On-scalp Yttrium-Iron Garnet sensor arrays for brain source localization: Cramér-Rao bound analysis

A new type of highly sensitive Magnetometers based on Yttrium-Iron Garnet films (YIGM) is being designed and applied to brain signals recording. We modelled sensor arrays based on YIGM and well-established OPM (QuSpin Inc.) and SQUID (Elekta Neuromag) arrays used for magnetoencephalography (MEG). In this simulation study we investigate the inverse problem accuracy bound depending on the sensor arrays configuration, types of magnetometers and instrumental noise level. We estimate the variance of the probability of dipole source localization using the Cramér-Rao Lower Bound (CRLB); the CRLB is calculated with use of the numerical approximation of the forward model and its partial derivative parametrization. Our models confirm that the development of multi-channel YIG-MEG systems requires noise reduction, rather than an increase in number of sensors, to outperform OPM and SQUID arrays. The modelling showed the advantages of YIG-based gradiometry compared to conventional SQUID-system.


Introduction
Magnetoencephalography (MEG) is a powerful non-invasive modality for functional research of neuronal electrical activity.Compared to other non-invasive modalities such as EEG (electroencephalography) and fMRI (functional magnetic resonance imaging), MEG provides researchers with the most extensive amount of information due to the transparency of biological tissues to magnetic fields.By combining this fact with its high temporal resolution and sensitivity to high frequencies, MEG allows to state and solve accurately the inverse problem of neuronal currents sources localization on the base of measurement of the magnetic fields produced by them [1,2].
At the same time MEG remains a rather unpopular technique due to the high operation costs.Conventional MEG are based on ultra-sensitive SQUID sensors (superconducting quantum interference devices) that operate at extremely low temperatures.Therefore the great expenses are connected with constant need in liquid helium.
However, recent technological advances in the field of low-frequency magnetometry -Optically Pumped Magnetometers (OPM) -could increase the availability of MEG [3,4].In parallel to OPM a new solid-state magnetometer based on Yttrium-Iron Garnet films (YIGM) is designed and tested for brain signal recording [5,6].Both, OPM and YIGM-based MEG systems are promising not only in terms of reliability and durability but also as sensors that can register various components of MEG signals which may bring a reasonable advantage [7].
The development of YIG-MEG system requires modelling of the sensor characteristics in order to obtain optimal array designs.In our previous study we modelled YIGM sensor arrays and compared them with conventional SQUID system (Elekta Neuromag) and custom OPM systems based on QZFM sensors (QuSpin Inc.) in terms of metrics describing the quality of the forward solution [6].As a continuation of this work, we investigate the inverse problem accuracy bound depending on the sensor arrays, types of magnetometers and instrumental noise level.
1.1.Metrics for estimation of sensor arrays efficiency Forward solution metrics.On the basis of the Gain matrix (see "Problem statement" for reference) we can consider the following: • the Signal-to-Noise ratio (SNR) estimates the sensitivity of each sensor with respect to sources in different cortical regions; • the Signal power (SP) reflects the integral source detectability by the particular sensor system; • the Total Shannon information capacity (TiC) of the sensor system indicates the overall quality of information gathered by it.
Refer to [6] for the detailed forward metrics calculation and sensor systems analysis.The aforementioned metrics provide no information regarding inverse solution accuracy of chosen sensor array while it is supposed to be the key parameter of any MEG-system.This paper is dedicated to estimation of accuracy of the inverse problem solution in terms of Cramér-Rao lower bound estimation of dipole localization variance with modelled sensor arrays of different types.
The Cramér-Rao lower bound of inverse solution.The Cramér-Rao inequality refers to methods of statistical processing of noisy experimental data, on the basis of which one or more parameters of the studied object/process should be calculated.Any physical measurements are affected by noise.The quality of the signal from statistical point of view can be described in terms of measured parameter variance: the lower the variance the higher the measurement quality.The Cramér-Rao inequality allows us to obtain a Lower Bound (CRLB) for the variance of the recovered parameters by expressing the parameter's variance through the Fisher information matrix [8] and the characteristics of the noise.
The Cramér-Rao approach was used to estimate the dipole localization error on the basis of the forward model analysis.Early works introducing CRLB for a cortical dipole parameters show the correspondence of this estimation to Monte-Carlo simulations of inverse problem solutions with various solvers available [9,10]; In those studies the validation of the approach were performed on simplified spherical head models [9,11,12].In further works the CRLB calculation for a realistic head model was introduced [4,10,13,14].These works analytically derive CRLB for a FEM head model, while in our paper we propose the numerical evaluation of CRLB on the basis of the Gain matrix derivatives.
We use the CRLB to compare the possible dipole localization accuracy obtained with different sensor types in various sensor configurations.The paper inherits the sensor system modelling approach from [6] while the idea of comparison of different sensor systems was taken from [14].The CRLB estimation does not depend on the chosen method of solving the inverse problem, since it relies only on the mathematical model, i.e. the forward problem.

Problem statement
The CRLB estimation is a mathematical approach to estimate the smallest possible variance of the position of a brain signal's emission point (primary current dipole) on the cortical surface.In order to create our model, we discretize the brain into current sources, i.e. distributed dipole model [1,15].The governing equation for magnetic induction is a Poisson-like elliptic PDE, derived by applying the quasi-static approximation to Maxwell's equations.The numerical solution of this equation enables us to predict the magnetic field recorded by appropriately positioned sensors [2].The modelling of the registered signal given known cortical current sources is called "Forward Problem".Likewise, we call the reconstruction of a set of current sources given the measured magnetic fields on sensors "Inverse Problem".Our article [6] proposes a formal and complete derivation of the forward and inverse problems.For the purpose of the CRLB estimation, the model can be reduced to: where b ∈ R Ns is a vector of the scalar measurements on sensors N s sensors, j ∈ R N d is a vector determining scalar dipole moments at N d points of the cortex within the common assumption of orthogonality of current dipoles with respect to the cortex surface.The matrix G in eq.( 1) is a numerical/discrete approximation of the forward MEG problem, so-called Gain matrix (or the Leadfield matrix).Each column of the G contains the finite impulse response (FIR) of each sensor in the array to the considered dipole position.For example, G si is the scalar signal of the s-th sensor from a single active source located at the i-th position.The Gain matrix composition is the most common approximation of the MEG forward problem solution.In this study the Gain matrix is calculated using Boundary elements method (BEM) implemented in MNE-python package [16].
Using the pseudo-inverse of matrix G calculated for the individual geometry of the head volume, it is possible to approximate a solution to the inverse problem: to obtain the location of active sources on the brain cortical surface.The inverse problem is essentially underdetermined, since the number of sources is much greater than the number of sensors (N d > N s ).However, the solution of the inverse problem has been studied and implemented by many researchers around the world [15,[17][18][19].
In the present study, we estimate a theoretical lower bound of the source localization error at a cortical point.This estimation is independent of the inverse solver and relies solely on the forward problem (model) For our solution, we obtain the head geometry by MRI and simulate sensor arrays around the head.Afterwards, by simulating the forward problem, we estimate the CRLB of the sources' localization.

Models
Head model.A high-resolution structural-MRI scan served as the anatomic model for the study.The scan was acquired using a ToshibaExcelArt Vantage 1.5 with T1-weighted protocol, with the following imaging parameters: 12 ms repetition time (TR), 5 ms echo time (TE), 20°rotation angle, 160 sagittal slices with a slice thickness of 1.0 mm, and a voxel size of 1.0×1.0×1.0 mm 3 .
The cortical structures of the brain were segmented, and the interface between gray and white matter was tessellated with FreeSurfer 4.3 software (Martinos Center for Biomedical Imaging) [20,21]).
A single-layer BEM of the cortex (gray matter) enclosed in an inner skull and scalp surfaces was constructed using MNE-python [16].For each cortical surface (hemisphere), the model contained 10242 dipoles (cortical current sources).We used fixed dipole orientations normal to the cortical surface.
Table 1.Parameters of all sensors used in this study (from left to right): instrumental noise σ, distance to scalp d, number of integration points N i , base of gradiometer and the size of the sensitive element, adopted from [6] Sensor type Sensor signal modelling.The YIGM sensor is a solid-state magnetometer based on thin films of yttrium-iron garnet and is currently under development at the Russian Quantum Center [5,22].The films are arranged to form a square plate 38×38 mm 2 .The thickness of the sensor is 2-3 mm due to the winding placed on the films.Magnetic fields can be measured along both plate's sides.We name these orientations X and Y respectively, see Fig. 1

(G).
The sensor can be located on the head in two possible ways: tangentially or normally to the scalp surface, Fig. 1(D) and (E), correspondingly.In case of tangential sensor orientation there is no principle choice along which of two sensitive axes (X or Y) the signal is measured (the YIGM sensor registers two tangential components of the signal with respect to scalp surface).
In case of normal sensor orientation, switching between X and Y sensitive axes of YIGM controls which component of magnetic field we measure, either tangential or normal one, see Fig. 1(E) and (F), respectively.
It could be possible to construct a gradiometer using two YIGM sensors.We modelled a YIG-Gradiometer (YIGG) by arranging two YIGMs oriented normally to the head, parallel to each other with a 5 mm gap (base), see Fig. 1(H).In this case, the tangential derivatives of the normal component ∂B ∂x and one tangential component of the magnetic induction vector can be obtained.
We modelled the sensor signal via approximate integration over a sensor block, using fixed number of integration points on it.Since SQUID and OPM sensors are well-established in MEG, for these sensor types the integration points (and the other parameters) are specified in various MEG/EEG-related open-source packages.In order to reduce the integration error for the YIGM sensor we use a dense grid for the integration: 5×5 points, N i = 25.The nodes in this grid are evenly distributed on the sensor plane.Consequently, the coordinates of the integration points depend on the linear size of the sensor, Fig. 1 (E)-(G), magenta markers.
We compared the YIGM and YIGG with OPM QZFM systems (QuSpin Inc., USA) and the Elekta Neuromag SQUID magnetometers and gradiometer (Elekta Oy, Finland).The OPM and SQUID parameters are reflected in the literature (see [23][24][25][26][27], for reference).Overall parameters of all modelled sensor types are summarized in Table 1 reproduced from [6].Note the rather small distance between the YIGM/YIGG sensor and the surface of the head (scalp) of 2 mm.
In our study we used two types of sensor arrangement.The first one is taken from the existing Elekta Neuromag MEG system (hereinafter denoted as "SQUID-like layout").The second one is defined by uniform distribution of sensors over the head surface (denoted as "uniform layout").Uniform sensor arrays.Another type of sensor layout used in this study was originally created for our implementation of the OPM-MEG system [5].In this layout, the sensors are evenly distributed over the measurement area (head surface).We mainly consider uniform layout of 35 sensors as the most realistic number of YIGM and OPM sensors available for research groups in near future.The YIGM, YIGG, and OPM arrays with their integration points for these layouts are shown in Fig. 1(F), (K) and (C), respectively.However, in some of our experiments we also considered uniform layouts of 333 and 45 sensors in order to provide more complete picture.
In our modelling we vary the size of square plate of YIGMs in order to provide more dense layouts without sensor overlapping (especially, in case of tangential positioning), refer to Table 2 for sensor plate size for particular sensor number.

Numerical evaluation of CRLB
The CRLB was calculated to estimate the accuracy of signal localization on the cortex, which is given by various layouts based on sensors of the YIGM type (magnetometers or gradiometers based on iron-yttrium garnet films) at different levels of instrumental noise.The standard layout of 102 triplets based on the SQUID type sensors (SQUID-MEG system Elekta Neuromag), as well as the layouts based on the magnetometers with optical pumping (OPMs) were considered as reference layouts.
Cramér-Rao lower bound derivation.The main goal of electromagnetic brain mapping is to characterize the sources of electrical activity (represented by dipoles oriented normally to the cortical hemisphere surface) based on a set of measurements and head and sensor models.Let Θ ∈ R P be a vector of parameters defining the source.As parameters of the electric dipole to be recovered, we can take for example its spatial coordinate ξ ∈ R 3 and its dipole moment d: θ = {ξ, d}.Our task is to estimate these parameters Θ using the Cramér-Rao inequality: where J(Θ) is the Fisher information matrix.Thus, the diagonal elements of the CRLB are estimates of the minimal variance for the parameter Θ i .In case of parameters Θ for dipoles oriented normally to the cortical surface, the Fisher matrix elements are calculated by the formula: where b ∈ R Ns is a vector containing projections of the magnetic induction vector on the sensitive axes of the N s sensors in the system.C is the covariance matrix of the sensor system noise [8,12,13].
For simplicity, we assume that the sensors are independent of each other, which leads to a diagonal form of the covariance matrix [7,13].The covariance matrix is represented as a unit diagonal with the multiplier of the variance of the instrumental noise of the sensors σ 2 , thus: The exact calculation of the partial derivatives in the eq.( 4) is possible only if an analytical expression of the magnetic induction is possible, i.e. in the case of a spherically symmetric conductor, that was demonstrated in [4,7,13,28].Since our goal is to use physiological models of the head, we propose to compute the partial derivatives of the sensor signal over the spatial dipole coordinate numerically, based on a computational triangulated grid.
The specified derivatives on spatial position of a source (dipole) will be calculated for nearest neighbouring dipoles fixed at grid vertices, Fig. 2. Consider a dipole located at point ξ v .The  1)), takes the form: ∂ξ ξ=ξv Let us consider the patch of the triangulated mesh, see Fig. 2. The neighbouring nodes (vertices) of the grid k m allow to approximately calculate derivatives of the sensor response along the directions ξ km − ξ v for each node ξ v in the form: where G sv is the response at sensor s from a single dipole located in vertex v. G skm -the response at sensor s from a single dipole located in vertex k m adjacent to vertex v, see Fig. 2. ∆ vkm is the distance between vertices k m and v.
Since we assume that dipole moment in the neighbouring vertices d km = 0 (we consider single-point current source), eq.( 5) takes the form: The gradient in current magnitude d v : The localization accuracy was evaluated as the worst estimate of the Cramér-Rao inequality for the nearest neighbouring vertices (knn): A formula for comparison of sensory systems a and b in terms of Cramér-Rao metric is accessed via their log ratio:

Overview
We evaluated the CRLB of 20k dipole sources uniformly distributed on the cortical tessellated surface.The Gain matrices for the considered sensor arrays were produced by BEM forward solver realized within MNE-python [16].On Fig. 3 we show the 102 magnetometers of Elekta SQUID system as a reference and compared it with 35 uniform and 102 SQUID-like layouts for both OPM and YIGM at an idealistic instrumental noise level of 3f T / √ Hz (SQUID instrumental noise, see Table 1).
Our evaluation yields, that with identical layouts at 3f T / √ Hz (Fig. 3, upper row) both OPM ans YIGM layouts compare favourably against reference SQUID system (less localization variance, blue and magenta), especially in the medial frontal, occipital and deeper brain regions (i.e.sulci).
However, because of physical and economical constraints, the construction of a 102 YIGM array is impractical.Therefore, we modelled as well the performance of 35 sensor arrays (Fig. 3, lower row), which shows a comparable performance to the Elekta SQUID system.A diminution of signal quality with sensor number reduction is expected due to the inverse problem strong under-determination.

YIGM Noise levels
Our further assessment aimed at the evaluation of different sensor numbers at different noise levels in order to examine the possibilities for further development of YIGM systems.Here and after we assess the CRLB difference of considered pair of layouts using their log ratio (identical CRLB yields a log ratio of 0, positive values means that the CRLB of current layout is lower than referent one), see "Numerical evaluation of CRLB", eq.( 9).On Fig. 4(A) we compare layouts of YIGM having different number of sensors with 102 OPM and 102 SQUID systems.The comparison reveals, that at current YIGM noise levels of 35f T / √ Hz, only unrealistic layouts of 333 YIGMs (black curve) would permit a CRLB comparable to that of the SQUIDs (magenta horizontal line).However, the performance of the YIGM drastically improves as the level of instrumental noise diminishes.At a noise level of OPMs, that is 10f T / √ Hz (vertical dashed red line), the performance of a 102-YIGM array (green curve) already equals that of the SQUIDs.If manufactured at the same level of instrumental noise as SQUIDs, 3f T / √ Hz (vertical dashed magenta line), the YIGM show a superior performance, even when used in very limited array: only 45 sensors would be sufficient to obtain a CRLB superior to the reference Elekta SQUID system.The comparison of YIGMs with OPMs is almost always in favor of the YIGMs.Only sparse layouts of YIGMs at high instrumental noise levels are underperforming with respect to the 102-OPM layout.
The localization of the relative CRLB shows that the YIGM achieve similar to SQUID localization error, except in the occipital part of the cortex, and the medial cortices.Fig. 4(B) shows that on other regions, e.g. the frontal cortex surface, the CRLB ratios are closer to 0. This fronto-occipital asymmetry can be explained by more uniform distance between the different parts of head surface and compact YIGM sensors compared to non-flexible SQUID helmet which cannot equidistantly cover the subject's head (and in our case SQUIDS are closer to the occipital cortices compared to frontal ones).

YIGG Noise levels
We simulated the CRLB of YIG Gradiometers that were emulated by considering differential measurements of two sensors.Our simulations were aimed at evaluation of the instrumental noise level required to obtain similar localization error with YIGG as with Elekta SQUID system.Fig. 5(A) shows the simulation of relative CRLB of YIGG layouts with different number of gradiometers at different YIGG instrumental noise levels compared to 204 Elekta SQUID gradiometric layout.Fig. 5 shows, that unlike YIGMs, YIGGs definitely outperform SQUIDbased gradiometers: at similar noise level of 10f T /cm √ Hz (vertical dashed red line) the 35 YIGG layout (blue curve) are slightly inferior to the standard SQUID system.Even a small increase in the number of sensor -the 45 YIGG layout (orange curve) -guarantees better performance from the YIGG rather than SQUID.When increasing the number of sensors, the comparison becomes yet more favourable to YIGGs: the green curve of the 102 YIGGs still surpasses the SQUID even at 35f T /cm √ Hz.And as shown by the vertical dashed magenta line, 102 YIGGs manufactured with the same instrumental noise as SQUID (3f T /cm √ Hz) would outperform them by a factor of 2.5.
A visualisation of the YIGG-SQUID comparison shows that the error is higher in the deeper regions i.e. the sulci and the medial surface of the brain: as shown on Fig. 5(B)-(C) the blue regions (strongly better performance of the YIGG) concentrate on the lateral surface whereas the gray-green regions (marginally better performance of the YIGG) are localised in the depths.Those findings are consistent with our previous findings on YIGGs in [6].

YIGM and YIGG Orientation
The YIGM sensors have the shape of a square plate.We defined two sensor position: normal and tangential to the head surface, with normal and tangential axes of the magnetic fields measurement.We compute relative CRLB for identical layouts with either tangential sensor (tg-x) or tangential axes orientation (nl-tg), normal sensor position with normal axis orientation was taken as referent (nl-nl).We compared the shape of the CRLB histogram, as modelled for each dipole source.In order to maintain readability, we show only the histogram's envelopes on Fig. 6(A).In all considered situations the envelopes are shifted towards negative values, indicating that tangential sensing performs worse than normal sensing.This result is especially marked for gradiometers (green).As seen on Fig. 6(A), the envelopes show a decrease of the localization variance lower bound.A visualisation of the quality loss shows that the decrease in CRLB is uniform across all brain regions, except the lower frontal part, in which the decrease is more apparent, see Fig. 6(B).

Discussion
This section presents a comprehensive discussion of our findings and the implications of our results.
Through our comparative study, we observed variations in the performance of different sensor types.Fig. 4 and Fig. 5 show that in the case of comparable noise levels and the same sensor layout, the YIGM/YIGG systems demonstrate lower CRLB when compared to OPM or SQUID.This follows from a lower distance between the scalp and the sensors in the YIGM/YIGG systems.The current noise level, however, leaves them far behind the reference systems.These results are quite similar to another simulation study [6] due to the same assumptions put on the sources of neuronal noise.6) show that reducing the sensors' intrinsic noise has a greater impact on the CRLB metric than increasing the number of channels, which gives us the direction of development of the YIGM sensor.From the Fig. (6) we can see that tangential orientations of sensors' sensitive axes gives us higher error in comparison to the normal orientations.This could be explained by major contribution of ohmic currents to the tangential component of the measured magnetic fields [1].The study, however, has been done using the MNE BEM forward solver, and the results should be verified by using more accurate FEM-based models [29], which will be one of the directions of the further work.Moreover, we have studied cases of all sensors oriented normally or tangentially, and need to consider possibility of optimization of layouts by sensors' sensitive axes orientations, like it was shown in the studies [7] and [14].
To develop the metric, we made a strong assumption regarding the neuronal noise sources: we assumed these sources to be independent of and equivalent to each other [7].Nevertheless, it is essential to highlight that the eq.( 3) for the CRLB gradient covariance enables us to incorporate various configurations of neuronal noise sources based on the noise covariance matrix.As a part of our ongoing research, one of our primary focuses will be on estimating the neuronal noise covariance matrix from the noise covariance observed on sensors through experimental measurements, which will conform to experimental reality better.
In our study the Cramér-Rao lower bound represents the minimum achievable variation when determining the position and strength of a single dipole within the cortex.However, it is important to acknowledge that the existence of a single active dipole in the brain is not a realistic scenario.To ensure accurate estimations, it is imperative to consider the potential contribution of other possible sources.Consequently, we will also investigate the influence of other active sources on the CRLB in the nearest future.
Several studies (see e.g.[6,30]) including the present investigation have demonstrated that full head coverage is not effective when dealing with a small number of channels/sensors.By computing the mean/median value of CRLB within a specific region of interest in the cortex, we are able to formulate and address an optimization problem concerning the coordinates and/or orientations of the sensors.This approach potentially enables us to achieve an optimal taskoriented design for the sensor system, a critical consideration given the prospect of employing compact novel magnetometer types such as OPMs and YIGMs.The formulation and resolution of this optimization problem constitute another focal point of our ongoing research.

Conclusion
In this study, we have successfully developed a numerical technique for estimating the Cramér-Rao lower bound (CRLB) for complex meshed objects.By discretizing the objects into meshes and incorporating appropriate statistical models, we have efficiently evaluated the Fisher Information Matrix, allowing us to derive the CRLB for multiple parameters of interest simultaneously.The technique's flexibility enables us to analyze complex structures with irregular geometries, and its computational efficiency makes it applicable to large-scale problems.Our approach addresses the challenges posed by complex geometries commonly encountered in real-world scenarios, such as medical imaging, geophysical exploration, and computer graphics.By applying this technique, we have obtained efficiency metrics for MEG (Magnetoencephalography) sensor arrangements and compared various layouts realized on different types of sensors: YIGM, SQUID, OPM.
Specifically, we assessed the capability of different sensor systems to recover cortical current sources parameters accurately using CRLB estimation as efficiency metric.The chosen metric serves as a crucial performance indicator for MEG systems, guiding the selection and optimization of sensor arrangement in practical settings.
We gained a deeper understanding of how sensor type and placement and intrinsic noise impact MEG inverse problem's solution accuracy.This information can assist researchers and practitioners in designing optimal sensor layouts tailored to specific experimental or clinical requirements.

Figure 1 .
Figure 1.Modelled SQUID (blue), OPM (magenta), YIGM (cyan), YIGG (grey) layouts examples and YIGM, YIGG sensor schematics.A) conventional Elekta SQUID system of 102 magnetometers and 204 planar gradiometers, 102 sensor triplets in a fixed helmet; B) SQUID-like 102 OPM layout; C) uniform 35 OPM sensor layout; D) SQUID-like 102 YIGM-tg layout with tangential sensor orientation; E) SQUID-like 102 YIGM-nl-tg layout with normal sensor orientation and sensitive axis oriented tangentially to scalp surface; F) uniform 35 YIGM-nl-nl layout with normal sensor orientation and sensitive axis oriented normally to scalp surface; G) YIGM schematic; H) schematic of gradiometer realized of two YIGMs; K) uniform 35 YIGG layout with normal sensor orientation and sensitive axes oriented normally to scalp surface;

Figure 2 .
Figure 2. Schematic of gradient calculation for vertex v (red) within k neighbouring nodes (yellow) of the tessellated cortical surface.

Figure 4 .
Figure 4.The log ratio of the CRLB for YIGM layouts with normal orientation and normal component registration at different YIGM noise levels.(cold colours code lower variance compared to reference, warm colours -higher one) (A) Relative CRLB for 102 SQUID magnetometers from Elekta layout was taken as referent (zero line, magenta); Relative CRLB for 102-OPM SQUID-like layout at 10 fT/Hz is shown as constant (red); Dependence of relative CRLB on YIGM instrumental noise is shown for various YIGM-nl-nl layouts: -for an idealistic 333 uniform layout (black); -for 102 SQUID-like layout (green); -for 45 uniform layout (orange); -for 35 uniform layout (blue); (B) Relative CRLB for 102-YIGM-nl-nl with realistic instrumental noise of 35f T / √ Hz compared to 102-SQUID layout at 3f T / √ Hz (marked as green ▽ on (A))

Figure 5 .
Figure 5. CRLB log ratio for YIGG layouts with normal orientation and normal component registration with different sensor number at different YIGG noise levels.(A) CRLB for 204 SQUID gradiometers from Elekta layout was taken as referent (zero line, magenta); Dependence of relative CRLB on YIGG noise is shown for various YIGG-nl-nl layouts: -for an idealistic 333 uniform layout (black); -for 102 SQUID-like layout (green);for 45 uniform layout (orange); -for 35 uniform layout (blue); (B) Relative CRLB for 102-YIGG-nl-nl with noise of 12f T /cm √ Hz compared to 102-SQUID layout with 3f T /cm √ Hz (dark green ▽ on (A)) (C) Relative CRLB for 102-YIGG-nl-nl with realistic noise of 35f T /cm √ Hz compared to 102-SQUID layouts with 3f T /cm √ Hz (light green ▽ on (A))

Figure 6 .
Figure 6.Comparison of YIGM sensors depending on the sensitive axis and sensor orientations.(A) Envelopes of log ratio CRLB distribution computed for 102 SQUID-like layouts with tangential sensor configurations; the identical layouts with 'nl-nl' configuration (normal sensor orientation and normal sensitive axis orientation) was taken as referents.-YIGM-nl-tg with normal sensor orientation and tangential sensitive axis compared to YIGM-nl-nl (blue) -YIGM-tg-x with tangential sensor orientation and tangential sensitive axis compared to YIGM-nl-nl (orange) -YIGG-nl-tg with normal gradiometer orientation and tangential component registration compared to YIGG-nl-nl (green) (B) Localization of the CRLB diminution for 102 YIGM oriented tangentially to the scalp (YIGM-tg-x) against their normal orientation with normal sensitive axis orientation (YIGM-nl-nl).