Localizing on-scalp MEG sensors using an array of magnetic dipole coils

Christoph Pfeiffer; Lau M Andersen; Daniel Lundqvist; Matti Hämäläinen; Justin F Schneiderman; Robert Oostenveld

doi:10.1371/journal.pone.0191111

. 2018 May 10;13(5):e0191111. doi: 10.1371/journal.pone.0191111

Localizing on-scalp MEG sensors using an array of magnetic dipole coils

Christoph Pfeiffer ^1,^*, Lau M Andersen ², Daniel Lundqvist ², Matti Hämäläinen ^2,^3,^4,⁵, Justin F Schneiderman ⁶, Robert Oostenveld ^2,⁷

Editor: Blake Johnson⁸

PMCID: PMC5944911 PMID: 29746486

Abstract

Accurate estimation of the neural activity underlying magnetoencephalography (MEG) signals requires co-registration i.e., determination of the position and orientation of the sensors with respect to the head. In modern MEG systems, an array of hundreds of low-T_c SQUID sensors is used to localize a set of small, magnetic dipole-like (head-position indicator, HPI) coils that are attached to the subject’s head. With accurate prior knowledge of the positions and orientations of the sensors with respect to one another, the HPI coils can be localized with high precision, and thereby the positions of the sensors in relation to the head. With advances in magnetic field sensing technologies, e.g., high-T_c SQUIDs and optically pumped magnetometers (OPM), that require less extreme operating temperatures than low-T_c SQUID sensors, on-scalp MEG is on the horizon. To utilize the full potential of on-scalp MEG, flexible sensor arrays are preferable. Conventional co-registration is impractical for such systems as the relative positions and orientations of the sensors to each other are subject-specific and hence not known a priori. Herein, we present a method for co-registration of on-scalp MEG sensors. We propose to invert the conventional co-registration approach and localize the sensors relative to an array of HPI coils on the subject’s head. We show that given accurate prior knowledge of the positions of the HPI coils with respect to one another, the sensors can be localized with high precision. We simulated our method with realistic parameters and layouts for sensor and coil arrays. Results indicate co-registration is possible with sub-millimeter accuracy, but the performance strongly depends upon a number of factors. Accurate calibration of the coils and precise determination of the positions and orientations of the coils with respect to one another are crucial. Finally, we propose methods to tackle practical challenges to further improve the method.

Introduction

Magnetoencephalography (MEG), which measures magnetic fields generated by neural currents, is a tool for non-invasive studies of human brain function. Via sampling of the magnetic fields around the head surface, one can determine neural activity in the brain with high temporal and spatial accuracy. Detection of these weak fields (on the order of 10⁻¹⁴ to 10⁻¹⁵ T) requires very sensitive magnetometers. State-of-the-art MEG systems employ hundreds of low critical temperature (low-T_c) superconducting quantum interference devices (SQUIDs). [1, 2]

Conventional low-T_c SQUIDs have to be cooled with liquid helium (4.2 K), which necessitates a well-insulated dewar. This results in a distance between sensor (cryogenic temperature) and head (room temperature) that is on the order of ca. 2–3 centimeters. As magnetic fields decay rapidly with distance, significant gains in signal-to-noise ratio (SNR) would be achieved if the distances between the sensors and the head were reduced.

Fixed-helmet sensor arrays, like the ones used in conventional state-of-the-art MEG systems, allow a high sensor density but are restricted to a single, fixed sensor arrangement. The helmet design and sensor arrangement are chosen to fit the majority of head sizes of a target population, in most cases adults. This excludes some subjects, e.g., with large heads. On the other hand, for subjects with heads significantly smaller than the helmet, a large fraction of the sensors is far away from the head. This is especially problematic for research on young children–a subject group of high interest in neuroscience (e.g. in studies of brain development).

New technologies, such as optically pumped magnetometers (OPMs) and high-T_c SQUIDs, allow us to get much closer (within a few millimeters) to the head; we refer to these techniques jointly as “on-scalp MEG”. Reducing the distance between sensor and source results in stronger signals and improved spatial resolution, allowing for an improved view on brain activity. Despite their typically higher sensor noise levels, several groups have shown such MEG systems can outperform state-of-the-art systems in terms of extracted information and localization accuracy [3–5] and several measurements have demonstrated their potential [6–11].

On-scalp MEG allows the construction of flexible arrays where the magnetic field sensors are positioned individually or in small modules on the head of the subject. This allows the head size variability in the population to be dealt with by adjusting the sensor array to the individual head–ensuring high SNR independent of head size.

While offering good coverage and improved sensor-to-head distance, adjustable sensor arrays have a practical issue that does not apply to fixed-helmet sensor arrays: how to determine the positions and orientations of all individual magnetometers with respect to the subject’s head. Accurate reconstruction of neural activity from a set of magnetic field measurements from outside the head requires a precise assessment of the location and orientation at which the field is sampled. This is especially important when recording closer to the head [4].

In a flexible system the location and orientation of each individual sensor has to be determined—in contrast to current systems, where the sensor array is fixed and only the whole array needs to be localized relative to the head. We propose to do this using localization coils similar to the ones used in commercial state-of-the-art MEG systems. In commercial systems, three to five localization coils are attached to the head of the subject. These are driven with well-defined, periodic currents at unique frequencies, generating a magnetic dipole field distribution at these frequencies for the different coils. The field strength of each coil detected at each sensor location is extracted from the sensor signals, e.g. by Fourier transform. When the sensor locations inside the helmet and the magnetic moments of the coils are known, the measured field distribution can be used to determine the positions of the coils relative to the sensors and, thereby, the location and orientation of the head with respect to the helmet.

Instead of localizing the dipolar coils attached to the head with a set of well-known (in respect to position and orientation) sensors, we propose to invert the procedure. We use a well-defined array of dipolar coils to localize the positions and orientations of individual sensors (see Fig 1).

Fig 1 — Left: classical head localization where a large array of sensors (blue; here: the low-T_c SQUID magnetometer locations of the Elekta Neuromag TRIUX helmet) localizes a set of small, dipolar coils (green) that are attached to the subject’s head. Right: example of the proposed magnetometer localization where an array of dipolar coils (green) is used to localize individual (or small arrays of) on-scalp magnetometers (red).

The aim of this study was to simulate the proposed sensor localization procedure with dipolar coils for flexible on-scalp MEG systems and–in order to guide future work—investigate the influence of different parameters on the localization accuracy. The parameters we varied are (i) accuracy of a priori knowledge of calibration, position, and orientation of the localization coils, (ii) magnetic moment of the localization coils, (iii) magnetometer noise, (iv) localization time, and (v) number of localization coils.

The sensor localization procedure proposed here is not needed for commercial state-of-the-art MEG systems, which contain a large number of magnetometers and/or gradiometers whose position is fixed and well calibrated. Such systems, however, provide a good reference and can hence be used to test the performance of the proposed procedure in practical experiments. Besides exploring the parameters of the localization coil configuration and the recording details, we investigated how accurately small magnetometer arrays (1 to 10 channels) with a known relative position and orientation can be localized using this approach.

Materials and methods

All simulations were done on the basis of geometrical data that was acquired from a MEG measurement in a previous study [11]. The involvement of the human volunteer was performed in accordance with the technical development prerogative of the Swedish law for ethical approval of research. The subject was informed about the purpose of the study and consented to participating in the previous as well as in this study.

We performed a number of localizations of individual magnetometers to assess statistical variability. The magnetometer positions were based on the layout of the commercial state-of-the-art low-T_c SQUID-based Elekta Neuromag “TRIUX” system (hereafter called low-T_c system). A corresponding set of on-scalp sensor positions were obtained by projecting the TRIUX sensor positions to a distance of 1mm from the subject’s scalp by finding the closest point on the scalp surface and moving it 1 mm in direction of the surface normal at that point. This distance was chosen to account for the minimum separation that can reasonably be achieved with on-scalp sensors [9]. Our particular choice of on-scalp sensor locations was motivated by it being evenly distributed over the head and its similarity to the low-T_c channel distribution, as shown in Fig 2. Each system has 102 possible magnetometer locations.

Fig 2 — Left: Magnetometer positions used for the low-T_c system (blue; based on the Elekta Neuromag TRIUX). Right: Magnetometer positions used for the on-scalp system (red). Localization is performed for each magnetometer individually.

We approximated the localization coils as magnetic dipoles to simplify the simulations. This is a valid assumption for coils that are flat and small in size compared to the distance to the magnetometers, like the ones that are typically used in commercial systems (e.g. CTF and Elekta Neuromag TRIUX, see Fig 3).

The low-T_c system allows for 12 localization coils to be configured in software and hardware and comes with localization coils in sets of 10 [11]. Consequently, we have been using 10 localization coils in previous measurements. For most simulations we used 10 locations based on the actual locations from a previous measurement. The coil locations on the head were obtained using a Polhemus Fastrak 3D digitization device (which is part of the standard co-registration procedure when using the low-T_c system). In optimal conditions, the Fastrak achieves accuracies of <1 mm. For practical measurements, however, position errors of around 2 mm are more common [12]. Fig 4 shows the standard localization coil positions on the subject’s head we used.

Magnetometer localization

The field from a magnetic dipole that is picked up by a magnetometer, i.e., the forward solution, is given by:

S (r_{j k}, m_{j}, {\hat{n}}_{k}) = B (r_{j k}, m_{j}) ∙ {\hat{n}}_{k} = \frac{μ_{0}}{4 π} (\frac{3 r_{j k} (m_{j} ∙ r_{j k})}{{| r_{j k} |}^{5}} - \frac{m_{j}}{{| r_{j k} |}^{3}}) ∙ {\hat{n}}_{k}

(Eq 1)

where r_jk is a vector describing the relative position of magnetometer k to the magnetic dipole j, m is the magnetic dipole moment vector and $\hat{n}$ is a unit vector describing the sensitive direction of the magnetometer. For convenience the forward solution is typically defined by the sensor’s spatial sensitivity to (here: magnetic) dipoles, the so-called lead field L_m.

S (r_{j k}, m_{j}, {\hat{n}}_{k}) = B (r_{j k}, \frac{m_{j}}{| m_{j} |}) | m_{j} | ∙ {\hat{n}}_{k} = L_{m} (r_{j k}, {\hat{m}}_{j}, {\hat{n}}_{k}) ∙ | m_{j} |

(Eq 2)

Localization of a magnetic dipole coil is achieved by fitting a magnetic dipole to the measured/simulated data. The error between the data and the forward solution for the magnetic dipole is the minimized with a non-linear optimization procedure by varying the position, orientation, and moment of the dipole. The error function e used for the optimization is:

e (m_{f i t}) = S_{d a t a} - L_{m} m_{f i t}

(Eq 3)

with S_data a vector containing the measured (or simulated) magnetometer signals, L_m a magnetometers-by-localization-coils lead field matrix, and m_fit a vector containing the dipole moment strengths of the localization coil(s).

For our approach we want to localize individual magnetometers with a set of localization coils instead of localizing coils with a sensor array as in current commercial low-T_c systems. We therefore exchange the roles of magnetometer and localization coils:

m = {L_{m}}^{+} * S

(Eq 4)

From symmetry follows that ${[L_{m} (r, \hat{m}, \hat{n})]}^{+} = L_{m} (- r, \hat{n}, \hat{m})$ i.e., the pseudo inverse of the lead field equals the lead field that would result if the magnetometers were generating the magnetic field and the localization coils sensing it.

By swapping in- and outputs the same dipole fitting functions used in classical head localization can therefore be used to localize a magnetometer with a set of localization coils. A requisite hereto is that positions, orientations, and amplitudes of the localization coils are known. In comparison: classical head localization in commercial full-head systems requires knowledge of position, orientation, and (input) sensitivity of the magnetometers.

The magnetic dipole fits were performed in MATLAB version R2015a (Natick, MA) using the FieldTrip toolbox [13].

Data generation

The data was generated by applying the forward solution to the signals driving the individual localization coils and summing the resulting magnetic fields that a magnetometer picks up. The localization coils were driven with sine wave signals at different frequencies from 218 Hz up in steps of 7 Hz, in line with the TRIUX system. The driving signal m of coil j is given by:

m_{j} (t) = m_{j, 0} * s i n (2 π f_{j} t)

(Eq 5)

where f is the frequency and m₀ the amplitude of the magnetic moment.

In addition to the signals from the localization coils, Gaussian noise $N \sim N (0, σ^{2})$ was added to the magnetometer signals.

S (t) = \sum_{j} {(L}_{m} m_{j} (t)) + N

(Eq 6)

Due to the use of different frequencies, the magnetic moment amplitudes of the individual localization coils can be extracted from the magnetometer signal through Fourier transform or phase-locked detection [14].

To reduce computation time, we chose the real positions of the localization coils as starting points for the optimization. This represents a fitting with an initial guess (i.e., from the digitization point) similar to the way the digitized locations of the coils are used in conventional head localization. Alternatively, a grid-search could be performed to find an initial starting point for the non-linear fit.

We localized the magnetometers individually and defined the localization error as the difference between the actual magnetometer location and the location obtained from the fit. The localization errors were then averaged over all magnetometers; this entire process was executed for each of the two systems.

Signal-to-noise ratio

To investigate the influence of the signal-to-noise ratio (SNR) on the localization accuracy, we performed simulations with different magnetometer noise levels and magnetic moment amplitudes. The noise was varied from σ = 0.1 to 500 fT/Hz^1/2, the magnetic moment amplitudes from m_j,0 = 0.25 nAm² to 250 nAm².

Since the Fourier transform is used for localization, the length of the recorded data also determines the SNR. Random noise reduces by averaging with the square root of the data length. If the signal does not change, the SNR therefore increases with increasing data length. Head localization with commercial systems is limited by practical considerations: averaging over time acts as a low pass filter, thereby limiting the speed with which movements of the head relative to the sensor array can be detected. Head localization is therefore typically performed with a sliding window of about one second.

Recently experimental procedures for low-T_c MEG were proposed where the relative positions of the head of the subject with respect to the sensors are restricted by the use of a 3-D printed or molded head cast [15, 16]. Since head movements are minimized by such an approach, the data length used for localization can be increased significantly to achieve higher SNR. Although on-scalp MEG systems are still at an experimental stage, we anticipate that these systems will have the sensors fixed relative to the head, as with EEG electrode caps or with the 3-D printed MEG head cast procedure (as in [16]). Therefore we expect that the head and sensors will not move relative to each other and that long stretches of data can be acquired with a constant localization coil signal.

To compare our simulations to present-day commercial MEG systems, we used data segments of one second with a sampling frequency of 1 kHz. Additionally, we performed a simulation with 10-minute data length to test how much the localization accuracy could be improved in systems where head movements are not a concern.

In the following simulations we used typical noise values for low-T_c and on-scalp magnetometers. While low-T_c SQUID magnetometers with noise levels around 0.15 fT/Hz^1/2 have been reported [17], these are, to date, not standard in MEG. Magnetometers in commercial systems like the TRIUX have typical noise levels of around 3 fT/Hz^1/2. While there have been several reports of OPMs and high-T_c SQUIDs with noise levels as low as a few fT/Hz^1/2 [18, 19], such high-performance sensors have only been produced in small quantities. We therefore employed a more conservative noise level of 20 fT/Hz^1/2 that may be more typical of commercially-produced on-scalp sensors. We furthermore used 10 nAm² for the magnetic dipole moment amplitude, consistent with the TRIUX system.

Calibration, position and orientation

As mentioned above, localizing the magnetometers requires knowledge of the magnetic moment, position and orientation of the localization coils. To investigate the influence of these parameters on the localization accuracy we simulated different types of inaccuracies in these parameters.

Instead of varying the magnetic moment we applied an error to the calibration values of the localization coils. In practice, the operator of a MEG system has control over the current applied to the localization coils, but has to rely on the accuracy of the calibration value for the conversion from current to magnetic moment. Errors in the calibration value thus translate to the magnetic moment. We therefore performed simulations with random calibration errors in a range of e_calib = ±0.1 to ±10%.

We furthermore simulated errors in the position and orientation of the localization coils. First, we added random position errors of e_pos = ±0.5 to ±5 mm in three orthogonal directions. Subsequently, we performed simulations with randomly applied orientation errors by rotating the localization coils e_ori = ±1 to ±10 degrees around the X-, Y-, and Z-axes.

We simulated each of the aforementioned error cases (calibration, position and orientation) ten times. The errors were then averaged over those repetitions as well as over the magnetometer locations.

Localization coil arrays

We also simulated the effects of using different numbers of localization coils. Here we followed two approaches corresponding to different experimental procedures for MEG recordings.

First, we assumed the case of recording the neuromagnetic field only on a small area of the head, e.g., to measure the dipolar pattern of a single source. In such a case, the localization coils can be placed close to that area to ensure good signals from all coils. We realized this by constructing a ring of localization coils with a 5 cm radius around each of the magnetometer positions (see Fig 5) and localizing all the magnetometers falling inside that ring. We simulated such ring-shaped localization arrays at all 94 sensor locations where such a ring fit on the head. This test was furthermore carried out with 4 to 10 localization coils.

Fig 5 — Placement of localization coils (green) in a small (here: 7-coil) array around an on-scalp magnetometer position (red).

Second, we assumed that MEG signals on the whole head are to be recorded in a single session. In such a measurement, a single localization array is required that should provide good coverage over the whole head. We simulated whole head arrays of 10, 12, 21 and 32 localization coils (see Fig 6). The layouts are based on the 10–10 EEG system locations and chosen with the aim of uniform coverage. With such whole-head localization arrays, we localized all magnetometer positions and averaged the localization error.

Fig 6 — Localization coil placement in 10- (top-left), 12- (top-right), 21- (bottom-left) and 32- (bottom-right) coil full-head arrays.

Small magnetometer arrays

Finally, we investigated how a small array of magnetometers would compare in terms of localization accuracy. OPMs can be designed such that a single vapor cell provides 4 measurement points on the corners [20] and high-T_c systems can be composed of several small cryostats that each contain multiple, tightly packed sensors. If the relative positions and orientations of a set of magnetometers are known, they can be fitted as a group, potentially improving accuracy. We simulated this for arrays of 1 to 10 magnetometers, where we used the nearest neighbors of each magnetometer to form a mini-array (see Fig 7). We determined the average localization error as a function of the numbers of magnetometers in the mini-array.

Fig 7 — Example of a small array of on-scalp magnetometers (red) that are fit as a group with a 32-coil localization array (green).