@article{Foerger2026IEEESensors, Author = {F. Foerger, M. Boberg, N. Hackelberg, P. Heinisch, K. Ostaszewski, J. Faltinath, P. Suskin, F. Thieben, F. Mohn, P. Jürß, M. Möddel and T. Knopp}, Title = {3-D Magnetic Field Camera With Subsecond Temporal Resolution.}, Journal = {IEEE Sensors Journal.}, Year = {(2026).}, Volume = {26.}, Number = {(1),}, Note = {article}, Doi = {https://doi.org/10.1109/JSEN.2025.3629803}, Url = {https://ieeexplore.ieee.org/document/11244237}, Abstract = {Accurate and efficient volumetric magnetic field measurements are essential for a wide range of applications. Conventional methods are often limited in terms of measurement speed and applicability or suffer from scaling problems at larger volumes. This work presents a proof-of-concept field camera designed to measure magnetic fields within a spherical volume at a frame rate of 10 Hz. The camera features an array of 3-D Hall magnetometers positioned according to a spherical t-design, allowing simultaneous magnetic field data acquisition from the surface of the sphere. The approach enables the efficient representation of all three components of the magnetic field inside the sphere using a sixth-degree polynomial, significantly reducing measurement time compared with sequential methods. This work details the design, calibration, and measurement methods of the field camera. To evaluate its performance, we compare it with a sequential single-sensor measurement by examining a magnetic gradient field. The obtained measurement uncertainties of approximately 1% demonstrate the feasibility of the approach and its potential applicability to a variety of future applications.} } @article{Dora:25, Author = {J. Dora, M. Möddel, S. Flenner, J. Reimers, B. Zeller-Plumhoff, C. G. Schroer, T. Knopp, and J. Hagemann}, Title = {Model-based autofocus for near-field phase retrieval.}, Journal = {Optics Express.}, Year = {(2025).}, Volume = {33.}, Number = {(4),}, Pages = {6641-6657}, Month = {Feb}, Note = {article, openaccess}, Publisher = {Optica Publishing Group:}, Doi = {10.1364/OE.544573}, Url = {https://opg.optica.org/oe/abstract.cfm?URI=oe-33-4-6641}, Abstract = {The phase problem is a well known ill-posed reconstruction problem of coherent lens-less microscopic imaging, where only the intensities of a complex wave-field are measured by the detector and the phase information is lost. For the reconstruction of sharp images from holograms in a near-field experimental setting, it is crucial to solve the autofocus problem, i.e., to precisely estimate the Fresnel number of the forward model. Otherwise, blurred out-of focus images that also can contain artifacts are the result. In general, a simple distance measurement at the experiment is not sufficiently accurate, thus the fine-tuning of the Fresnel number has to be done prior to the actual reconstructions. This can be done manually or automatically by an estimation algorithm. To automatize the process, as needed, e.g., for in-situ/operando experiments, different focus criteria have been widely studied in literature but are subjected to certain restrictions. The methods often rely on image analysis of the reconstructed image, making them sensitive to image noise and also neglecting algorithmic properties of the applied phase retrieval. In this paper, we propose a novel criterion, based on a model-matching approach, which improves autofocusing by also taking the underlying reconstruction algorithm, the forward model and the measured hologram into account. We derive a common autofocusing framework, based on a recent phase-retrieval approach and a downhill-simplex method for the automatic optimization of the Fresnel number. We further demonstrate the robustness of the framework on different data sets obtained at the nano imaging endstation of P05 at PETRA III (DESY, Hamburg) operated by Helmholtz-Zentrum Hereon.} } @article{Faltinath2025natural, Author = {J. Faltinath, F. Mohn, F. Foerger, M. Möddel, and T. Knopp}, Title = {Natural Frequency Dependence of Magneto-Mechanical Resonators on Magnet Distance.}, Journal = {IEEE Sensors Journal.}, Year = {(2025).}, Volume = {25.}, Number = {(20),}, Pages = {38073-38081}, Note = {article, openaccess, mmr}, Doi = {https://doi.org/10.1109/JSEN.2025.3600007}, Url = {https://ieeexplore.ieee.org/document/11139087}, Abstract = {The precise derivation of physical quantities like temperature or pressure at arbitrary locations is useful in numerous contexts, e.g., medical procedures or industrial process engineering. The novel sensor technology of magneto-mechanical resonators (MMRs), based on the interaction of a rotor and stator permanent magnet, allows for the combined tracking of the sensor position and orientation while simultaneously sensing an external measurand. Hence, the quantity is coupled to the torsional oscillation frequency, e.g., by varying the magnet distance. In this article, we analyze the (deflection angle-independent) natural frequency dependence of MMR sensors on the rotor-stator distance and evaluate the performance of theoretical models. The three presented sensors incorporate magnets of spherical and/or cylindrical geometry and can be operated at adjustable frequencies within the range of 61.9–307.3 Hz. Our proposed method to obtain the natural frequency demonstrates notable robustness to variations in the initial deflection amplitudes and quality factors, resulting in statistical errors on the mean smaller than 0.05%. We find that the distance–frequency relationship is well-described by an adapted dipole model accounting for material and manufacturing uncertainties. Their combined effect can be compensated by an adjustment of a single parameter, which drives the median model deviation generally below 0.2%. Our depicted methods and results are important for the design and calibration process of new sensor types utilizing the MMR technique.} } @article{scheffler2025efficient, Author = {K. Scheffler, L. Meyn, F. Foerger, M. Boberg, M. Möddel, and T. Knopp}, Title = {Efficient measurement and representation of magnetic fields in tomographic imaging using ellipsoidal harmonics.}, Journal = {Communications Physics.}, Year = {(2025).}, Volume = {8.}, Number = {(112),}, Month = {January}, Note = {article, openaccess, magneticfield}, Publisher = {Nature:}, Doi = {10.1038/s42005-025-02012-5}, Keywords = {magnetic resonance imaging, magnetic particle imaging, magneticfield }, Abstract = {Given the pivotal role of magnetic fields in modern medicine, there is an increasing necessity for a precise characterization of their strength and orientation at high spatial and temporal resolution. As source-free magnetic fields present in tomographic imaging can be described by harmonic polynomials, they can be efficiently represented using spherical harmonic expansions, which allows for measurement at a small set of points on a sphere surrounding the field of view. However, the majority of closed-bore systems possess a cylindrical field of view, making a sphere an inadequate choice for coverage. Ellipsoids represent a superior geometrical choice, and the theory of ellipsoidal harmonic expansions can be applied to magnetic fields in an analogous manner. Despite the mathematical principles underpinning ellipsoidal harmonics being well-established, their utilization in practical applications remains relatively limited. In this study, we present an effective and flexible approach to measuring and representing magnetic fields present in tomographic imaging, which draws upon the theory of ellipsoidal harmonic expansions.} } @article{boberg2025SHE, Author = {M. Boberg, T. Knopp, and M. Möddel}, Title = {Unique compact representation of magnetic fields using truncated solid harmonic expansions.}, Journal = {European Journal of Applied Mathematics.}, Year = {(2025).}, Pages = {1-28}, Month = {Jan}, Note = {article, magneticfield, openaccess}, Doi = {10.1017/S0956792524000883}, Url = {https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/article/unique-compact-representation-of-magnetic-fields-using-truncated-solid-harmonic-expansions/4654E5547EE13A3894CD42342782231C#article}, Abstract = {Precise knowledge of magnetic fields is crucial in many medical imaging applications such as magnetic resonance imaging (MRI) or magnetic particle imaging (MPI), as they form the foundation of these imaging systems. Mathematical methods are essential for efficiently analysing the magnetic fields in the entire field-of-view. In this work, we propose a compact and unique representation of the magnetic fields using real solid spherical harmonic expansions, which can be obtained by spherical t-designs. To ensure a unique representation, the expansion point is shifted at the level of the expansion coefficients. As an application scenario, these methods are used to acquire and analyse the magnetic fields of an MPI system. Here, the field-free-point of the spatial encoding field serves as the unique expansion point.} } @article{Merbach2025, Author = {T. Merbach, F. Kexel, J. Faltinath, M. Möddel, M. Schlüter, T. Knopp, F. Mohn}, Title = {Wireless and passive pressure detection using magneto-mechanical resonances in process engineering.}, Journal = {Measurement Science and Technology.}, Year = {(2025).}, Volume = {36.}, Number = {(8),}, Pages = {085109}, Month = {aug}, Note = {article, mmr}, Doi = {10.1088/1361-6501/adf2c8}, Url = {https://dx.doi.org/10.1088/1361-6501/adf2c8}, Abstract = {A custom-developed magneto-mechanical resonator (MMR) for wireless pressure measurement is investigated for potential applications in process engineering. The MMR sensor utilises changes in the resonance frequency caused by pressure on a flexible 3D printed membrane. The thickness of the printed membrane plays a crucial role in determining the performance and sensitivity of MMRs and can be tailored to meet the requirements of specific applications. The study includes static and dynamic measurements to determine the pressure sensitivity and temporal resolution of the sensor. The results show a minimum sensitivity of and are in agreement with theoretical calculations and measurements. The maximum sensor readout frequency is 2 Hz in this study. Additionally, the temperature dependence of the sensor is investigated, revealing a significant dependence of the resonance frequency on temperature. The developed MMR offers a promising and versatile method for precise pressure measurements in process engineering environments.} } @article{TsandaDoseRobustness2024CT, Author = {A. Tsanda, H. Nickisch, T. Wissel, T. Klinder, T. Knopp, and M. Grass}, Title = {Dose robustness of deep learning models for anatomic segmentation of computed tomography images.}, Journal = {Journal of Medical Imaging.}, Year = {(2024).}, Volume = {11.}, Number = {(4),}, Pages = {044005}, Note = {article}, Publisher = {SPIE:}, Doi = {10.1117/1.JMI.11.4.044005}, Url = {https://doi.org/10.1117/1.JMI.11.4.044005}, Keywords = {Low-Dose Computed Tomography, Semantic Segmentation, Denoising, Deep Learning}, Abstract = {PurposeThe trend towards lower radiation doses and advances in computed tomography (CT) reconstruction may impair the operation of pretrained segmentation models, giving rise to the problem of estimating the dose robustness of existing segmentation models. Previous studies addressing the issue suffer either from a lack of registered low- and full-dose CT images or from simplified simulations.ApproachWe employed raw data from full-dose acquisitions to simulate low-dose CT scans, avoiding the need to rescan a patient. The accuracy of the simulation is validated using a real CT scan of a phantom. We consider down to 20% reduction of radiation dose, for which we measure deviations of several pretrained segmentation models from the full-dose prediction. In addition, compatibility with existing denoising methods is considered.ResultsThe results reveal the surprising robustness of the TotalSegmentator approach, showing minimal differences at the pixel level even without denoising. Less robust models show good compatibility with the denoising methods, which help to improve robustness in almost all cases. With denoising based on a convolutional neural network (CNN), the median Dice between low- and full-dose data does not fall below 0.9 (12 for the Hausdorff distance) for all but one model. We observe volatile results for labels with effective radii less than 19 mm and improved results for contrasted CT acquisitions.ConclusionThe proposed approach facilitates clinically relevant analysis of dose robustness for human organ segmentation models. The results outline the robustness properties of a diverse set of models. Further studies are needed to identify the robustness of approaches for lesion segmentation and to rank the factors contributing to dose robustness.} } @article{Foerger2024AIS, Author = {F. Foerger, M. Boberg, J. Faltinath, T. Knopp, M. Möddel}, Title = {Design and Optimization of a Magnetic Field Generator for Magnetic Particle Imaging with Soft Magnetic Materials.}, Journal = {Advanced Intelligent Systems.}, Year = {(2024).}, Volume = {6.}, Number = {(11),}, Note = {article}, Doi = {https://doi.org/10.1002/aisy.202400017}, Url = {https://advanced.onlinelibrary.wiley.com/doi/full/10.1002/aisy.202400017}, Abstract = {Magnetic field generators are a key component of Magnetic Particle Imaging (MPI) systems, and their power consumption is a major obstacle on the path to human-sized scanners. Despite their importance, a focused discussion of these generators is rare, and a comprehensive description of the design process is currently lacking. This work presents a methodology for the design and optimization of selection field generators operating with soft magnetic materials outside the linear regime in the context of MPI. Key elements are a mathematical model of magnetic field generators, a formalism for defining field sequences, and a relationship between power consumption and field sequence. These are used to define the design space of a field generator given its system requirements and constraints. The design process is then formulated as an optimization problem. Subsequently, this methodology is then utilized to design a new magnetic field generator specifically for cerebral imaging studies. The optimization result outperforms our existing MPI field generator in terms of power consumption and field of view size, providing a proof-of-concept for the entire methodology. As the approach is very general, it can be extended beyond the MPI context to other areas such as magnetic manipulation of medical devices and micro-robotics.} } @article{mohn_resonant_2024, Author = {F. Mohn, F. Foerger, F. Thieben, M. Möddel, I. Schmale, T. Knopp and M. Graeser}, Title = {Resonant Inductive Coupling Network for Human-Sized Magnetic Particle Imaging.}, Journal = {Review of Scientific Instruments.}, Year = {(2024).}, Volume = {95.}, Number = {(4),}, Pages = {044701}, Note = {article, openaccess, brainimager}, Doi = {10.1063/5.0192784}, Keywords = {Mohn}, Abstract = {In magnetic particle imaging, a field-free region is maneuvered throughout the field of view using a time-varying magnetic field known as the drive-field. Human-sized systems operate the drive-field in the kHz range and generate it by utilizing strong currents that can rise to the kA range within a coil called the drive field generator. Matching and tuning between a power amplifier, a band-pass filter, and the drive-field generator is required. Here, for reasons of safety in future human scanners, a symmetrical topology and a transformer called an inductive coupling network are used. Our primary objectives are to achieve floating potentials to ensure patient safety while attaining high linearity and high gain for the resonant transformer. We present a novel systematic approach to the design of a loss-optimized resonant toroid with a D-shaped cross section, employing segmentation to adjust the inductance-to-resistance ratio while maintaining a constant quality factor. Simultaneously, we derive a specific matching condition for a symmetric transmit--receive circuit for magnetic particle imaging. The chosen setup filters the fundamental frequency and allows simultaneous signal transmission and reception. In addition, the decoupling of multiple drive field channels is discussed, and the primary side of the transformer is evaluated for maximum coupling and minimum stray field. Two prototypes were constructed, measured, decoupled, and compared to the derived theory and method-of-moment based simulations.} } @article{mohn_characterization_2024, Author = {F. Mohn, K. Scheffler, J. Ackers, A. Weimer, F. Wegner, F. Thieben, M. Ahlborg, P. Vogel, M. Graeser, and T. Knopp}, Title = {Characterization of the Clinically Approved MRI Tracer Resotran for Magnetic Particle Imaging in a Comparison Study.}, Journal = {Physics in Medicine & Biology.}, Year = {(2024).}, Volume = {69.}, Number = {(13),}, Pages = {135014}, Note = {article, openaccess}, Doi = {10.1088/1361-6560/ad5828}, Abstract = {Abstract Objective. The availability of magnetic nanoparticles (MNPs) with medical approval for human intervention is fundamental to the clinical translation of magnetic particle imaging (MPI). In this work, we thoroughly evaluate and compare the magnetic properties of an magnetic resonance imaging (MRI) approved tracer to validate its performance for MPI in future human trials. Approach. We analyze whether the recently approved MRI tracer Resotran is suitable for MPI. In addition, we compare Resotran with the previously approved and extensively studied tracer Resovist, with Ferrotran, which is currently in a clinical phase III study, and with the tailored MPI tracer Perimag. Main results. Initial magnetic particle spectroscopy (MPS) measurements indicate that Resotran exhibits performance characteristics akin to Resovist, but below Perimag. We provide data on four different tracers using dynamic light scattering, transmission electron microscopy, vibrating sample magnetometry measurements, MPS to derive hysteresis, point spread functions, and a serial dilution, as well as system matrix based MPI measurements on a preclinical scanner (Bruker 25/20 FF), including reconstructed images. Significance. Numerous approved MNPs used as tracers in MRI lack the necessary magnetic properties essential for robust signal generation in MPI. The process of obtaining medical approval for dedicated MPI tracers optimized for signal performance is an arduous and costly endeavor, often only justifiable for companies with a well-defined clinical business case. Resotran is an approved tracer that has become available in Europe for MRI. In this work, we study the eligibility of Resotran for MPI in an effort to pave the way for human MPI trials.} } @article{Orsi2024CavityMicroscope, Author = {F. Orsi, N. Sauerwein, R. P. Bhatt, J. Faltinath, E. Fedotova, N. Reiter, T. Cantat-Moltrecht, and J.-P. Brantut}, Title = {Cavity Microscope for Micrometer-Scale Control of Atom-Photon Interactions.}, Journal = {PRX Quantum.}, Year = {(2024).}, Volume = {5.}, Number = {(4),}, Pages = {040333}, Note = {article, openaccess, instrumentation}, Doi = {10.1103/PRXQuantum.5.040333}, Url = {https://link.aps.org/doi/10.1103/PRXQuantum.5.040333}, Abstract = {Cavity quantum electrodynamics offers the possibility of observing and controlling the motion of a few or individual atoms, enabling the realization of various quantum technological tasks such as quantum enhanced metrology or quantum simulation of strongly correlated matter. A core limitation of these experiments lies in the mode structure of the cavity field, which is hard coded in the shape and geometry of the mirrors. As a result, most applications of cavity QED trade spatial resolution for enhanced sensitivity. Here, we propose and demonstrate a cavity-microscope device capable of controlling in space and time the coupling between atoms and light in a single-mode high-finesse cavity, reaching a spatial resolution an order of magnitude lower than the cavity-mode waist. This is achieved through local Floquet engineering of the atomic level structure, imprinting a corresponding atom-field coupling. We illustrate this capability by engineering micrometer-scale coupling, using cavity-assisted atomic measurements and optimization. Our system forms an optical device with a single optical axis, has the same footprint and complexity as a standard Fabry-Perot cavity or confocal lens pair, and can be used for any atomic species. This technique opens a wide range of perspectives, from ultrafast cavity-enhanced midcircuit readout to the quantum simulation of fully connected models of quantum matter such as the Sachdev-Ye-Kitaev model.} } @article{thieben_system_2024, Author = {F. Thieben, F. Foerger, F. Mohn, N. Hackelberg, M. Boberg, J.-P. Scheel, Möddel, M. Graeser, and T. Knopp}, Title = {System Characterization of a Human-Sized 3D Real-Time Magnetic Particle Imaging Scanner for Cerebral Applications.}, Journal = {Communications Engineering.}, Year = {(2024).}, Volume = {3.}, Number = {(1),}, Pages = {47}, Note = {article, openaccess, brainimager}, Doi = {10.1038/s44172-024-00192-6}, Keywords = {Mohn}, Abstract = {Abstract Since the initial patent in 2001, the Magnetic Particle Imaging community has endeavored to develop a human-applicable Magnetic Particle Imaging scanner, incorporating contributions from various research fields. Here we present an improved head-sized Magnetic Particle Imaging scanner with low power consumption, operated by open-source software and characterize it with an emphasis on human safety. The focus is on the evaluation of the technical components and on phantom experiments for brain perfusion. We achieved 3D single- and multi-contrast imaging at 4 Hz frame rate. The system characterization includes sensitivity, resolution, perfusion and multi-contrast experiments as well as field measurements and sequence analysis. Images were acquired with a clinically approved tracer and within human peripheral nerve stimulation thresholds. This advanced scanner holds potential as a tomographic imager for diagnosing conditions such as ischemic stroke (different stages) or intracranial hemorrhage in environments lacking electromagnetic shielding, such as the intensive care unit.} } @article{Dora:24, Author = {J. Dora, M. Möddel, S. Flenner, C. G. Schroer, T. Knopp, and J. Hagemann}, Title = {Artifact-suppressing reconstruction of strongly interacting objects in X-ray near-field holography without a spatial support constraint.}, Journal = {Optics Express.}, Year = {(2024).}, Volume = {32.}, Number = {(7),}, Pages = {10801-10828}, Month = {Mar}, Note = {article, openaccess}, Publisher = {Optica Publishing Group:}, Doi = {10.1364/OE.514641}, Url = {https://opg.optica.org/oe/abstract.cfm?URI=oe-32-7-10801}, Abstract = {The phase problem is a well known ill-posed reconstruction problem of coherent lens-less microscopic imaging, where only the squared magnitude of a complex wavefront is measured by a detector while the phase information of the wave field is lost. To retrieve the lost information, common algorithms rely either on multiple data acquisitions under varying measurement conditions or on the application of strong constraints such as a spatial support. In X-ray near-field holography, however, these methods are rendered impractical in the setting of time sensitive in situ and operando measurements. In this paper, we will forego the spatial support constraint and propose a projected gradient descent (PGD) based reconstruction scheme in combination with proper preprocessing and regularization that significantly reduces artifacts for refractive reconstructions from only a single acquired hologram without a spatial support constraint. We demonstrate the feasibility and robustness of our approach on different data sets obtained at the nano imaging endstation of P05 at PETRA III (DESY, Hamburg) operated by Helmholtz-Zentrum Hereon.} } @article{scheffler2024solving, Author = {K. Scheffler, M. Boberg, and T. Knopp}, Title = {Solving the MPI reconstruction problem with automatically tuned regularization parameters.}, Journal = {Physics in Medicine & Biology.}, Year = {(2024).}, Volume = {69.}, Number = {(4),}, Month = {January}, Note = {article, openaccess}, Doi = {10.1088/1361-6560/ad2231}, Keywords = {Image reconstruction, magnetic particle imaging, regularization }, Abstract = {In the field of medical imaging, Magnetic Particle Imaging (MPI) poses a promising non-ionizing tomographic technique with high spatial and temporal resolution. In MPI, iterative solvers are used to reconstruct the particle distribution out of the measured voltage signal based on a system matrix. The amount of regularization needed to reconstruct an image of good quality differs from measurement to measurement, depending on the MPI system and the measurement settings. Finding the right choice for the three major parameters controlling the regularization is commonly done by hand and requires time and experience. In this work, we study the reduction to a single regularization parameter and propose a method that enables automatic reconstruction. The method is qualitatively and quantitatively validated on several MPI data sets showing promising results.} } @article{Nawwas2024PMB, Author = {L. Nawwas, M. Möddel and T. Knopp }, Title = {Analysis of leakage artifacts and their impact on convergence of algebraic reconstruction in multi-contrast magnetic particle imaging.}, Journal = {Physics in Medicine & Biology.}, Year = {(2024).}, Volume = {69.}, Number = {(21),}, Pages = {1-15}, Month = {October}, Note = {article, artifact}, Doi = {10.1088/1361-6560/ad7e77}, Url = {https://iopscience.iop.org/article/10.1088/1361-6560/ad7e77}, Keywords = {article}, Abstract = {Objective. Magnetic particle imaging (MPI) is a tracer-based medical imaging modality with great potential due to its high sensitivity, high spatiotemporal resolution, and ability to quantify the tracer concentration. Image reconstruction in MPI is an ill-posed problem, which the use of regularization methods can address. Multi-contrast MPI reconstructs the signal from different tracer materials or environments separately, resulting in multi-channel images that enable quantification of, for example, temperature or viscosity. Single- and multi-contrast MPI reconstructions produce different kinds of artifacts. The objective of this work is threefold: first, to present the concept of multi-contrast specific MPI channel leakage artifacts; second, to ascertain the source of these leakage artifacts; and third, to introduce a method for their reduction. Approach. A definition for leakage artifacts is established, and a quantification method is proposed. A comprehensive analysis is conducted to establish a connection between the properties of the multi-contrast MPI system matrix and the leakage artifacts. Moreover, a two-step measurement and reconstruction method is introduced to reduce channel leakage artifacts between multi-contrast MPI channels. Main results. The severity of these artifacts correlates with the system matrix shape and condition number and depends on the similarity of the corresponding frequency components. Using the proposed two-step method on both semi-simulated and measured data a significant leakage reduction and speed up the convergence of the multi-contrast MPI reconstruction was observed. Significance. The multi-contrast system matrix analysis we conducted is essential for understanding the source of the channel leakage artifacts and finding methods to reduce them. Our proposed two-step method is expected to improve the potential for real-time multi-contrast MPI applications.} } @article{maass2024equilibrium, Author = {M. Maass, T. Kluth, C. Droigk, H. Albers, K. Scheffler, A. Mertins, and T. Knopp}, Title = {Equilibrium Model With Anisotropy for Model-Based Reconstruction in Magnetic Particle Imaging.}, Journal = {IEEE Transactions on Computational Imaging.}, Year = {(2024).}, Volume = {10.}, Pages = {1588 - 1601}, Month = {November}, Note = {article, openaccess, model-based}, Doi = {10.1109/TCI.2024.3490381}, Keywords = {Mathematical models, Computational modeling, Imaging, Anisotropic magnetoresistance, Magnetization, Image reconstruction, Magnetic moments, Data models, Complexity theory, Particle measurements, Magnetic particle imaging, anisotropic equilibrium model, model-based reconstruction, Lissajous-type excitation}, Abstract = {Magnetic particle imaging is a tracer-based tomographic imaging technique that allows the concentration of magnetic nanoparticles to be determined with high spatio-temporal resolution. To reconstruct an image of the tracer concentration, the magnetization dynamics of the particles must be accurately modeled. A popular ensemble model is based on solving the Fokker-Plank equation, taking into account either Brownian or Néel dynamics. The disadvantage of this model is that it is computationally expensive due to an underlying stiff differential equation. A simplified model is the equilibrium model, which can be evaluated directly but in most relevant cases it suffers from a non-negligible modeling error. In the present work, we investigate an extended version of the equilibrium model that can account for particle anisotropy. We show that this model can be expressed as a series of Bessel functions, which can be truncated based on a predefined accuracy, leading to very short computation times, which are about three orders of magnitude lower than equivalent Fokker-Planck computation times. We investigate the accuracy of the model for 2D Lissajous magnetic particle imaging sequences and show that the difference between the Fokker-Planck and the equilibrium model with anisotropy is sufficiently small so that the latter model can be used for image reconstruction on experimental data with only marginal loss of image quality, even compared to a system matrix-based reconstruction.} } @COMMENT{Bibtex file generated on 2026-5-13 with typo3 si_bibtex plugin. Data from https://www.tuhh.de/ibi/publications }