A Meshfree Solver for the MEG Forward Problem

Noninvasive estimation of brain activity via magnetoencephalography (MEG) involves an inverse problem whose solution requires an accurate and fast forward solver. To this end, we propose the Method of Fundamental Solutions (MFS) as a meshfree alternative to the Boundary Element Method (BEM). The solution of the MEG forward problem is obtained, via the Method of Particular Solutions (MPS), by numerically solving a boundary value problem for the electric scalar potential, derived from the quasi-stationary approximation of Maxwell’s equations. The magnetic field is then computed by the Biot-Savart law. Numerical experiments have been carried out in a realistic single-shell head geometry. The proposed solver is compared with a state-of-the-art BEM solver. A good agreement and a reduced computational load show the attractiveness of the meshfree approach

Data?driven model optimization for optically pumped magnetometer sensor arrays

© 2019 The Authors. Human Brain Mapping published by Wiley Periodicals, Inc. Optically pumped magnetometers (OPMs) have reached sensitivity levels that make them viable portable alternatives to traditional superconducting technology for magnetoencephalography (MEG). OPMs do not require cryogenic cooling and can therefore be placed directly on the scalp surface. Unlike cryogenic systems, based on a well-characterised fixed arrays essentially linear in applied flux, OPM devices, based on different physical principles, present new modelling challenges. Here, we outline an empirical Bayesian framework that can be used to compare between and optimise sensor arrays. We perturb the sensor geometry (via simulation) and with analytic model comparison methods estimate the true sensor geometry. The width of these perturbation curves allows us to compare different MEG systems. We test this technique using simulated and real data from SQUID and OPM recordings using head-casts and scanner-casts. Finally, we show that given knowledge of underlying brain anatomy, it is possible to estimate the true sensor geometry from the OPM data themselves using a model comparison framework. This implies that the requirement for accurate knowledge of the sensor positions and orientations a priori may be relaxed. As this procedure uses the cortical manifold as spatial support there is no co-registration procedure or reliance on scalp landmarks

Using reciprocity for relating the simulation of transcranial current stimulation to the EEG forward problem

To explore the relationship between transcranial current stimulation (tCS) and the electroencephalography (EEG) forward problem, we investigate and compare accuracy and efficiency of a reciprocal and a direct EEG forward approach for dipolar primary current sources both based on the finite element method (FEM), namely the adjoint approach (AA) and the partial integration approach in conjunction with a transfer matrix concept (PI). By analyzing numerical results, comparing to analytically derived EEG forward potentials and estimating computational complexity in spherical shell models, AA turns out to be essentially identical to PI. It is then proven that AA and PI are also algebraically identical even for general head models. This relation offers a direct link between the EEG forward problem and tCS. We then demonstrate how the quasi-analytical EEG forward solutions in sphere models can be used to validate the numerical accuracies of FEM-based tCS simulation approaches. These approaches differ with respect to the ease with which they can be employed for realistic head modeling based on MRI-derived segmentations. We show that while the accuracy of the most easy to realize approach based on regular hexahedral elements is already quite high, it can be significantly improved if a geometry-adaptation of the elements is employed in conjunction with an isoparametric FEM approach. While the latter approach does not involve any additional difficulties for the user, it reaches the high accuracies of surface-segmentation based tetrahedral FEM, which is considerably more difficult to implement and topologically less flexible in practice. Finally, in a highly realistic head volume conductor model and when compared to the regular alternative, the geometry-adapted hexahedral FEM is shown to result in significant changes in tCS current flow orientation and magnitude up to 45° and a factor of 1.66, respectively

Numerical modeling in electro- and magnetoencephalography

This Thesis concerns the application of two numerical methods, Boundary Element Method (BEM) and Finite Element Method (FEM) to forward problem solution of bioelectromagnetic source localization in the brain. The aim is to improve the accuracy of the forward problem solution in estimating the electrical activity of the human brain from electric and magnetic field measurements outside the head. Electro- and magnetoencephalography (EEG, MEG) are the most important tools enabling us to gather knowledge about the human brain non-invasively. This task is alternatively named brain mapping. An important step in brain mapping is determining from where the brain signals originate. Using appropriate mathematical models, a localization of the sources of measured signals can be performed. A general motivation of this work was the fact that source localization accuracy can be improved by solving the forward problem with higher accuracy. In BEM studies, accurate representation of model geometry using higher order elements improves the solution of the forward problem. In FEM, complex conductivity information can be incorporated into numerical model. Using Whitney-type finite elements instead of using singular sources such as point dipoles, primary and volume currents are represented as continuous sources. With comparison to analytical solutions available in simple geometries such as sphere, the studied numerical methods show improvements in the forward problem solution of bioelectromagnetic source imaging.reviewe

Techniques for imaging small impedance changes in the human head due to neuronal depolarisation

A new imaging modality is being developed, which may be capable of imaging small impedance changes in the human head due to neuronal depolarization. One way to do this would be by imaging the impedance changes associated with ion channels opening in neuronal membranes in the brain during activity. The results of previous modelling and experimental studies indicated that impedance changes between 0.6%and 1.7% locally in brain grey matter when recorded at DC. This reduces by a further of 10% if measured at the surface of the head, due to distance and the effect of the resistive skull. In principle, this could be measured using Electrical Impedance Tomography (ElT) but it is close to its threshold of detectability. With the inherent limitation in the use of electrodes, this work proposed two new schemes. The first is a magnetic measurement scheme based on recording the magnetic field with Superconducting Quantum Interference Devices (SQUIDs), used in Magnetoencephalography (MEG) as a result of a non-invasive injection of current into the head. This scheme assumes that the skull does not attenuate the magnetic field. The second scheme takes into consideration that the human skull is irregular in shape, with less and varying conductivity as compared to other head tissues. Therefore, a key issue is to know through which electrodes current can be injected in order to obtain high percentage changes in surface potential when there is local conductivity change in the head. This model will enable the prediction of the current density distribution at specific regions in the brain with respect to the varying skull and local conductivities. In the magnetic study, the head was modelled as concentric spheres, and realistic head shapes to mimic the scalp, skull, Cerebrospinal Auid (CSF) and brain using the Finite Element Method (FEM). An impedance change of 1 % in a 2cm-radius spherical volume depicting the physiological change in the brain was modelled as the region of depolarisation. The magnetic field, 1 cm away from the scalp, was estimated on injecting a constant current of 100 µA into the head from diametrically opposed electrodes. However, in the second scheme, only the realistic FEM of the head was used, which included a specific region of interest; the primary visual cortex (V1). The simulated physiological change was the variation in conductivity of V1 when neurons were assumed to be firing during a visual evoked response. A near DC current of 100 µA was driven through possible pairs of 31 electrodes using ElT techniques. For a fixed skull conductivity, the resulting surface potentials were calculated when the whole head remained unperturbed, or when the conductivity of V1 changed by 0.6%, 1 %, and 1.6%. The results of the magnetic measurement predicted that standing magnetic field was about 10pT and the field changed by about 3fT (0.03%) on depolarization. For the second scheme, the greatest mean current density through V1 was 0.020 ± 0.005 µAmm-2, and occurred with injection through two electrodes positioned near the occipital cortex. The corresponding maximum change in potential from baseline was 0.02%. Saline tank experiments confirmed the accuracy of the estimated standing potentials. As the noise density in a typical MEG system in the frequency band is about 7fT/√Hz, it places the change at the limit of detectability due to low signal to noise ratio. This is therefore similar to electrical recording, as in conventional ElT systems, but there may be advantages to MEG in that the magnetic field direcdy traverses the skull and instrumentation errors from the electrode-skin interface will be obviated. This has enabled the estimation of electrode positions most likely to permit recording of changes in human experiments and suggests that the changes, although tiny, may just be discernible from noise

Spherical harmonic based noise rejection and neuronal sampling with multi-axis OPMs

In this study we explore the interference rejection and spatial sampling properties of multi-axis Optically Pumped Magnetometer (OPM) data. We use both vector spherical harmonics and eigenspectra to quantify how well an array can separate neuronal signal from environmental interference while adequately sampling the entire cortex. We found that triaxial OPMs have superb noise rejection properties allowing for very high orders of interference (L=6) to be accounted for while minimally affecting the neural space (2dB attenuation for a 60-sensor triaxial system). We show that at least 11th order (143 spatial degrees of freedom) irregular solid harmonics or 95 eigenvectors of the lead field are needed to model the neural space for OPM data (regardless of number of axes measured). This can be adequately sampled with 75-100 equidistant triaxial sensors (225-300 channels) or 200 equidistant radial channels. In other words, ordering the same number of channels in triaxial (rather than purely radial) configuration may give significant advantages not only in terms of external noise rejection but also by minimizing cost, weight and cross-talk