and for the development of the numerical model. The generated magnetic field is adjustable within the range of 0 to 20 millitesla. The device is employed in magnetotherapy treatments. All analyses were conducted for a sinusoidally varying magnetic field with a frequency of 50 Hz.
To improve the clarity of the presented results, the main axis of the applicator was aligned with the vertical (z) axis of the coordinate system (Figure 1b). Specifically, the height of the applicator is set to h = 0.22 m, and the inner radius is RIN = 0.145 m. The eddy current distribution is obtained in a numerical model based on a 0.4 m section of the limb—extending significantly beyond the coil itself. This provides a substantial margin that goes well beyond the region in which the results are analyzed qualitatively—namely, a 0.16 m segment of bone.
Therefore, to perform calculations of the eddy current distribution, all simulation parameters have to be compatible with the device. Here, we use a magnetic field applicator and a solenoidal coil. In the first step, measurements of the magnetic field distribution inside the solenoidal coil were carried out.
The results are presented in Figure 2. The current supplied to the coil was selected so that the simulated magnetic field distribution matched the measurement results obtained inside the solenoid. The figure shows the field distribution on the surface of the arm (whose position remains unchanged in the eddy current calculations) as well as within a selected section of the y = 0 plane.
To complement the presented data, the z-component of the magnetic field (the dominant quantity inside the applicator) is shown in Figure 3a. Here, non-negative values indicate vector directions aligned with the z-axis of the coordinate system. The x-component is shown in Figure 3b—this component constitutes only a fraction of the z-component, and its vector direction depends on the distance from the coil windings. Consistently, with the presented plane (z = 0) kept unchanged, it should be noted that the y-component is negligibly small.
Then, the simulation results were compared with the measurement results. The table summarizes the measurement and simulation results. Here, the z-component of magnetic field is presented.
It can therefore be concluded that the simulation parameters have been calibrated correctly in relation to the real device what presents Figure 4. The absolute value of the magnetic component reaches 17 mT. The distribution of eddy currents in the limb will be computed based on the obtained magnetic field component distribution.

4. Eddy Currents in Human Bodies

The quantities that can be measured are presented in the previous paragraph. This is where field distributions can only be calculated through simulations. Or, in very simple cases, calculated analytically with a basis on more or less complex field models [31]. Eddy current distributions belong to these groups. At this point, it is also worth mentioning the wide range of available field analysis methods [32] depending on the complexity of the issue being considered.
In this case as well, depending on the complexity of the analyzed region, various approaches are possible. In certain cases, the problem can be reduced to the computation of integral formulas [33,34]. In others, where it is necessary to account for and distinguish between the electrical conductivities of individual tissues, the following pair of potentials are involved to obtain eddy current distribution: the vector magnetic potential and the scalar electric potential. And solving Poisson’s equation [35], after the boundary conditions are set, we obtain the following:
∇·(γ∇V) = −∇ · (γ ∂A/∂t)
where V—electric scalar potential, A—magnetic vector potential [36], and γ—electric conductivity.
Details regarding the formulation of the boundary conditions are provided in [37]. The density of eddy currents is a superposition received from the (1) distribution of V, and the distribution of the magnetic vector potential (its time derivative):
J = −γ∇V − γ (∂A/∂t)
Assuming that all results, i.e., J, V, A, reach aRMS value. Theresults presented in the article are absolute values of density current vectors. The problem is solved with the Finite Difference Method in a self-developed solver implemented in MATLAB R2021b.
It is assumed that the analyzed region, including the human limbs, contains no metallic implants and, in particular, no components with ferromagnetic properties. Consequently, the entire constructed model is linear, which allows for the scaling of the system’s response.
Additionally, a characteristic feature of eddy currents in the human body is that they induce a so-called secondary magnetic field, which is negligible compared to the external magnetic field (induced by external coil). Under these conditions, it is not necessary to adjust the calculated distribution of eddy currents (also referred to as secondary currents) induced by the secondary magnetic field.
It should be emphasized that the model considered here is linear, owing to the absence of metallic elements, especially those characterized by high magnetic permeability. This implies that the model preserves scalability; that is, scaling the coil current results in a proportional scaling of the magnetic field distribution, which in turn leads to a proportional scaling of the eddy current distribution.

5. Results

5.1. Quantitative Analysis

Figure 5a presents the resulting distribution of eddy current density in the limb. The distribution is shown for several selected cross-sections. With a magnetic field locally reaching up to 17 mT, the induced current density in the arm reaches up to 28.7 mA/m2. Even a preliminary analysis of the results reveals that the highest current densities occur in muscle tissue, and these values increase with the distance from the central (longitudinal) axis of the limb. This is, of course, due to the high electrical conductivity of muscle tissue compared to other tissues present in the analyzed region.
In contrast, in the bones—namely the ulna and the radius—the current density reaches up to 3.80 mA/m2, as shown in Figure 5b. It is important to emphasize that this corresponds to approximately 13% of the maximum current density observed in the limb. Thus, regardless of the applied magnetic field strength, or even the frequency (as long as we remain within the low-frequency range), this ratio remains constant. This is a significant observation, as it provides a basis for estimating safe or permissible current levels in bone tissue based on the total induced current in the limb.
Naturally, the investigation of methods for applying the magnetic field in such a way as to maximize the ratio of current density in bone tissue relative to other tissues remains an open research question.

5.2. Qualitative Analysis

Figure 6 presents the average eddy current density values in the bone, measured along successive two-centimeter segments from top to bottom. The highest values are observed in the central sections, which can be attributed to the distribution of the magnetic field component—these are maximal in the center of the coil. The results clearly show that the current density decreases with increasing distance from the central segments.
However, it is noteworthy that over a segment as long as 16 cm, the average values do not fall below 50% of the value in the central segment. This should be considered very promising, especially given the relatively simple construction of the applicator.
Considering bone-related conditions that affect the entire length of the bone, maintaining a uniform current density distribution becomes important. At this point, it is worth considering and proposing concepts that could improve the homogeneity of such a distribution across different sections—or ideally, across the entire length of the bone.
The most intuitive idea is to design an applicator with significantly greater depth (referring to height h, as shown in Figure 1b). However, this solution is not sufficient, primarily because the arm cannot be inserted too deeply without exposing the chest to the magnetic field.
A second, more promising approach is to divide the coil windings into independently powered sections, allowing control over the magnetic field distribution inside the applicator. Such a strategy requires further, extended research, including optimization procedures, where the evaluation criterion may be the uniformity of current density in different segments of the bone.

5.3. Specific Absorption Rate and ICNIRP Limitations

The Specific Absorption Rate (SAR), defined as (3), is negligibly small in the case of magnetotherapy using extremely low frequency (ELF) electromagnetic fields.
The SAR (Specific Absorption Rate) depends on the electric field (or current density) and the mass density of the tissue and tissue’s electric conductivity. Since the relation between current and electric field expresses the formula j = γE, SAR may be expressed in a form involving eddy current and also as follows:
SAR = γE2/ρ = j2/γ ρ
where γ—electric conductivity and ρ—tissue mass density (in kg/m3).
Assuming the tissue mass densities as provided by ITIS [38], at the levels of 1090, 1908, and 911 kg/m3 for muscles, bone, and SAT, the obtained SAR values are, respectively, as follows: 16.5 × 10−6, 0.38 × 10−6, and 0.13 × 10−6 W/kg. It is therefore negligibly small, especially considering that even at radio frequencies, the SAR limit for the general public is 0.08 W/kg, according to the International Commission on Non-Ionizing Radiation Protection guidelines [39].
According to the same commission (ICNIRP), the exposure limit at 50 Hz for the general public is set at 0.2 mT [40]. Therefore, the magnetic field values used in medical procedures significantly exceed the limits recommended for everyday exposure. However, this is justified, as each therapy is carried out under medical supervision, and every patient is individually qualified for the treatment.

5.4. Coil Arrays

In this section, we present the results obtained for a group of applicators. Four applicators were used. It is worth noting that even with such a small group, there is already a wide range of possibilities regarding their positioning, the amplitude, and the initial phase of the sinusoidal current.
In such a case, a complex optimization process is required, starting with the definition of an objective function. One possible criterion is the ratio between the average eddy current distribution in the bones and the maximum current value observed within the entire analyzed region.
Out of many arbitrarily chosen applicator configurations, four arrangements yielding the best results according to the proposed objective function were selected. In this case, identical coils powered with the same current amplitude were used; 16 current direction configurations (in-phase and anti-phase) were considered. The most favorable configuration, with the indicated current direction, is shown in Figure 7a.
It should be emphasized that the process of selecting the position, coil structure, and even the current amplitude requires the use of evolutionary algorithms and highly computationally demanding calculations. This represents a very promising direction for further research.
Figure 8a presents the resulting distribution of eddy current density in the limb. The distribution is shown for several selected cross-sections. With a magnetic field locally reaching up to 17 mT, the induced current density in the arm reaches up to 28.6 mA/m2.
In the bones—namely the ulna and the radius—the current density reaches up to 2.65 mA/m2, as shown in Figure 8b. This corresponds to approximately 9% of the maximum current density observed in the limb. This constitutes the first difference in relation to the single large applicator described in the article. The remaining differences include strong, localized eddy current concentration in the muscle tissue and clearly lower (averaged) current values in the bones, as shown in Figure 7b.

6. Discussion

The excited magnetic field inside the applicator is predictable. The presented results show that the distribution of the magnetic field vector magnitude is locally uniform. In contrast, the distribution of eddy currents behaves quite differently. The induced current distribution depends not only on the magnetic field distribution and frequency (i.e., the dynamics of field variation), but also on the electrical conductivity and geometry of the conductive region.
As a result, significant differences in current density can even be observed locally—for example, within a single cross-section. The current density depends on the conductivity of a given tissue, but also on its position and the types of tissues it borders. In simplified terms, it can also be said that the eddy current density tends to increase near the outer layers of the limb.
Recent work by Prattico [41] demonstrated the effectiveness of combining U-Net and LSTM architectures to model time-varying electromagnetic absorption in dynamic media. This approach could be adapted to predict patient-specific eddy current distributions during therapy, accounting for anatomical changes or motion. Integrating such spatiotemporal deep learning models with the current framework may enable real-time adaptive simulation and optimization of magnetotherapy protocols.
Future work may explore the use of SPAD-based optical sensors to monitor physiological responses, such as tissue perfusion or metabolic activity, during magnetotherapy. Such integration could support real-time feedback and validation of simulation results.
Although the present study did not incorporate surface electromyography (sEMG), the high current densities observed in muscle regions suggest potential neuromuscular activation during therapy. Future research may integrate sEMG acquisition and AI-based analysis to correlate simulated current distributions with functional muscle responses, providing experimental validation and physiological relevance to the numerical results.
A potential extension of the presented model could involve hybrid stimulation scenarios, such as the combination of low-frequency magnetotherapy with ultrasound-based techniques. This multimodal approach has been suggested in the literature [42,43] as potentially synergistic in promoting bone regeneration. While not addressed in the current study, future work may adapt the numerical model to simulate the spatial and temporal interactions of both magnetic and acoustic fields within bone tissue.
While the present study does not explicitly optimize magnetic stimulation parameters for osteogenic outcomes, the modeling framework can be extended to support such objectives. Incorporating biological response data—such as intensity thresholds for promoting osteogenesis—would allow for a more targeted design of applicators and stimulation protocols. Future research may focus on integrating system-level constraints and biological feedback to bridge the gap between numerical predictions and clinical efficacy. Group of coils, doubtless, give us, even in optimization procedures, a wide range of possibilities. But, in this context, we need to consider many optional coil types and their spatial arrangement with respect to both the limb and one another.

7. Conclusions

An appropriate model and software serve as valuable tools for visualizing the distribution of eddy currents, which can address important expectations from patients. The results obtained from simulations may also be used for quantitative analysis, which is essential when studying the effects of induced current on selected therapeutic outcomes within patient groups.
The correlation presented in this article—between the magnetic field component generated by a specific device and the eddy currents induced in the body of a particular patient—represents a first step toward therapy individualization. However, such an approach requires extending the software capabilities to include the ability to construct field models based on patient-specific medical data, such as MRI or X-ray imaging.
Future extensions of this work could include the integration of impedance tomography (EIT) data to personalize conductivity distributions in the simulated tissue. Such hybrid modeling may improve accuracy and allow for tracking changes in bone health during therapy.
Furthermore, future studies could explore the integration of AI-based multisensory monitoring systems, similar to those used in dynamic industrial environments, for real-time assessment of physiological responses during magnetotherapy. Such an approach would be particularly valuable in large-scale clinical research or in designing treatment protocols for broader patient populations, where individualized monitoring and adaptive control could significantly enhance therapeutic outcomes.

Author Contributions

Conceptualization, P.S., M.S., and P.K.; methodology, P.S., M.S., and P.K.; software, P.S. and M.S.; validation, P.S., M.S., and P.K.; formal analysis, P.S., M.S., and P.K.; investigation, P.S., M.S., and P.K.; resources, P.S., M.S., and P.K.; data curation, P.S., M.S., and P.K.; writing—original draft preparation, P.S., M.S., and P.K.; writing—review and editing, P.S., M.S., and P.K.; visualization, P.S.; supervision, P.S., M.S., and P.K.; project administration, P.S., M.S., and P.K.; funding acquisition, P.S., M.S., and P.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Ethical review and approval were waived for this study, as it did not involve human participants, identifiable personal data, or animal subjects.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are not publicly available.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:
TMSTranscranial Magnetic Stimulation
EMFElectromagnetic Field
ECExcitation Coil

References

  1. Lunca, E.; Istrate, M.; Salceanu, A. Comparative Analysis of the Extremely Low-Frequency Magnetic Field Exposure from Overhead Power Lines. Environ. Eng. Manag. J. 2013, 12, 1145–1152. [Google Scholar] [CrossRef]
  2. Gas, P. Essential Facts on the History of Hyperthermia and their Connections with Electromedicine. Prz. Elektrotech. 2011, 87, 37–40. [Google Scholar]
  3. Marchewka, R.; Trzmiel, T.; Hojan, K. Effect of Low-Frequency Magnetic Field Stimulation on Physical Performance and Inflammation in Post-Stroke Patients: A Feasibility and Safety Study. Appl. Sci. 2025, 15, 3182. [Google Scholar] [CrossRef]
  4. Turner, C.; Pavalko, F. Mechanotransduction and Functional Response of the Skeleton to Physical Stress: The Mechanisms and Mechanics of Bone Adaptation. J. Orthop. Sci. 1998, 3, 346–355. [Google Scholar] [CrossRef]
  5. Kazerooni, A.F.; Rabbani, M.; Yazdchi, M.R. Simulation of the Effects of Electric and Magnetic Loadings on Internal Bone Remodeling. In Proceedings of the 5th Kuala Lumpur International Conference on Biomedical Engineering 2011, Kuala Lumpur, Malaysia, 20–23 June 2011; Springer: Berlin/Heidelberg, Germany, 2011; Volume 35, pp. 458–462. [Google Scholar]
  6. López, E.; Ibarz, E.; Herrera, A.; Puértolas, S.; Gabarre, S.; Más, Y.; Mateo, J.; Gil-Albarova, J.; Gracia, L. A predictive mechanical model for evaluating vertebral fracture probability in lumbar spine under different osteoporotic drug therapies. Comput. Methods Programs Biomed. 2016, 131, 37–50. [Google Scholar] [CrossRef]
  7. Roberts, M.G.; Graham, J. Image Texture in Dental Panoramic Radiographs as a Potential Biomarker of Osteoporosis. IEEE Trans. Biomed. Eng. 2013, 60, 2384–2392. [Google Scholar] [CrossRef]
  8. Gupta, K.; Gupta, P.; Singh, G.K.; Kumar, S.; Singh, R.K.; Srivastava, R.N. Changes in Electrical Properties of Bones as a Diagnostic Tool for Measurement of Fracture Healing. Hard Tissue 2013, 2, 3. [Google Scholar] [CrossRef]
  9. Jayanand, S.; Behari, J. Effect of Electrical Stimulation in Mineralization and Collagen Enrichment of Osteoporotic Rat Bones. In Proceedings of the International Conference on Microwave, Jaipur, India, 21–24 November 2008; pp. 568–571. [Google Scholar]
  10. Sert, C.; Deniz, M.; Duz, M.Z.; Aksen, F.; Kaya, A. The Preventive Effect on Bone Loss of 50-Hz, 1-mT Electromagnetic Field in Ovariectomized Rats. J. Bone Miner. Metab. 2002, 20, 345–349. [Google Scholar] [CrossRef] [PubMed]
  11. Mori, H.; Kagaya, H.; Inamoto, Y.; Izumi, S.I.; Yashima, K.; Takagi, T. Numerical Analysis of Eddy Current Distribution in Submental Region Induced by Magnetic Stimulation for Treating Dysphagia. IEEE Trans. Neural Syst. Rehabil. Eng. 2020, 28, 1178–1186. [Google Scholar] [CrossRef] [PubMed]
  12. Sandurkov, B.; Rapp, J.; Hemmert, W.; Gleich, B. Low Energy Magnetic Stimulation of the Phrenic Nerve-A Simulation Study. Biomed. Phys. Eng. Express 2023, 9, 057001. [Google Scholar] [CrossRef]
  13. Oltean-Dan, D.; Dogaru, G.B.; Apostu, D.; Mester, A.; Benea, H.R.C.; Paiusan, M.G.; Popa, C.O.; Jianu, E.M.; Bodizs, G.I.; Berce, C.; et al. Enhancement of Bone Consolidation Using High-Frequency Pulsed Electromagnetic Fields (HF-PEMFs): An Experimental Study on Rats. Bosn. J. Basic Med. Sci. 2019, 19, 201–209. [Google Scholar] [CrossRef]
  14. Liu, J.; Ren, W.; Wang, S.; Yang, J.; Zhang, H.; Zeng, Y.; Yin, D.; Shang, P. The Effects and Mechanisms of Electromagnetic Fields on Bone Remodeling: From Clinical to Laboratory. J. Orthop. Transl. 2025, 52, 14–26. [Google Scholar] [CrossRef]
  15. Wang, J.; Tang, N.; Xiao, Q.; Zhang, L.; Li, Y.; Li, J.; Wang, J.; Zhao, Z.; Tan, L. Pulsed Electromagnetic Field May Accelerate In Vitro Endochondral Ossification. Bioelectromagnetics 2015, 36, 35–44. [Google Scholar] [CrossRef]
  16. Walewicz, K.; Halski, T.; Dymarek, R.; Taradaj, J. Do Magnetic Field Applications Lead to Improved Bone Union in Light of Evidence-Based Medicine Principles? Analysis of the Scientific Evidence from Basic Science to Clinical Research. Physiother. Rev. 2023, 27, 17–22. [Google Scholar] [CrossRef]
  17. Martinez-Rondanelli, A.; Martinez, J.P.; Moncada, M.E.; Manzi, E.; Pinedo, C.R.; Cadavid, H. Electromagnetic Stimulation as Coadjuvant in the Healing of Diaphyseal Femoral Fractures: A Randomized Controlled Trial. Colomb. Med. 2014, 45, 67–71. [Google Scholar] [CrossRef]
  18. Ciesla, A.; Kraszewski, W.; Tadeusiewicz, R. Visualization of Magnetic Field Generated by Portable Coil Designed for Magnetotherapy. Prz. Elektrotech. 2012, 88, 127–131. [Google Scholar]
  19. Krawczyk, A.; Miaskowski, A.; Łada-Tondyra, E.; Ishihara, Y. Healing of Orthopaedic Diseases by Means of Electromagnetic Field. Acta Tech. Jaurinensis 2011, 4, 357–364. [Google Scholar]
  20. Bottauscio, O.; Arduino, A.; Chiampi, M.; Zilberti, L. Simplified Modeling of Implanted Medical Devices with Metallic Filamentary Closed Loops Exposed to Low or Medium Frequency Magnetic Fields. Comput. Methods Programs Biomed. 2023, 229, 107316. [Google Scholar] [CrossRef]
  21. Syrek, P.; Barbulescu, R. Parametric Curves to Trace the TMS Coils Windings. In Proceedings of the 10th International Symposium on Advanced Topics in Electrical Engineering (ATEE), Bucharest, Romania, 23–25 March 2017; pp. 386–391. [Google Scholar]
  22. Worbs, T.; Rumi, B.; Madsen, K.H.; Thielscher, A. Realistic Electric Field Characterization of Clinically Used Deformable Large TMS Coils in a Large Cohort. Brain Stimul. 2025, 18, 1174–1183. [Google Scholar] [CrossRef]
  23. Berger, T.A.; Mantell, K.; Haigh, Z.; Perera, N.; Alekseichuk, I.; Opitz, A. Comprehensive Evaluation of U-Net-Based Transcranial Magnetic Stimulation Electric Field Estimations. Sci. Rep. 2025, 15, 12204. [Google Scholar] [CrossRef] [PubMed]
  24. Saxena, Y.; Lewis, C.; Lee, J.; Franke, L.; Alam, M.S.; Tashli, M.; Atulasimha, J.; Hadimani, R. Optimizing A Deep Learning Model for the Prediction of Electric Field Induced by Transcranial Magnetic Stimulation for Mild to Moderate Traumatic Brain Injury Patients. AIP Adv. 2024, 14, 1–9. [Google Scholar] [CrossRef]
  25. Christ, A.; Kainz, W.; Hahn, E.G.; Honegger, K.; Zefferer, M.; Neufeld, E.; Rascher, W.; Janka, R.; Bautz, W.; Chen, J.; et al. The Virtual Family—Development of surface-based anatomical models of two adults and two children for dosimetric simulations. Phys. Med. Biol. 2010, 55, 23–38. [Google Scholar] [CrossRef]
  26. Fang, Q.; Boas, D. Tetrahedral mesh generation from volumetric binary and gray-scale images. In Proceedings of the IEEE International Symposium on Biomedical Imaging, Boston, MA, USA, 28 June–1 July 2009; pp. 1142–1145. [Google Scholar]
  27. Syrek, P. Uncertainty problem as illustrated by magnetotherapy. ACES J. 2019, 34, 1445–1452. [Google Scholar]
  28. Sawicki, B.; Krupa, A. Experiments with models of variability for biological tissues. Prz. Elektrotech. 2016, 92, 83–86. [Google Scholar] [CrossRef]
  29. Gabriel, S.; Lau, R.W.; Gabriel, C. The dielectric properties of biological tissues: III Parametric models of the dielectric spectrum of tissues. Phys. Med. Biol. 1996, 41, 2271–2293. [Google Scholar] [CrossRef]
  30. Stubendekova, A.; Janousek, L. Non-destructive testing of conductive material by eddy current air probe based on swept frequency. J. Electr. Eng. 2015, 66, 174–177. [Google Scholar] [CrossRef]
  31. Noetscher, G.M.; Butson, C.R. A Simple Absolute Estimate of Peak Eddy Currents Induced by Transcranial Magnetic Stimulation Using the GR Model. IEEE Trans. Magn. 2013, 49, 4999–5003. [Google Scholar] [CrossRef]
  32. Diao, Y.; Joseph, W.; Poljak, D.; Giaccone, L.; Kodera, S.; Laakso, I.; Yamazaki, K.; Li, K.; Sasaki, K.; Tanghe, E.; et al. Recent Advances and Future Perspective in Computational Bioelectromagnetics for Exposure Assessments. Bioelectromagnetics 2025, 46, e70002. [Google Scholar] [CrossRef] [PubMed]
  33. Cvetković, M.; Totić, A. Magnetotherapy Modeling Using a Simplified and Anatomical Human Body Models. In Proceedings of the SpliTech2024, SpliTech, Split, Croatia, 25–28 June 2024; pp. 1–6. [Google Scholar]
  34. Cvetković, M.; Sučić, B. Analysis of Magnetotherapy Device-Induced Fields Using Cylindrical Human BodyModel. Electronics 2024, 13, 849. [Google Scholar] [CrossRef]
  35. Yang, S.; Xu, G.; Wang, L.; Zhang, X. Effect of the different winding methods of coil on electromagnetic field during transcranial magnetic stimulation. In Proceedings of the 2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Vancouver, BC, Canada, 20–25 August 2008; pp. 4270–4273. [Google Scholar]
  36. Yeung, S.H.; Pradhan, R.; Feng, X.; Zheng, Y. Focused Magnetic Resonance Coupling Coils for Electromagnetic Therapy Applications. IEEE Trans. Biom. Eng. 2015, 62, 2602–2610. [Google Scholar] [CrossRef]
  37. Plonecki, P.; Starzynski, P.; Wincenciak, S.; Sawicki, B. Modelling of eddy-currents using electric scalar potential. In Proceedings of the XV ISTET International Symposium on Theoretical Engineering, Lübeck, Germany, 22–24 June 2009; pp. 294–298. [Google Scholar]
  38. IT’IS Foundation. Tissue Properties Database V5.0: Density (kg/m3) Data for Human Tissues. Available online: https://itis.swiss/virtual-population/tissue-properties/database/density/ (accessed on 21 August 2025).
  39. International Commission on Non-Ionizing Radiation Protection (ICNIRP). Guidelines for Limiting Exposure to Electromagnetic Fields (100 kHz to 300 GHz). Health Phys. 2020, 118, 483–524. [Google Scholar] [CrossRef] [PubMed]
  40. International Commission on Non-Ionizing Radiation Protection (ICNIRP). Guidelines for Limiting Exposure to Time-Varying Electric and Magnetic Fields (1 Hz to 100 kHz). Health Phys. 2010, 99, 818–836. [Google Scholar] [CrossRef] [PubMed]
  41. Pratticò, D.; Laganá, F.; Oliva, G.; Fiorillo, A.; Pullano, S.; Calcagno, S.; De Carlo, D.; La Foresta, F. Integration of LSTM and U-Net Models for Monitoring Electrical Absorption with a System of Sensors and Electronic Circuits. IEEE Trans. Instrum. Meas. 2025, 74, 1–11. [Google Scholar] [CrossRef]
  42. Warden, S.J. A new direction for ultrasound therapy in sports medicine. Sports Med. 2003, 33, 95–107. [Google Scholar] [CrossRef] [PubMed]
  43. Pilla, A.A. Mechanisms and therapeutic applications of time-varying and static magnetic fields. In Biological and Medical Aspects of Electromagnetic Fields, 3rd ed.; Barnes, F.S., Greenebaum, B., Eds.; CRC Press: Boca Raton, FL, USA, 2006; pp. 351–412. [Google Scholar]
Applsci 15 09892 g001
Figure 1. Magnetotherapy applicator: (a) View of the device during magnetic field component measurements; (b) dimensioned setup used in the simulations.
Figure 1. Magnetotherapy applicator: (a) View of the device during magnetic field component measurements; (b) dimensioned setup used in the simulations.
Applsci 15 09892 g001
Applsci 15 09892 g002
Figure 2. Magnetic field distribution inside the applicator (excitation coil). The figure shows the distribution of absolute value of the magnetic flux density on the surface of the hand and in a selected plane aligned with the elbow and the wrist. The figure indicates that the magnetic field component remains relatively homogeneous in terms of magnitude across various heights of the applicator.
Figure 2. Magnetic field distribution inside the applicator (excitation coil). The figure shows the distribution of absolute value of the magnetic flux density on the surface of the hand and in a selected plane aligned with the elbow and the wrist. The figure indicates that the magnetic field component remains relatively homogeneous in terms of magnitude across various heights of the applicator.
Applsci 15 09892 g002
Applsci 15 09892 g003
Figure 3. Distribution of selected magnetic field components: (a) z-component; (b) x-component. The figures supplement Figure 2 and present the distribution of selected components of the magnetic flux density on the surface of the hand and in a selected plane aligned with the elbow and the wrist. The distribution of magnetic field z-component is very similar to that of the magnetic flux density magnitude (absolute value) with comparable values, suggesting that it constitutes the dominant component of the field. The x-component of the magnetic field increases progressively toward the applicator walls (reaching increasingly extreme positive and negative values).
Figure 3. Distribution of selected magnetic field components: (a) z-component; (b) x-component. The figures supplement Figure 2 and present the distribution of selected components of the magnetic flux density on the surface of the hand and in a selected plane aligned with the elbow and the wrist. The distribution of magnetic field z-component is very similar to that of the magnetic flux density magnitude (absolute value) with comparable values, suggesting that it constitutes the dominant component of the field. The x-component of the magnetic field increases progressively toward the applicator walls (reaching increasingly extreme positive and negative values).
Applsci 15 09892 g003
Applsci 15 09892 g004
Figure 4. Distribution of the z-component of the magnetic field along two lines parallel to the z-axis (RMSvalues).
Figure 4. Distribution of the z-component of the magnetic field along two lines parallel to the z-axis (RMSvalues).
Applsci 15 09892 g004
Applsci 15 09892 g005
Figure 5. Eddy currents distribution within 16 mm part of upper limb: (a) selected slices (all tissues); (b) current density on bones surface.
Figure 5. Eddy currents distribution within 16 mm part of upper limb: (a) selected slices (all tissues); (b) current density on bones surface.
Applsci 15 09892 g005
Applsci 15 09892 g006
Figure 6. The average eddy current density in bones within 20-mm segments of the limb (cross-sectional planes are taken perpendicular to the local z-axis, spanning the range from z = –0.04 (m) to z = 0.12 (m)). Each 20-mm segment has an average value between 1.2 and 2.5 mA/m2.
Figure 6. The average eddy current density in bones within 20-mm segments of the limb (cross-sectional planes are taken perpendicular to the local z-axis, spanning the range from z = –0.04 (m) to z = 0.12 (m)). Each 20-mm segment has an average value between 1.2 and 2.5 mA/m2.
Applsci 15 09892 g006
Applsci 15 09892 g007
Figure 7. (a) The magnetic field generated by 4 applicators (4 excitation coils). The figure shows the distribution of absolute value of the magnetic flux density and the direction of the current supplying the coils for the 0-degree phase; (b) the average eddy current density in bones, within 20-mm segments of the limb (cross-sectional planes are taken perpendicular to the local z-axis, spanning the range from z = −0.04 (m) to z = 0.12 (m)). Each 20-mm segment has an average value between 0.62 and 0.92 mA/m2.
Figure 7. (a) The magnetic field generated by 4 applicators (4 excitation coils). The figure shows the distribution of absolute value of the magnetic flux density and the direction of the current supplying the coils for the 0-degree phase; (b) the average eddy current density in bones, within 20-mm segments of the limb (cross-sectional planes are taken perpendicular to the local z-axis, spanning the range from z = −0.04 (m) to z = 0.12 (m)). Each 20-mm segment has an average value between 0.62 and 0.92 mA/m2.
Applsci 15 09892 g007
Applsci 15 09892 g008
Figure 8. Eddy currents distribution within upper limb generated by 4 applicators: (a) selected slices (all tissues); (b) current density on bones surface.
Figure 8. Eddy currents distribution within upper limb generated by 4 applicators: (a) selected slices (all tissues); (b) current density on bones surface.
Applsci 15 09892 g008
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Syrek, P.; Skowron, M.; Kapustka, P. Eddy Current Distribution in Magnetotherapy of Bones: A Qualitative and Quantitative Study. Appl. Sci. 2025, 15, 9892. https://doi.org/10.3390/app15189892

AMA Style

Syrek P, Skowron M, Kapustka P. Eddy Current Distribution in Magnetotherapy of Bones: A Qualitative and Quantitative Study. Applied Sciences. 2025; 15(18):9892. https://doi.org/10.3390/app15189892

Chicago/Turabian Style

Syrek, Przemyslaw, Mikolaj Skowron, and Piotr Kapustka. 2025. "Eddy Current Distribution in Magnetotherapy of Bones: A Qualitative and Quantitative Study" Applied Sciences 15, no. 18: 9892. https://doi.org/10.3390/app15189892

APA Style

Syrek, P., Skowron, M., & Kapustka, P. (2025). Eddy Current Distribution in Magnetotherapy of Bones: A Qualitative and Quantitative Study. Applied Sciences, 15(18), 9892. https://doi.org/10.3390/app15189892

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop