Advances in Vascular Diagnostics using Magnetic Particle Imaging (MPI) for Blood Circulation Assessment.

Rapid and accurate assessment of conditions characterized by altered blood flow, cardiac blood pooling, or internal bleeding is crucial for diagnosing and treating various clinical conditions. While widely used imaging modalities such as magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound offer unique diagnostic advantages, they fall short for specific indications due to limited penetration depth and prolonged acquisition times. Magnetic particle imaging (MPI), an emerging tracer-based technique, holds promise for blood circulation assessments, potentially overcoming existing limitations with reduction in background signals and high temporal and spatial resolution, below the millimeter scale. Successful imaging of blood pooling and impaired flow necessitates tracers with diverse circulation half-lives optimized for MPI signal generation. Recent MPI tracers show potential in imaging cardiovascular complications, vascular perforations, ischemia, and stroke. The impressive temporal resolution and penetration depth also position MPI as an excellent modality for real-time vessel perfusion imaging via functional MPI (fMPI). This review summarizes advancements in optimized MPI tracers for imaging blood circulation and analyzes the current state of pre-clinical applications. This work discusses perspectives on standardization required to transition MPI from a research endeavor to clinical implementation and explore additional clinical indications that may benefit from the unique capabilities of MPI.

Pharmaceutical Quality by Design Approach to Develop High-Performance Nanoparticles for Magnetic Hyperthermia.

Magnetic hyperthermia holds significant therapeutic potential, yet its clinical adoption faces challenges. One obstacle is the large-scale synthesis of high-quality superparamagnetic iron oxide nanoparticles (SPIONs) required for inducing hyperthermia. Robust and scalable manufacturing would ensure control over the key quality attributes of SPIONs, and facilitate clinical translation and regulatory approval. Therefore, we implemented a risk-based pharmaceutical quality by design (QbD) approach for SPION production using flame spray pyrolysis (FSP), a scalable technique with excellent batch-to-batch consistency. A design of experiments method enabled precise size control during manufacturing. Subsequent modeling linked the SPION size (6-30 nm) and composition to intrinsic loss power (ILP), a measure of hyperthermia performance. FSP successfully fine-tuned the SPION composition with dopants (Zn, Mn, Mg), at various concentrations. Hyperthermia performance showed a strong nonlinear relationship with SPION size and composition. Moreover, the ILP demonstrated a stronger correlation to coercivity and remanence than to the saturation magnetization of SPIONs. The optimal operating space identified the midsized (15-18 nm) Mn0.25Fe2.75O4 as the most promising nanoparticle for hyperthermia. The production of these nanoparticles on a pilot scale showed the feasibility of large-scale manufacturing, and cytotoxicity investigations in multiple cell lines confirmed their biocompatibility. In vitro hyperthermia studies with Caco-2 cells revealed that Mn0.25Fe2.75O4 nanoparticles induced 80% greater cell death than undoped SPIONs. The systematic QbD approach developed here incorporates process robustness, scalability, and predictability, thus, supporting the clinical translation of high-performance SPIONs for magnetic hyperthermia.

Post-synthesis Oxidation of Superparamagnetic Iron Oxide Nanoparticles to Enhance Magnetic Particle Imaging Performance.

This study investigates the impact of post-synthesis oxidation on the performance of superparamagnetic iron oxide nanoparticles (SPIONs) in magnetic particle imaging (MPI), an emerging technology with applications in diagnostic imaging and theranostics. SPIONs synthesized from iron oleate were subjected to a post-synthesis oxidation treatment with a 1% Oxygen in Argon mixture. MPI performance, gauged via signal intensity and resolution using a MOMENTUM™ scanner, was correlated to the nanoparticles' physical and magnetic properties. Post-synthesis oxidation did not alter physical attributes like size and shape, but significantly enhanced magnetic properties. Saturation magnetization increased from 52% to 93% of the bulk value for magnetite, leading to better MPI performance in terms of signal intensity and resolution. However, the observed MPI performance did not fully align with predictions based on the ideal Langevin model, indicating the need for considering factors like relaxation and shape anisotropy. The findings underscore the potential of post-synthesis oxidation as a method to fine-tune magnetic properties of SPIONs and improve MPI performance, and the need for reproducible synthesis methods that afford finely tuned control of nanoparticle size, shape, and magnetic properties.

Magnetic Cellular Backpacks for Spatial Targeting, Imaging, and Immunotherapy.

Adoptive cell transfer (ACT) therapies are growing in popularity due to their ability to interact with diseased tissues in a specific manner. Disc-shaped particles, or "backpacks", that bind to cellular surfaces show promise for augmenting the therapeutic potential of adoptively transferred cells by resisting phagocytosis and locally releasing drugs to maintain cellular activity over time. However, many ACTs suffer from limited tumor infiltration and retention and lack a method for real-time spatial analysis. Therefore, we have designed biodegradable backpacks loaded with superparamagnetic iron oxide nanoparticles (SPIONs) to improve upon current ACT strategies by (i) controlling the localization of cell-backpack complexes using gradient magnetic fields and (ii) enabling magnetic particle imaging (MPI) to track complexes after injection. We show that magnetic backpacks bound to macrophages and loaded with a proinflammatory drug, resiquimod, maintain anticancer phenotypes of carrier macrophages for 5 days and create cytokine "factories" that continuously release IL-12. Furthermore, we establish that forces generated by gradient magnet fields are sufficient to displace cell-backpack complexes in physiological settings. Finally, we demonstrate that MPI can be used to visualize cell-backpack complexes in mouse tumors, enabling a potential strategy to track the biodistribution of ACTs in real time.

Development of a bioactive tunable hyaluronic-protein bioconjugate hydrogel for tissue regenerative applications.

Every year, there are approximately 500 000 peripheral nerve injury (PNI) procedures due to trauma in the US alone. Autologous and acellular nerve grafts are among current clinical repair options; however, they are limited largely by the high costs associated with donor nerve tissue harvesting and implant processing, respectively. Therefore, there is a clinical need for an off-the-shelf nerve graft that can recapitulate the native microenvironment of the nerve. In our previous work, we created a hydrogel scaffold that incorporates mechanical and biological cues that mimic the peripheral nerve microenvironment using chemically modified hyaluronic acid (HA). However, with our previous work, the degradation profile and cell adhesivity was not ideal for tissue regeneration, in particular, peripheral nerve regeneration. To improve our previous hydrogel, HA was conjugated with fibrinogen using Michael-addition to assist in cell adhesion and hydrogel degradability. The addition of the fibrinogen linker was found to contribute to faster scaffold degradation via active enzymatic breakdown, compared to HA alone. Additionally, cell count and metabolic activity was significantly higher on HA conjugated fibrinogen compared previous hydrogel formulations. This manuscript discusses the various techniques deployed to characterize our new modified HA fibrinogen chemistry physically, mechanically, and biologically. This work addresses the aforementioned concerns by incorporating controllable degradability and increased cell adhesivity while maintaining incorporation of hyaluronic acid, paving the pathway for use in a variety of applications as a multi-purpose tissue engineering platform.

Inter-user Comparison for Quantification of Superparamagnetic Iron Oxides with Magnetic Particle Imaging Across Two Institutions Highlights a Need for Standardized Approaches.

PURPOSE: Magnetic particle imaging (MPI) is being explored in biological contexts that require accurate and reproducible quantification of superparamagnetic iron oxide nanoparticles (SPIONs). While many groups have focused on improving imager and SPION design to improve resolution and sensitivity, a few have focused on improving quantification and reproducibility of MPI. The aim of this study was to compare MPI quantification results by two different systems and the accuracy of SPION quantification performed by multiple users at two institutions. PROCEDURES: Six users (3 from each institute) imaged a known amount of Vivotrax + (10 µg Fe), diluted in a small (10 µL) or large (500 µL) volume. These samples were imaged with or without calibration standards in the field of view, to create a total of 72 images (6 users × triplicate samples × 2 sample volumes × 2 calibration methods). These images were analyzed by the respective user with two region of interest (ROI) selection methods. Image intensities, Vivotrax + quantification, and ROI selection were compared across users, within and across institutions. RESULTS: MPI imagers at two different institutes produce significantly different signal intensities, that differ by over 3 times for the same concentration of Vivotrax + . Overall quantification yielded measurements that were within [Formula: see text] 20% from ground truth; however, SPION quantification values obtained at each laboratory were significantly different. Results suggest that the use of different imagers had a stronger influence on SPION quantification compared to differences arising from user error. Lastly, calibration conducted from samples in the imaging field of view gave the same quantification results as separately imaged samples. CONCLUSIONS: This study highlights that there are many factors that contribute to the accuracy and reproducibility of MPI quantification, including variation between MPI imagers and users, despite pre-defined experimental setup, image acquisition parameters, and ROI selection analysis.

Dual imaging agent for magnetic particle imaging and computed tomography.

Magnetic particle imaging (MPI) is a novel biomedical imaging modality that allows non-invasive, tomographic, and quantitative tracking of the distribution of superparamagnetic iron oxide nanoparticle (SPION) tracers. While MPI possesses high sensitivity, detecting nanograms of iron, it does not provide anatomical information. Computed tomography (CT) is a widely used biomedical imaging modality that yields anatomical information at high resolution. A multimodal imaging agent combining the benefits of MPI and CT imaging would be of interest. Here we combine MPI-tailored SPIONs with CT-contrast hafnium oxide (hafnia) nanoparticles using flash nanoprecipitation to obtain dual-imaging MPI/CT agents. Co-encapsulation of iron oxide and hafnia in the composite nanoparticles was confirmed via transmission electron microscopy and elemental mapping. Equilibrium and dynamic magnetic characterization show a reduction in effective magnetic diameter and changes in dynamic magnetic susceptibility spectra at high oscillating field frequencies, suggesting magnetic interactions within the composite dual imaging tracers. The MPI performance of the dual imaging agent was evaluated and compared to the commercial tracer ferucarbotran. The dual-imaging agent has MPI sensitivity that is ∼3× better than this commercial tracer. However, worsening of MPI resolution was observed in the composite tracer when compared to individually coated SPIONs. This worsening resolution could result from magnetic dipolar interactions within the composite dual imaging tracer. The CT performance of the dual imaging agent was evaluated in a pre-clinical animal scanner and a clinical scanner, revealing better contrast compared to a commercial iodine-based contrast agent. We demonstrate that the dual imaging agent can be differentiated from the commercial iodine contrast agent using dual energy CT (DECT) imaging. Furthermore, the dual imaging agent displayed energy-dependent CT contrast arising from the combination of SPION and hafnia, making it potentially suitable for virtual monochromatic imaging of the contrast agent distribution using DECT.

Magnetic Nanoparticles Improve Flow Rate and Enable Self-Healing in Covalent Adaptable Networks.

Covalent adaptable networks (CANs) combine the mechanical and chemical stability of thermosets with the reprocessability of thermoplastics through the incorporation of stimuli-responsive dynamic crosslinks. To allow for processing through induction heating, we have created associative CANs that include fillers in the polymer matrix for efficient heat transfer. While the inclusion of inorganic fillers often decreases flow rate in CANs and complicates reprocessing of the material, the presence of Fe3O4 nanoparticles had no detrimental effect on flow behavior in a vinylogous urethane vitrimer, an observation we attribute to the catalytic nature of nanoparticles on the dynamic exchange chemistry. We employed two methods of nanoparticle incorporation: blending bare nanoparticles and crosslinking chemically modified nanoparticles. The vitrimers with covalently crosslinked nanoparticles exhibited a decreased relaxation time compared to those with blended nanoparticles. The magnetic character of the Fe3O4 nanoparticles enabled self-healing of the vitrimer composite materials upon exposure to an alternating electromagnetic field during induction heating.

Inter-user comparison for quantification of superparamagnetic iron oxides with magnetic particle imaging across two institutions highlights a need for standardized approaches.

Purpose: Magnetic particle imaging (MPI) is being explored in biological contexts that require accurate and reproducible quantification of superparamagnetic iron oxide nanoparticles (SPIONs). While many groups have focused on improving imager and SPION design to improve resolution and sensitivity, few have focused on improving quantification and reproducibility of MPI. The aim of this study was to compare MPI quantification results by two different systems and the accuracy of SPION quantification performed by multiple users at two institutions. Procedures: Six users (3 from each institute) imaged a known amount of Vivotrax+ (10 µg Fe), diluted in a small (10 µL) or large (500 µL) volume. These samples were imaged with or without calibration standards in the field of view, to create a total of 72 images (6 users x triplicate samples x 2 sample volumes x 2 calibration methods). These images were analyzed by the respective user with two region of interest (ROI) selection methods. Image intensities, Vivotrax+ quantification, and ROI selection was compared across users, within and across institutions. Results: MPI imagers at two different institutes produce significantly different signal intensities, that differ by over 3 times for the same concentration of Vivotrax+. Overall quantification yielded measurements that were within ± 20% from ground truth, however SPION quantification values obtained at each laboratory were significantly different. Results suggest that the use of different imagers had a stronger influence on SPION quantification compared to differences arising from user error. Lastly, calibration conducted from samples in the imaging field of view gave the same quantification results as separately imaged samples. Conclusions: This study highlights that there are many factors that contribute to the accuracy and reproducibility of MPI quantification, including variation between MPI imagers and users, despite pre-defined experimental set up, image acquisition parameters, and ROI selection analysis.

Application of magnetic particle imaging to evaluate nanoparticle fate in rodent joints.

Nanoparticles are a promising approach for improving intra-articular drug delivery and tissue targeting. However, techniques to non-invasively track and quantify their concentration in vivo are limited, resulting in an inadequate understanding of their retention, clearance, and biodistribution in the joint. Currently, fluorescence imaging is often used to track nanoparticle fate in animal models; however, this approach has limitations that impede long-term quantitative assessment of nanoparticles over time. The goal of this work was to evaluate an emerging imaging modality, magnetic particle imaging (MPI), for intra-articular tracking of nanoparticles. MPI provides 3D visualization and depth-independent quantification of superparamagnetic iron oxide nanoparticle (SPION) tracers. Here, we developed and characterized a polymer-based magnetic nanoparticle system incorporated with SPION tracers and cartilage targeting properties. MPI was then used to longitudinally assess nanoparticle fate after intra-articular injection. Magnetic nanoparticles were injected into the joints of healthy mice, and evaluated for nanoparticle retention, biodistribution, and clearance over 6 weeks using MPI. In parallel, the fate of fluorescently tagged nanoparticles was tracked using in vivo fluorescence imaging. The study was concluded at day 42, and MPI and fluorescence imaging demonstrated different profiles in nanoparticle retention and clearance from the joint. MPI signal was persistent over the study duration, suggesting NP retention of at least 42 days, much longer than the 14 days observed based on fluorescence signal. These data suggest that the type of tracer - SPIONs or fluorophores - and modality of imaging can affect interpretation of nanoparticle fate in the joint. Given that understanding particle fate over time is paramount for attaining insights about therapeutic profiles in vivo, our data suggest MPI may yield a quantitative and robust method to non-invasively track nanoparticles following intra-articular injection on an extended timeline.

An anatomically correct 3D-printed mouse phantom for magnetic particle imaging studies.

We report anatomically correct 3D-printed mouse phantoms that can be used to plan experiments and evaluate analysis protocols for magnetic particle imaging (MPI) studies. The 3D-printed phantoms were based on the Digimouse 3D whole body mouse atlas and incorporate cavities representative of a liver, brain tumor, and orthotopic breast cancer tumor placed in anatomically correct locations, allowing evaluation of the effect of precise doses of MPI tracer. To illustrate their use, a constant tracer iron mass was present in the liver for the breast (200 µgFe) and brain tumor (10 µgFe) model, respectively, while a series of decreasing tracer iron mass was placed in the tumor region. MPI scans were acquired in 2D and 3D high sensitivity and high sensitivity/high resolution (HSHR) modes using a MOMENTUM imager. A thresholding algorithm was used to define regions of interest (ROIs) in the scans and the tracer mass in the liver and tumors was calculated by comparison of the signal in their respective ROI against that of known mass fiducials that were included in each scan. The results demonstrate that this approach to image analysis provides accurate estimates of tracer mass. Additionally, the results show how the limit of detection in MPI is sensitive to the details of tracer distribution in the subject, as we found that a greater tracer mass in the liver cavity resulted in poorer sensitivity in tumor regions. These experiments illustrate the utility of the reported 3D-printed anatomically correct mouse phantoms in evaluating methods to analyze MPI scans and plan in vivo experiments.

Examining thein vivofunctionality of the magnetically aligned regenerative tissue-engineered electronic nerve interface (MARTEENI).

Objective. Although neural-enabled prostheses have been used to restore some lost functionality in clinical trials, they have faced difficulty in achieving high degree of freedom, natural use compared to healthy limbs. This study investigated thein vivofunctionality of a flexible and scalable regenerative peripheral-nerve interface suspended within a microchannel-embedded, tissue-engineered hydrogel (the magnetically aligned regenerative tissue-engineered electronic nerve interface (MARTEENI)) as a potential approach to improving current issues in peripheral nerve interfaces.Approach. Assembled MARTEENI devices were implanted in the gaps of severed sciatic nerves in Lewis rats. Both acute and chronic electrophysiology were recorded, and channel-isolated activity was examined. In terminal experiments, evoked activity during paw compression and stimulus response curves generated from proximal nerve stimulation were examined. Electrochemical impedance spectroscopy was performed to assess the complex impedance of recording sites during chronic data collection. Features of the foreign-body response (FBR) in non-functional implants were examined using immunohistological methods.Main results. Channel-isolated activity was observed in acute, chronic, and terminal experiments and showed a typically biphasic morphology with peak-to-peak amplitudes varying between 50 and 500µV. For chronic experiments, electrophysiology was observed for 77 days post-implant. Within the templated hydrogel, regenerating axons formed minifascicles that varied in both size and axon count and were also found to surround device threads. No axons were found to penetrate the FBR. Together these results suggest the MARTEENI is a promising approach for interfacing with peripheral nerves.Significance. Findings demonstrate a high likelihood that observed electrophysiological activity recorded from implanted MARTEENIs originated from neural tissue. The variation in minifascicle size seen histologically suggests that amplitude distributions observed in functional MARTEENIs may be due to a combination of individual axon and mini-compound action potentials. This study provided an assessment of a functional MARTEENI in anin vivoanimal model for the first time.

Superferromagnetic Nanoparticles Enable Order-of-Magnitude Resolution & Sensitivity Gain in Magnetic Particle Imaging.

Magnetic nanoparticles have many advantages in medicine such as their use in non-invasive imaging as a Magnetic Particle Imaging (MPI) tracer or Magnetic Resonance Imaging contrast agent, the ability to be externally shifted or actuated and externally excited to generate heat or release drugs for therapy. Existing nanoparticles have a gentle sigmoidal magnetization response that limits resolution and sensitivity. Here it is shown that superferromagnetic iron oxide nanoparticle chains (SFMIOs) achieve an ideal step-like magnetization response to improve both image resolution & SNR by more than tenfold over conventional MPI. The underlying mechanism relies on dynamic magnetization with square-like hysteresis loops in response to 20 kHz, 15 kAm-1 MPI excitation, with nanoparticles assembling into a chain under an applied magnetic field. Experimental data shows a "1D avalanche" dipole reversal of every nanoparticle in the chain when the applied field overcomes the dynamic coercive threshold of dipole-dipole fields from adjacent nanoparticles in the chain. Intense inductive signal is produced from this event resulting in a sharp signal peak. Novel MPI imaging strategies are demonstrated to harness this behavior towards order-of-magnitude medical image improvements. SFMIOs can provide a breakthrough in noninvasive imaging of cancer, pulmonary embolism, gastrointestinal bleeds, stroke, and inflammation imaging.

Development of a magnetically aligned regenerative tissue-engineered electronic nerve interface for peripheral nerve applications.

Peripheral nerve injuries can be debilitating to motor and sensory function, with severe cases often resulting in complete limb amputation. Over the past two decades, prosthetic limb technology has rapidly advanced to provide users with crude motor control of up to 20° of freedom; however, the nerve-interfacing technology required to provide high movement selectivity has not progressed at the same rate. The work presented here focuses on the development of a magnetically aligned regenerative tissue-engineered electronic nerve interface (MARTEENI) that combines polyimide "threads" encapsulated within a magnetically aligned hydrogel scaffold. The technology exploits tissue-engineered strategies to address concerns over traditional peripheral nerve interfaces including poor axonal sampling through the nerve and rigid substrates. A magnetically templated hydrogel is used to physically support the polyimide threads while also promoting regeneration in close proximity to the electrode sites on the polyimide. This work demonstrates the utility of magnetic templating for use in tuning the mechanical properties of hydrogel scaffolds to match the stiffness of native nerve tissue while providing an aligned substrate for Schwann cell migration in vitro. MARTEENI devices were fabricated and implanted within a 5-mm-long rat sciatic-nerve transection model to assess regeneration at 6 and 12 weeks. MARTEENI devices do not disrupt tissue remodeling and show axon densities equivalent to fresh tissue controls around the polyimide substrates. Devices are observed to have attenuated foreign-body responses around the polyimide threads. It is expected that future studies with functional MARTEENI devices will be able to record and stimulate single axons with high selectivity and low stimulation regimes.

Fast nanoparticle rotational and translational diffusion in synovial fluid and hyaluronic acid solutions.

Nanoparticles are under investigation as diagnostic and therapeutic agents for joint diseases, such as osteoarthritis. However, there is incomplete understanding of nanoparticle diffusion in synovial fluid, the fluid inside the joint, which consists of a mixture of the polyelectrolyte hyaluronic acid, proteins, and other components. Here, we show that rotational and translational diffusion of polymer-coated nanoparticles in quiescent synovial fluid and in hyaluronic acid solutions is well described by the Stokes-Einstein relationship, albeit with an effective medium viscosity that is much smaller than the macroscopic low shear viscosity of the fluid. This effective medium viscosity is well described by an equation for the viscosity of dilute polymer chains, where the additional viscous dissipation arises because of the presence of the polymer segments. These results shed light on the diffusive behavior of polymer-coated inorganic nanoparticles in complex and crowded biological environments, such as in the joint.

Tracking adoptive T cell immunotherapy using magnetic particle imaging.

Adoptive cellular therapy (ACT) is a potent strategy to boost the immune response against cancer. ACT is effective against blood cancers but faces challenges in treating solid tumors. A critical step for the success of ACT immunotherapy is to achieve efficient trafficking and persistence of T cells to solid tumors. Non-invasive tracking of the accumulation of adoptively transferred T cells to tumors would greatly accelerate development of more effective ACT strategies. We demonstrate the use of magnetic particle imaging (MPI) to non-invasively track ACT T cells in vivo in a mouse model of brain cancer. Magnetic labeling did not impair primary tumor-specific T cells in vitro, and MPI allowed the detection of labeled T cells in the brain after intravenous or intracerebroventricular administration. These results support the use of MPI to track adoptively transferred T cells and accelerate the development of ACT treatments for brain tumors and other cancers.

Long circulating tracer tailored for magnetic particle imaging.

Superparamagnetic iron oxide nanoparticle (SPION) tracers possessing long blood circulation time and tailored for magnetic particle imaging (MPI) performance are crucial for the development of this emerging molecular imaging modality. Here, single-core SPION MPI tracers coated with covalently bonded polyethyelene glycol (PEG) brushes were obtained using a semi-batch thermal decomposition synthesis with controlled addition of molecular oxygen, followed by an optimized PEG-silane ligand exchange procedure. The physical and magnetic properties, MPI performance, and blood circulation time of these newly synthesized tracers were compared to those of two commercially available SPIONs that were not tailored for MPI but are used for MPI: ferucarbotran and PEG-coated Synomag®-D. The new tailored tracer has MPI sensitivity that is ~3-times better than the commercial tracer ferucarbotran and much longer circulation half-life than both commercial tracers (t1/2=6.99 h for the new tracer, vs t1/2=0.59 h for ferucarbotran, and t1/2=0.62 h for PEG-coated Synomag®-D).

Emerging Biomedical Applications Based on the Response of Magnetic Nanoparticles to Time-Varying Magnetic Fields.

Magnetic nanoparticles are of interest for biomedical applications because of their biocompatibility, tunable surface chemistry, and actuation using applied magnetic fields. Magnetic nanoparticles respond to time-varying magnetic fields via physical particle rotation or internal dipole reorientation, which can result in signal generation or conversion of magnetic energy to heat. This dynamic magnetization response enables their use as tracers in magnetic particle imaging (MPI), an emerging biomedical imaging modality in which signal is quantitative of tracer mass and there is no tissue background signal or signal attenuation. Conversion of magnetic energy to heat motivates use in nanoscale thermal cancer therapy, magnetic actuation of drug release, and rapid rewarming of cryopreserved organs. This review introduces basic concepts of magnetic nanoparticle response to time-varying magnetic fields and presents recent advances in the field, with an emphasis on MPI and conversion of magnetic energy to heat.