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Thermal motion of charge carriers in a conducting object causes magnetic field noise that interferes with sensitive measurements nearby the conductor. In this paper, we describe a method to compute the spectral properties of the thermal magnetic noise from arbitrarily-shaped thin conducting objects. We model divergence-free currents on a conducting surface using a stream function and calculate the magnetically independent noise-current modes in the quasi-static regime. We obtain the power spectral density of the thermal magnetic noise as well as its spatial correlations and frequency dependence. We describe a numerical implementation of the method; we model the conducting surface using a triangle mesh and discretize the stream function. The numerical magnetic noise computation agrees with analytical formulas. We provide the implementation as a part of the free and open-source software package bfieldtools.

In this paper, we analyze spatial sampling of electro- (EEG) magnetoencephalography (MEG), where the electric or magnetic field is typically sampled on a curved surface such as the scalp. Using simulated measurements, we study the spatial-frequency content in EEG as well as in on- and off-scalp MEG. The analysis suggests that on-scalp MEG would generally benefit from three times more samples than EEG or off-scalp MEG. Based on the theory of Gaussian processes and experimental design, we suggest an approach to obtain sampling locations on surfaces that are optimal with respect to prior assumptions. Additionally, the approach allows to control, e.g., the uniformity of the sampling locations in the grid. By simulating the performance of grids constructed with different priors, we show that for a low number of spatial samples, model-informed non-uniform sampling can be beneficial. For a large number of samples, uniform sampling grids yield nearly the same total information as the model-informed grids., Comment: arXiv admin note: substantial text overlap with arXiv:1912.05401

Surface currents provide a general way to model static magnetic fields in source-free volumes. To facilitate the use of surface currents in magneto-quasistatic problems, we have implemented a set of computational tools in a Python package named bfieldtools. In this work, we describe the physical and computational principles of this toolset. To be able to work with surface currents of arbitrary shape, we discretize the currents on triangle meshes using piecewise-linear stream functions. We apply analytical discretizations of integral equations to obtain the magnetic field and potentials associated with the discrete stream function. In addition, we describe the computation of the spherical multipole expansion and a novel surface-harmonic expansion for surface currents, both of which are useful for representing the magnetic field in source-free volumes with a small number of parameters. Last, we share examples related to magnetic shielding and surface-coil design using the presented tools., Comment: Updated document style, some references and wordings. 15 pages, 8 figures

We present a novel open-source Python software package, bfieldtools, for magneto-quasistatic calculations with current densities on surfaces of arbitrary shape. The core functionality of the software relies on a stream-function representation of surface-current density and its discretization on a triangle mesh. Although this stream-function technique is well-known in certain fields, to date the related software implementations have not been published or have been limited to specific applications. With bfieldtools, we aimed to produce a general, easy-to-use and well-documented open-source software. The software package is written purely in Python; instead of explicitly using lower-level languages, we address computational bottlenecks through extensive vectorization and use of the NumPy library. The package enables easy deployment, rapid code development and facilitates application of the software to practical problems. In this paper, we describe the software package and give an extensive demonstration of its use with an emphasis on one of its main applications -- coil design., Comment: Updated abstract, document style and reference format. 13 pages, 7 figures

Magnetic fields associated with currents flowing in tissue can be measured non-invasively by means of zero-field-encoded ultra-low-field magnetic resonance imaging (ULF MRI) enabling current density imaging (CDI) and possibly conductivity mapping of human head tissues. Since currents applied to a human are limited by safety regulations and only a small fraction of the current passes through the relatively high-resistive skull, a sufficient signal-to-noise ratio (SNR) may be difficult to obtain when using this method. In this work, we study the relationship between the image SNR and the SNR of the field reconstructions from zero-field-encoded data. We evaluate these results for two existing ULF MRI scanners, one ultra-sensitive single-channel system and one whole-head multi-channel system, by simulating sequences necessary for current-density reconstruction. We also derive realistic current-density and magnetic-field estimates from finite-element-method simulations based on a three-compartment head model. We found that existing ULF-MRI systems reach sufficient SNR to detect intra-cranial current distributions with statistical uncertainty below 10%. However, they also reveal that image artifacts influence the reconstruction quality. Further, our simulations indicate that current-density reconstruction in the scalp requires a resolution less than 5 mm and demonstrate that the necessary sensitivity coverage can be accomplished by multi-channel devices., Comment: 18 pages, 8 figures. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 686865

With a hybrid MEG--MRI device that uses the same sensors for both modalities, the co-registration of MRI and MEG data can be replaced by an automatic calibration step. Based on the highly accurate signal model of ultra-low-field (ULF) MRI, we introduce a calibration method that eliminates the error sources of traditional co-registration. The signal model includes complex sensitivity profiles of the superconducting pickup coils. In ULF MRI, the profiles are independent of the sample and therefore well-defined. In the most basic form, the spatial information of the profiles, captured in parallel ULF-MR acquisitions, is used to find the exact coordinate transformation required. We assessed our calibration method by simulations assuming a helmet-shaped pickup-coil-array geometry. Using a carefully constructed objective function and sufficient approximations, even with low-SNR images, sub-voxel and sub-millimeter calibration accuracy was achieved. After the calibration, distortion-free MRI and high spatial accuracy for MEG source localization can be achieved. For an accurate sensor-array geometry, the co-registration and associated errors are eliminated, and the positional error can be reduced to a negligible level., Comment: 11 pages, 8 figures. This work is part of the BREAKBEN project and has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 686865

Superconducting QUantum-Interference Devices (SQUIDs) make magnetic resonance imaging (MRI) possible in ultra-low microtesla-range magnetic fields. In this work, we investigate the design parameters affecting the signal and noise performance of SQUID-based sensors and multichannel magnetometers for MRI of the brain. Besides sensor intrinsics, various noise sources along with the size, geometry and number of superconducting detector coils are important factors affecting the image quality. We derive figures of merit based on optimal combination of multichannel data, analyze different sensor array designs, and provide tools for understanding the signal detection and the different noise mechanisms. The work forms a guide to making design decisions for both imagingand sensor-oriented readers.