Influence of unfused cranial bones on magnetoencephalography signals in human infants

Abstract

Objective:

To clarify the effects of unfused cranial bones on magnetoencephalography (MEG) signals during early development.

Methods:

In a simulation study, we compared the MEG signals over a spherical head model with a circular hole mimicking the anterior fontanel to those over the same head model without the fontanel for different head and fontanel sizes with varying skull thickness and conductivity.

Results:

The fontanel had small effects according to three indices. The sum of differences in signal over a sensor array due to a fontanel, for example, was < 6% of the sum without the fontanel. However, the fontanel effects were extensive for dipole sources deep in the brain or outside the fontanel for larger fontanels. The effects were comparable in magnitude for tangential and radial sources. Skull thickness significantly increased the effect, while skull conductivity had minor effects.

Conclusion:

MEG signal is weakly affected by a fontanel. However, the effects can be extensive and significant for radial sources, thicker skull and large fontanels. The fontanel effects can be intuitively explained by the concept of secondary sources at the fontanel wall.

Significance:

The minor influence of unfused cranial bones simplifies MEG analysis, but it should be considered for quantitative analysis.

1. Introduction

It has been thought from early on in magnetoencephalography (MEG) research that MEG signals are relatively unaffected by the conductivity, thickness, and shape of the structures overlying the human brain (Geselowitz, 1970; Plonsey, 1974; 1982; Cohen and Cuffin, 1983). This general conclusion is valid not only for spherically symmetric head models (Grynszpan and Geselowitz, 1973), but also applies fairly accurately even when the skull shape deviates from sphericity (Hämäläinen and Sarvas, 1989). This characteristic of MEG has been important for human neuroimaging research because it has greatly simplified accurate modeling of MEG signals (Hämäläinen et al., 1993).

Although this property has been useful for analyzing adult MEG data, it is still unclear how unfused cranial bones in human infants may alter the magnetic field. Both experimental and theoretical results indicate that the presence of skull defects generally alters MEG signals less than EEG signals (Barth et al., 1986; Flemming et al., 2005; Lew et al., 2013; Lau et al., 2014; Lau et al., 2016). Removing a part of the skull such that the missing bone region is filled with air has been found to have minimal, if any, effects on MEG signals (Okada et al., 1999). However, the conductivity of the fontanels and sutures are higher than the skull conductivity. In this case the skull defects can alter MEG signals more strongly than when they are filled with a poorly conducting material (van den Broek et al., 1998; Lew et al., 2013; Lau et al., 2014; Lau et al., 2016). Such a skull opening can alter the spatial pattern of the MEG signals, the effect depending on the relative location of the source and the edge of the defect. It is also unclear how MEG signals are altered by the fontanels of varying size and sutures of varying gap. The infant skull has the anterior and posterior fontanels at the junctions of the bones, which can be 3–4 cm along the coronal suture (Pritchard et al., 1956). The sutures are present between the adjacent cranial bones during the infancy. The mean widths of the coronal and lamboidal sutures are 3–4 mm between 0 and 60 days after a full-term birth (Hansman, 1966; Erasmie and Ringertz, 1976); the width can be 10 mm in children with hydrocephalus (Erasmie and Ringertz). The sutures may not be fully closed until in the teens. In addition, as the skull matures, the conductivity of the skull becomes lower and the skull gets thicker (Gibson et al., 2000). These two factors can modify the effects of fontanels and sutures.

Our aim in this study was to provide a general understanding of how MEG signals may be altered by unfused cranial bones so that our analytical approach will be useful for a wide range of future pediatric studies. Toward this end, we first carried out a systematic analysis of the effect produced by an opening in the skull mimicking the anterior fontanel as a function of fontanel size across a range of head sizes for the pediatric population below one year of age. We also studied how the conductivity and thickness of the skull modifies the effect. We then used a theoretical framework based on the concept of secondary sources at the fontanel wall to fully explain all of our major results, in some cases quantitatively, and the earlier results obtained by others cited above.

2. Methods

2.1. Head model

The infant head was modeled as a spherical, concentrically homogeneous volume conductor, with a hole in the skull representing the anterior fontanel. The head consisted of four compartments: scalp, skull, cerebrospinal fluid (CSF), and brain (Fig. 1A). The radii of the compartment boundaries were based on an infant head growth chart (Guo et al., 1988) for five age groups (0, 3, 6, 9, and 12 months after full-term birth) (Table 1). The thickness of the scalp, skull, and CSF compartments were kept constant across all age models. However, to examine the effect of thickening of the skull during the development, the radius of the skull-scalp boundary was varied by 1 mm to create skull thickness of 2 and 3 mm. A three-dimensional volume model with a 1-mm voxel resolution was created using custom Python code. The model contained a set of labels representing the four tissue types. The anterior fontanel was modeled as a circular hole in the skull centered on the vertex of the head with a linear diameter d_f (Fig. 1A). d_f ranged between 5, 10, 20, 30, 40, 50, 60, 70 and 80 mm, covering a full range (Popich and Smith, 1972; Duc and Largo, 1986) in order to examine the effect of fontanel size.

Fig. 1.

Configuration for simulating MEG signals in the presence of a skull defect (fontanel). (A) Spherical head model with a fontanel. The four compartments used for the 3D model of the infant head conductivity geometry. The radii for different age models are given in Table 1. (B) Sensor array layout of the 270-channel BabyMEG infant MEG system, viewed from the right and from above. The spherical scalp surface of the 12-month-old infant model is depicted with the sensor array. A fontanel of 80-mm diameter is shown in dark gray. (C) Schematic of the source plane depicted with the outline of the scalp surface and the sensor array. The three orientations of the dipoles are indicated with sample arrows: yz-tangential (left), x-tangential (middle), and radial (right), viewed from the right side and from the top of the head. The x-axis points from the subject’s left to the right, the y-axis points to the front, and the z-axis points up.

Table 1.

Dimensions of the spherical head model for infants of 0, 3, 6, 9, and 12 months of postnatal age. Two radii for the skull-scalp boundary are for two skull thicknesses (2 and 3 mm).

Age		0 month	3 month	6 month	9 month	12 month
Radius (mm)	Rscalp	55	61	65	68	70
	Rskull	52 (53)	58 (59)	62 (63)	65 (66)	67 (68)
	RCSF	50	56	60	63	65
	Rbrain	48	54	58	61	63

We used the Finite Element Method (FEM) for the magnetic field computations. A node-shifted hexahedral finite-element mesh was created for each age (Wolters et al., 2007). The mesh resolution was 1 mm in the nodal distance except for the node-shifted elements. Electric conductivity values were assigned to each element according to the tissue type: fontanel - σ_f = 0.3 S/m; skull - σ_s = 0.04 or 0.03 S/m, and CSF - σ_CSF = 1.8 S/m (Pant et al., 2011; Lew et al., 2013). The conductivity of the scalp and brain were same as for the fontanel.

2.2. MEG sensor array

The MEG signals produced by the dipole sources were computed for a sensor array similar to the whole-head pediatric MEG system (“BabyMEG”) we have developed for human brain development research (Okada et al., 2016). BabyMEG has an ellipsoidal helmet that accommodates heads up to 95% of the boys at 36 months of age (based on the standard head-growth chart provided by the National Center for Health Statistics, USA: http://www.cdc.gov/growthcharts/). The sensor array has two layers of magnetometers (270, 10-mm diameter single-axis magnetometers in the inner layer and 105 units of three-axis magnetometers on 20 mm³ cubes in the outer layer) just inside the helmet within the vacuum section of the cryostat with the magnetometers in the inner layer located at 6–11 mm (mean 8.5 mm) from the outer surface of the helmet. The sensor array in our simulation study was similar to the inner layer of this instrument in shape. It had 270 10-mm diameter coils spaced 18 mm apart, measuring the magnetic field component normal to the helmet surface (Fig. 1B). Near the simulated fontanels, this normal component was close to the radial direction in the spherical head model. The distance between the magnetometer array and the vertex of the head was 8 mm, close to the mean gap (8.5 mm) for BabyMEG. The sensor array was left-right centered over the model fontanel, but the sensor closest to the center of the fontanel was located slightly posteriorly with respect to the fontanel center.

2.3. Source space

The source space consisted of a circular disk in the mid-sagittal (yz) plane concentric with the compartment boundaries of the model (Fig. 1C). The radius of the source space was 2 mm less than that of the brain. The simulated sources were located at evenly spaced points 1 mm apart from each other. Three orthogonal current dipoles were placed at each location: radial from the center of the head, and two tangential with respect to the sphere model. One of the tangential dipoles was selected to be parallel (“yz-tangential”) and the other normal (“x-tangential”) to the source disk. Thus, for dipoles just under the wall of a fontanel, the yz-tangential dipoles were oriented perpendicular to the wall at that location, whereas the x-tangential dipoles were oriented parallel to the wall.

2.4. Computation of forward solutions

The magnetic field produced by each dipole in the source space was computed by using the SimBio-NeuroFEM package (SimBio, 2012) with the Venant dipole model (Lew et al., 2009). The simulated dipole strength was always 25 nAm, representing a typical value found for somatosensory evoked activity in a previous infant MEG study (Pihko et al., 2009). A spherical head without a fontanel was used as the reference model for assessing effects of the fontanel on MEG.

2.5. Difference measures

We used three measures to describe the effects of the model fontanel on MEG. We denote the MEG signals in m sensors outside the reference model with intact skull by v_r = (v_r,1 … v_r,m)^T and those outside a model with a fontanel by v_f = (v_f,1 … v_f,m)^T, respectively. A difference measure D₁ was defined as the L₁-norm of the difference between v_r and v_f:

$$D_1={\sum }_{i=1}^m\left|v_{f,i}-v_{r,i}\right|={\Vert v_f-v_r\Vert }_1.$$ (1)

Note that D₁ is not normalized, and thus has physical units. Changes in both the magnitude and the topography of the MEG signals contribute to D₁.

The relative magnitude difference MAG_rel and the relative difference measure RDM* (Meijs et al., 1989) are commonly used to selectively describe changes in the MEG signal magnitude and topography, respectively. MAG_rel was defined here as:

$$MA{G}_{rel}=\sqrt{v_f^T{v}_f/v_r^T{v}_r}-1.$$ (2)

Thus, MAG_rel > 0 when the MEG signal is larger in the presence of the fontanel, MAG_rel = 0 when the fontanel has no effect, and MAG_rel < 0 when the magnitude is smaller. Defining the difference of the normalized topographies $d=v_f/\sqrt{v_f^T{v}_f}-v_r/\sqrt{v_r^T{v}_r}$,

$$RDM*=\sqrt{d^{T}d}.$$ (3)

The values of RDM* range from 0 (identical topographies) to 2 (topographies of opposite sign). Note that both MAG_rel and RDM* are ill-defined when the MEG signals in the reference model vanish, which is the case for radially oriented dipoles and dipoles that are close to the center in a spherical head model. Therefore, for radial primary sources, we only calculated D₁, which is well defined also when the magnetic field in the reference model is zero.

3. Results

3.1. Change in the MEG signals quantified by D₁

The difference measure D₁ was largest for superficial cortical sources (Fig. 2), which was expected since they generate the strongest signals and D₁ was not normalized by the signal strength (the right-most column in Fig. 2). The field produced by the fontanel was <6% of the maximum reference field. For tangential dipoles (Fig. 2A, B), D₁ was largest when the dipole was just below the center of a small fontanel (d_f = 20 mm). For a large fontanel (d_f = 80 mm), D₁ was >1 pT when the primary source was below the opening; the maximum effect was seen for dipoles near the wall of the fontanel. The field produced by the fontanel was zero along a locus below the fontanel wall for the yz-tangential dipoles.

Fig. 2.

The measure D₁ of the difference in the MEG signals for head models with and without a fontanel, shown for three fontanel diameters (d_f = 20, 40, and 80 mm). D₁ is color coded on the source space for (A) tangential primary current dipolar sources parallel (yz-tangential) to the source plane, (B) tangential primary current dipolar sources perpendicular (x-tangential) to the source plane, and (C) radial dipoles. The rightmost column shows the strength (L₁ norm) of the signal for the dipoles in the reference model. Note that the tangential orientation is defined relative to the head model, not relative to the fontanel wall. Units of the scales are in pT. The dipole moment of all dipoles was 25 nAm. The fontanel is depicted with a gray arc segment in each case. The 6-month model was used, with skull conductivity 0.04 S/m and thickness 2 mm.

The radial dipoles (Fig. 2C) produced no magnetic signal in the reference model as expected; however, in the presence of a fontanel the dipoles produced a magnetic field at the sensors. The signal was largest when the radial dipole was just below the wall of the fontanel for all fontanel sizes and vanished for dipoles under the center of the fontanel. Notably, the maximum signal (~3 pT) was comparable to that of the MEG signal attributable to the fontanel effect for the tangential dipoles. This field in the presence of the fontanel may be significant since radial sources are expected to produce no field over a spherically concentric inhomogeneous conductor and relatively weak fields over a realistically shaped closed head.

3.2. Change in the magnitude of MEG signals quantified by MAG_rel

Figure 3 shows MAG_rel for tangential dipoles. For both the yz- and x-tangential dipoles, MAG_rel values were negative in the region below the fontanel, indicating that the fontanel reduced the MEG signal produced by the primary dipole source. With d_f = 20 mm, this signal attenuation was largest for dipoles near the brain surface. With the larger fontanels attenuation of comparable magnitude was observed for deeper sources. Nevertheless, the effect was small, <5% for the yz-tangential dipoles and <3% for the x-tangential dipoles, except near the center of the model head, where MAG_rel is ill-defined. Positive values of MAG_rel, indicating larger MEG signals in the presence of the fontanel, were seen for yz-tangential dipoles located just outside the region covered by the fontanel (Fig. 3A).

Fig. 3.

The MAG_rel measure, which describes the magnitude of the effect of the fontanel on the MEG signals, is shown as a function of the primary source current dipole location for three fontanel sizes for tangential dipoles (A) parallel (yz-tangential) and (B) perpendicular (x-tangential) to the source plane. Because MAG_rel becomes unstable when the MEG signal in the reference model approaches zero, a region around the center of the sphere model was masked out. Model details as in Fig. 2.

3.3. Change in the spatial pattern of MEG signals quantified by RDM*

The RDM* was generally small (Fig. 4), <0.01 for yz-tangential dipoles and <0.03 for x-tangential dipoles, except near the center of the head, where RDM* is ill-defined. For yz-tangential dipoles (Fig. 4A), there was a locus starting just outside the region below the edge of the fontanel where the RDM* was zero. For x-tangential dipoles (Fig. 4B), the RDM* was maximum along this locus. Although the effect of the fontanel was small, the effect was present for dipoles located in a widely distributed region of the source space deep in the brain and outside of the fontanel zone for the larger fontanels.

Fig. 4.

The RDM* measure, which quantifies changes in the topography of the MEG signals due to the presence of the fontanel, is shown for three fontanel sizes for tangential dipoles (A) parallel (yz-tangential), and (B) perpendicular (x-tangential) to the source plane. Model details as in Fig. 2.

3.4. Profiles of the fontanel effects for superficial sources

To better see the effect of the fontanel on the MEG signals for superficial sources near the fontanel, we show the profiles of the difference measures for sources along the outer rim of the source disk. Figure 5A shows the profiles for D1. For the tangential dipoles the largest D₁ values were found for dipoles under the center of the fontanel for smaller fontanels, but the maximum began to shift toward the wall for larger fontanels. For radial dipoles D₁ was largest when the dipole was near the wall of the fontanel. The full width at half maximum (FWHM) of the effect for the radial dipoles was 10–15° of angular distance from the wall of the fontanel. The maximum value of the field due to the fontanel was ~2 pT for yz- and x-tangential dipoles, whereas the value was as much as 3 pT for the radial dipoles. The field attributable to the fontanel was <4% of the field for the same tangential dipoles in the reference model head without a fontanel. However, the field due to the fontanel for radial sources may be relatively strong since radial sources do not produce any MEG signals in the reference model.

Fig. 5.

Profiles of the fontanel effect on MEG signals for dipoles distributed along the superficial arc of the source plane. (A) D₁ in units of pT for 25-nAm dipoles, (B) MAG_rel and (C) RDM*. The angle θ was defined between the vertical centerline in the source disk and the position vector of the dipole on the outmost rim of the source space. The straight lines on the top of each plot indicate the extent of the fontanel for d_f = 10, 20, 40, and 80 mm. The 9-month model was used, with skull conductivity 0.04 S/m and thickness 2 mm.

The MAG_rel profiles for the yz-tangential dipoles (Fig. 5B, left) revealed that the field was enhanced (MAG_rel > 0) when they were just outside, but reduced (MAG_rel < 0) when they were inside the region covered by the fontanel. The field for the x-tangential dipoles was reduced for dipoles outside and inside the fontanel region (Fig. 5B, middle). The attenuation was maximum near the center of the small fontanels, consistent with the experimental data of Lau et al (2014). As the fontanel became larger, the attenuation maximum shifted toward dipole locations close to the wall of the fontanel. The maximum value of MAG_rel was small, being <5% of the reference value, for these superficial dipoles.

The topography of the field as measured by RDM* was affected by the fontanel as well, with the effect being greater for larger fontanels (Fig. 5C). However, the effect was small (<0.01) for yz-tangential dipoles with the minimum approaching zero when the dipole was located just below the wall of the fontanel as was seen by the locus of zero fontanel effect in Fig. 4A. The effect for the x-tangential dipole was slightly larger (~0.03) with the maximum effect for the dipoles below the wall as was seen Fig. 4B.

3.5. Effects of skull thickness and conductivity

We studied how the thickness and the conductivity of the skull influence the fontanel effect in MEG. In a spherically symmetric head model, these parameters have no effect on the MEG signals; however, they affect the external field when the fontanel is present. Figure 6 shows the maximum D₁ for the yz-tangential dipoles, averaged across all the head sizes (age models). This value increased as the fontanel diameter (d_f) increased from 5 to 20 mm and then decreased when d_f further increased. The averaged maximum D₁ increased by ~50% (from 2.1 to 3.2 pT) when the skull thickness changed from 2 to 3 mm (50% increase) at d_f = 20 mm. The effect of conductivity was much smaller: D₁ increased only by <4%, when the conductivity changed from 0.03 to 0.04 S/m (33% change).

Fig. 6.

Influence of skull thickness and skull conductivity on the fontanel effect. Maximum change in MEG signal (max D₁) due to the presence of a fontanel for yz-tangential dipoles is shown as a function of the fontanel diameter d_f. Forward models with two different values for the skull thickness (2 or 3 mm) and skull conductivity (0.04 or 0.03 S/m) were examined. The average and standard deviation over the five post-natal age models (i.e., head sizes) are shown.

3.6. Developmental projectory of the fontanel effect in MEG

To illustrate the possible impact of a fontanel on MEG signals during the first year of life, we investigated the combined effect of varying the fontanel size and the head size on the difference in the magnetic field due to the inclusion of the fontanel. Figure 7A shows the maximum value for D₁ for yz-tangential dipoles as a function of the fontanel size for the five different age models between 0 and 12 months of age. The values peaking for d_f = 20 mm were in all cases <5% of the maximum signal in the reference model. All age models showed a similar pattern, indicating that the fontanel size has more impact than the head size on the MEG signal.

Fig. 7.

Projection of the effect of a fontanel on MEG signals during development. (A) The maximum value of the difference measure D₁ among all yz-tangential dipole locations is shown as a function of the fontanel size. Data from five different age models (0, 3, 6, 9, and 12 postnatal months) are superimposed. The skull conductivity was 0.04 S/m, thickness 2 mm. The dashed line indicates the 5% level of the maximal L₁-norm of the magnetic signal in the reference model, averaged over the five age models. (B) Projection of developmental changes in the fontanel effect in MEG signals. The maximum value of D₁ is shown for a likely sequence of fontanel and head size combinations during the first year of life. Note the relatively constant effect despite of changes in the head and fontanel sizes.

Figure 7B shows maximum D₁ for five combinations of head and fontanel sizes, chosen based on the infant head and fontanel growth chart (Duc and Largo, 1986; Guo et al., 1988). Despite the changes in the size of the head and the size of the fontanel, the effect of a fontanel on MEG appeared to be small (<4%) and relatively constant during the first 9 months after the full-term birth and then decreasing notably at 12 months when the fontanel had a diameter of 5 mm.

3.7. Volume current distribution in the presence of a fontanel

We computed the volume current distribution, as shown in Fig. 8, for each condition of simulation to help understand the effects of the fontanel on the three types of indices of alterations in MEG signals. The magnitude of the field for each case is summarized by the L₁-norm of the measurements, ${\sum }_{i=1}^m\left|v_{i,f}\right|$, for the fontanel model and ${\sum }_{i=1}^m\left|v_{i,r}\right|$ for the reference model, and D₁ for the difference field. A map of the absolute values of the difference |v_i,f – v_i,r| over the sensor array is also shown in Fig. 8, column 4. The distribution of the electric potentials in these examples is shown in Figure A1 in Appendix A. Note that the values of D₁ in Figs. 8A and D are larger by 50% than the corresponding values in Figs. 2–5 because the thickness of the skull was 3 mm for Fig. 8 and 2 mm for Figs. 2–5. In Fig. 6, the maximum value of D₁ was correspondingly 50% larger for skull thickness of 3 mm compared to 2 mm.

Fig. 8.

Volume currents generated by a primary current dipole. The volume current distributions are shown for a head with (column 1) and without (column 2) a fontanel, as well as their difference (column 3). Column 4 shows the MEG field of the difference shown in column 3. Skull is shown in black, scalp and fontanel in gray, and CSF in white. The cone arrows indicate the volume current field, the primary current dipole is depicted by the large yellow arrow. Four cases are illustrated: (A) a tangential dipole just below the center of the fontanel, (B) a tangential dipole just below the wall of the fontanel, (C) a radial dipole just below the center of the fontanel, and (D) a radial dipole just below the wall of the fontanel. For the tangential dipoles (A, B) the fontanel diameter is d_f = 20 mm and the maximum color scale is 50 nA/mm², maximum of the difference MEG field is 0.20 pT. For the radial dipoles (C, D) d_f = 40 mm and maximum color scale 30 nA/mm², and maximum of the difference MEG field is 0.20 pT. The 9-month model was used, with skull conductivity 0.04 S/m with thickness 3 mm.

When there is a tangential primary source below the center of the fontanel (Fig. 8A), the volume current in the opening (Fig. 8A, column 1) is stronger than the volume current in the same region without the fontanel (Fig. 8A, column 2) as seen in the difference map (Fig. 8A, column 3) because the conductivity of the opening is higher than the conductivity of the skull. The corresponding value of D₁ was 3.2 pT for the 3-mm thick skull consistent with the 50% smaller value of D₁ (~2 pT) for the 2-mm thick skull in Fig. 5. The values of MAG_rel and RDM* were maximum when the yz-tangential dipole was below the middle of the fontanel (Fig. 5).

When the tangential dipole is below the wall of the fontanel (Fig. 8B), the volume current in the fontanel region is stronger than the volume current in the reference model as seen in the difference map (Fig. 8B, column 3). However, interestingly, D₁ = 0.1 pT, which is virtually zero within the FEM discretization error. Correspondingly, the MEG difference map (Fig. 8B, column 4) shows no field. The difference MEG map in Fig. 8A, column 4, showed a spread of the magnetic field over many sensors, indicating that the MEG sensor diameter and sensor spacing were small enough to capture the field variation in our simulations. Thus, this zero field difference in Fig. 8B is not due to the sensor and source geometries in our analysis. Similarly, MAG_rel and RDM* showed no difference between the fontanel and reference head models when the primary dipole was below the fontanel wall (Figs. 3–5).

When there is a radial primary source below the center of the fontanel (Fig. 8C), the volume current in the opening is stronger than the current in the same region in the reference head model (Fig. 8C, column 3), just as for the tangential dipole. However, there is no MEG signal. The D₁ value of 0.08 pT in Fig. 8C is within the FEM discretization error as for Fig. 8B. The magnetic field pattern for the difference field shows a nearly zero field (Fig. 8C, column 4). This is a well-known phenomenon for MEG signals.

For a radial primary dipole just below the wall of the model fontanel (Fig. 8D) the difference in the volume current is strongest in the region near the fontanel wall (Fig. 8D, column 3). The magnitude of the volume current in this region is stronger than in the reference model. In contrast to the virtually zero D₁ for the tangential dipole (Fig. 8B), this difference in the volume current produces an MEG signal (D₁ = 4.6 pT) that is ~50% larger than for the tangential dipole (Fig. 8A). As seen in Figs. 2C and 5A, the value of D₁ is maximum for the primary dipole just below the fontanel wall, but the effect extends over ~20 degrees of arc away from the wall. The comparison of the results for D₁ in Fig. 8B and 8D shows that the volume current in the fontanel region is different between the fontanel and reference models for both the tangential dipole in Fig. 8B and the radial dipole in Fig. 8D, but D₁ is remarkably different for these two cases. All of these contrasting results can be explained within a single framework using the concept of secondary sources as detailed below.

4. Discussion

4.1. Using secondary currents for discussing the effects of unfused cranial bones on MEG

In the quasistatic approximation, the magnetic field B can be computed from the Ampere-Laplace law

$$\mathit{B}(\mathit{r})=\frac{{\mu }_0}{4\pi }{\int }_V\frac{\mathit{J}\left({\mathit{r}}^{\prime }\right)\times \mathit{R}}{R^3}dv,$$ (4)

where R = r – r′ and J(r) = J^p(r) + J^v(r) = J^p(r) – σ(r)∇V(r). The primary currents J^p are the “battery” of the circuit, associated with neural activity. This battery sets up the potential distribution V whose gradient drives the volume currents J^v throughout the conducting medium with electrical conductivity σ(r). In our simulations, the primary current source is always a current dipole Q and thus the field due to the primary current only is B^p(r) = (μ₀/4π)Q × R/R³.

In this study, we consider the head as a piece-wise homogeneous volume conductor. In this model, the scalp, skull, CSF and brain are separate compartments with different conductivities. For such a conductor, Eq. (4) can be transformed into the Geselowitz equation (Geselowitz, 1970):

$$\mathit{B}(r)={\mathit{B}}^p+{\mathit{B}}^{\mathit{v}}={\mathit{B}}^p(\mathit{r})+\frac{{\mu }_o}{4\pi }{\sum }_{ij}\left({\sigma }_i-{\sigma }_j\right){\int }_{S_{ij}}V\left({\mathit{r}}^{\prime }\right)\frac{\mathit{R}}{R^3}\times d{\mathit{S}}_{ij}^{\prime }.$$ (5)

From (5) it is directly evident that in the calculation of the magnetic field, J^v can be replaced by an equivalent surface current distribution (σ_j – σ_i)V(r′) = VΔσ_ij oriented normal to the surfaces S_ij, which separate regions i and j with conductivities σ_i and σ_j. These equivalent currents are often called secondary sources but, in fact, they are only a mathematical tool, which simplifies the solution of the forward problem. The secondary currents can also be used to more easily understand the effects of an inhomogeneous conductor to the magnetic field. Previously, Lau et al. (2014, 2016) have explained effects of skull defects on MEG signals in terms of volume current patterns. In the following, we will use the secondary source concept to interpret the effects of the fontanel.

4.2. Explanation of results of the present study

4.2.1. Schematic illustration of secondary sources for spherical fontanel head models

Figure 9 illustrates the fontanel effects in terms of the secondary source concept. The primary dipole Q is shown by a bold arrow in each panel (Figs. 9A – yz dipole, 9B – x dipole, 9C – radial dipole). The fontanel is viewed from above looking into the fontanel. The small arrows schematically represent secondary dipoles as vectors at the wall of the model fontanel. Note that all these vectors are directed from the region of the skull to the region of the fontanel when the potential on the wall is positive and toward the opposite direction when the potential is negative. The length and polarity of each arrow approximately represent the signed magnitude of the secondary source given by ΔσV, where V is the potential at the boundary and Δσ > 0 is the difference in conductivity between the fontanel and skull (in this study Δσ = σ_fontanel – σ_skull = 0.3 – 0.03 S/m > 0; note the order of subtraction). There are secondary sources located not only at the fontanel walls, but at all the boundaries, including the boundary between the brain and the CSF and the scalp-air boundary. Figure 9 shows only those secondary sources that are essential for explaining our results and earlier results. The secondary sources at the air-scalp or CSF-fontanel boundaries are radial in our spherical head model and do not produce the radial component, B_r, of the magnetic field outside the head. Because in our simulations, the sensors were measuring field components close to the radial orientation the simplified representation of secondary sources in Fig. 9 may be adequate for the purpose of explaining the fontanel effects in our study. Appendix A describes the basis for this simplification.

Fig. 9.

Schematic illustration of secondary currents along the wall of a fontanel produced by three types of primary sources: (A) a tangential dipole perpendicular to the fontanel wall (corresponding to the yz-tangential case in our simulations), (B) a tangential dipole parallel to the fontanel wall (corresponding to the x-tangential case), and (C) a radial dipole. Each circle represents a circular skull defect (fontanel) viewed from above (cf. bottom row of Fig. 1C). The source disk is located in the mid-sagittal plane, horizontally oriented and perpendicular to the plane of the figure. The dipole is located either just outside (left column), just inside (middle), or at the center (right) of the region covered by the fontanel. The secondary currents are located at the boundary, oriented normal to it. The thick arrow indicates the primary current dipole; the radial dipoles point up from the plane of the paper. The potential V at the wall of the fontanel, shown by + or − signs, depends on the orientation and direction of the primary source. The thin arrows indicate the magnitude (schematically, not to scale) and direction of the secondary sources along the edge of the fontanel. Secondary currents at the other boundaries (top and bottom of the fontanel, brain-CSF, CSF-skull, skull-scalp, and scalp-air) were assumed radially oriented and thus to contribute only little to the near-radial component of the magnetic field measured in the present simulations. The magnitude of the secondary sources is proportional to V(σ_f – σ_s). The sign of the secondary sources depends on the product of these two terms. The conductivity of the tissue within the fontanel (σ_f) is assumed to be greater than the conductivity of the surrounding skull (σ_s), i.e., σ_f - σ_s > 0. In the present study σ_s = 0.03 or 0.04 S/m and σ_f = 0.3 S/m and thus σ_f - σ_s = 0.27 or 0.26 S/m.

4.2.2. Tangential dipoles perpendicular to the fontanel wall

When a primary dipole Q was oriented perpendicular to the fontanel wall in our simulations (Figs. 3A and 5, left), the MAG_rel for superficial sources was positive (B was enhanced) for the dipoles outside the fontanel, zero for the dipoles below the wall of the fontanel, and negative (B was reduced) for the Qs under the fontanel. This can be understood in terms of the secondary sources shown in Fig. 9A. When the Q is outside the fontanel region, the secondary sources are oriented in the same direction as the Q, thereby increasing the magnitude of B. When the Q is below the wall of the fontanel, the V on the wall above is close to zero and thus the moments of the secondary sources are close to zero. This leads to the locus of the null difference close to the edge of the fontanel in Figs. 2–5 and the virtually zero value of D₁ (= 0.1 pT) in Fig. 8B. When the Q is under the fontanel, the secondary sources are opposite to Q in direction (Fig. 9A, middle and right), thus making the B weaker and MAG_rel <0. (Figs. 3A and 5B, left). D₁ and RDM* can be understood accordingly.

When a Q is in the region of the cortex close to the fontanel wall (Fig. 9A, center), the fontanel effect depends on the conductivity of the fontanel σ_f. When it is higher than that of the skull (σ_f = 0.3 S/m and σ_s = 0.04–0.03 S/m), B is reduced (Fig. 5B, left). When σ_f < σ_s, the secondary sources enhance the B since the direction of the secondary sources then becomes the same as the direction of the Q. The enhancement is, however, weak or nonexistent in such a case, as found experimentally (Okada et al., 1999), since Δσ becomes very small. Okada et al. (1999) found that somatic evoked MEG signals were not affected by a large ellipsoidal opening filled with air in the skull of a juvenile swine when the active tissue was relatively shallow.

4.2.3. Tangential dipoles parallel to the fontanel wall

The results for a Q parallel to the fontanel wall can be understood with the aid of Fig. 9B. In this case, the B was weaker than in the reference model (MAG_rel <0) regardless of whether the Q was inside or outside the cortex below the fontanel (Fig. 5B, middle). When the Q is close to the wall (Fig. 9B, left and middle), the net direction of the secondary sources in all these conditions is opposite to that of the Q, and thus the B is attenuated. The attenuation is maximum when the Q is in the middle of a small fontanel, consistent with the empirical results of Lau et al (2014). The effect on the signal topography (RDM*) was largest for dipoles near the wall of the fontanel (Figs. 4B, 5C). This is because the secondary sources at the wall close to the Q are oriented perpendicular to the Q.

Lau et al. (2014; 2016) observed rotations of the spatial pattern of the B_r, which depended on the relative location and orientation of the Q and the edges of the skull defect. The secondary source concept could be helpful for evaluating the effect of skull defects generally on the rotation of the MEG signal topography in these cases. In our simulations, however, the fontanel was circular and the Q was always aligned with the symmetry axis of the fontanel; therefore, rotation was not expected.

4.2.4. Radial dipoles

For radial primary sources B = 0 outside a spherically symmetric conductor (Grynszpan and Geselowitz, 1973). In adult head models with realistically shaped but intact skull, MEG is relatively insensitive to quasi-radial sources (Ahlfors et al., 2010a). When the symmetry is broken by a skull defect, however, there is a notable B when the Q is close to the edge of the defect (Figs. 2C and 5A, right). The magnitudes of B for radial dipoles near a fontanel measured by D₁ were larger than those for the tangential case (Figs. 2, 5, 8). When the Q was just below the wall, D₁ was ~3 pT skull thickness of 2 mm (Fig. 5A) and 4.6 pT for thickness of 3 mm (Fig. 8D). These values are about 50% larger than the maximum value of 2 pT (Figs. 5A, 6) and 3 pT (Fig. 6) for the yz-tangential dipoles. Since B = 0 for a radial Q outside a spherically symmetric conductor, this field produced by a skull defect is significant. This implies that one cannot ignore radial primary sources such as those located in the crown of the cortical gyri near skull defects of one type or another, especially in quantitative analyses.

According to the secondary source concept, the fontanel affects B most strongly when a radial Q is close to the wall because V on the wall above the Q is at maximum when it is close to the wall; as it moves toward the center of the fontanel the net secondary source effect approaches zero when there is symmetry in the shape of the fontanel as it is the case in our example (Fig. 9C, right). The difference B is virtually zero for the yz-tangential Q (Fig. 8B, column 4), but it is relatively strong for the radial Q below the wall (Fig. 8D, column 4). This clear contrast can be easily explained by the secondary source approach because V is small on the wall above the yz-tangential dipole but V for a radial Q is maximum when it is close to the wall. The secondary source approach can also explain the source localization results of van den Broek et al. (1998) who found in a simulation study that the estimated equivalent dipoles for radial dipoles next to a skull defect were tangentially oriented and located inside the skull defect at the wall closest to the source. An equivalent source representing all the secondary sources on the wall just above the radial Q should be located just inside the skull defect due to the curvature of the defect as one can see from Fig. 9C.

4.2.5. Skull conductivity and thickness

The concept of secondary sources is useful for quantitatively explaining the effects of skull conductivity and thickness on MEG signals (Fig. 6). The magnitude of secondary source is proportional to VΔσ. Thus, the magnitude changes by 4% when the fontanel conductivity is 0.3 S/m and the skull conductivity changes by 33% from 0.03 to 0.04 S/m, assuming that V is not substantially affected by this change (Appendix A). The maximum D₁ differed by this amount (4%) for these two values of skull conductivity in Fig. 6. The combined magnitude of the secondary sources in a cross-sectional area of the wall with area A is VΔσA; thus, the magnitude increases by 50% when the thickness increases from 2 mm to 3 mm. Consistent with this, the maximum D₁ was larger by 50% in Fig. 6. In our simulation, the walls of the fontanel were strictly speaking not completely radial. Appendix B shows that this slanting is expected to have minimal effect on the radial component of the magnetic field commonly measured in MEG. The sensor array in our simulations measured the component that was nearly radial. Thus, the effects of skull conductivity and thickness can be understood simply from the above analysis.

4.2.6. Fontanel size

The fontanel effects seen in all the results strongly depend on fontanel size. The alteration of MEG signal was seen only when the Q was close to the wall when the fontanel was small. As the fontanel size increased, the effect became quite extensive with clear effects seen for dipoles deep in the brain or outside the region of the fontanel (Figs. 3 and 4). Thus, the fontanel effect might be quite important for analyzing MEG signals produced by active tissues in a large region of the brain when the fontanel is large and its conductivity is much higher than that of the skull.

In terms of the secondary source concept, the fontanel size effects are entire due to the potentials on the wall since Δσ does not change. When the secondary sources on the side wall of the fontanel are the predominant sources of the fontanel effects, as it is the case here, the fontanel size effect can be understood in terms of V on the wall. Consider, for example, any yz-tangential or radial dipoles outside the fontanel region. They can produce secondary sources all around the wall that are oriented toward the center of the fontanel (Figs. 9A, and C, left). The potential V on the wall produced by such a Q tends to become more similar in magnitude and thus the secondary sources tend to cancel each other more as the Q moves farther away from the fontanel (Ahlfors et al., 2010b). This mutual cancellation is weaker for larger fontanels and thus the fontanel effect extends farther outside of the fontanel region for larger fontanels. A similar situation exists for x-tangential Q’s. The alteration of MEG signal measured by D₁ increased with fontanel diameter, reaching its peak for a diameter around 20 mm and then decreased for larger diameters (Figs. 6 and 7). This dependence on the diameter is due to the increase and then the decrease in V at the fontal wall due to the dependence of the potential distribution in the fontanel on its diameter.

4.3. Effects of unfused cranial bones on MEG signals

Cranial bones are unfused during the early stages of development to allow for the large expansion of brain volume after the full-term birth. The different cranial bones are connected together by the sutures that consist of the soft tissues of the dura and scalp (Reddy et al., 1990; Cohen, 1993). The size of the fontanel varies across healthy newborns. As reviewed in the Introduction, the fontanel can be several centimeters wide, whereas the sutures may be several millimeters wide in typically developing babies; they may be closed in children with craniostenosis and quite wide in children with hydrocephalus. Our study shows that the fontanel effects on MEG are generally small (maximum value of D₁ < 6%). We did not analyze the corresponding effects of the sutures. Extrapolating from the theoretical study of Huang et al. (1990, Fig. 9), a suture of 2–4 mm width produces a maximum distortion of 2–4%, similar to the fontanel effect, when the suture consists of a tissue with a conductivity of 0.2 S/m and the skull is 0.03 S/m.

The present results showed that the fontanel effect depends on size of the fontanel, conductivity and thickness of the skull and conductivity of the tissue making up the fontanel. Although the effects on MEG signals due to unfused cranial bones appear to be relatively small, the fontanel effect was seen for dipoles well outside of the fontanel and deep in the brain when the fontanel was relatively large (Figs. 3 and 4). This widespread effect of the fontanel may be important for quantitative analysis of MEG signals. Although radial dipoles in a sphere do not produce MEG signals on their own, they can produce an MEG signal that are comparable or even larger in magnitudes to the signals produced by tangential dipoles when there is a fontanel. Thus, the effects of the radial dipoles may not be ignored in any quantitative analysis of MEG signals, especially when they are produced by shallow cortical sources located in the gyri of the neocortex closest to the MEG sensors. Our analysis indicates that one needs to take into account the conductivity and thickness of the skull in predicting the fontanel effect for any head. The effects are quantitatively understandable from the definition of secondary sources. The conductivity of the tissue filling the fontanel affects the volume current as shown by the previous work filling a hole in the skull with air, saline or a medium with an intermediate value (Okada et al 1999; Flemming et al 2005; Lau et al., 2014, 2016). When the opening in the skull of a rabbit was filled with a highly conductive material with a conductivity value of 1.0 S/m, the fontanel reduced the MEG signal by as much as 20–25% of the value with an intact skull (Lau et al., 2014). These values are much smaller than the enhancement of EEG signals by 200–1000%, but they are nevertheless large. However, the secondary source concept predicts that the effect should be less when the opening is filled with a material having a conductivity closer to the actual value, which is comparable to that of the scalp (0.3–0.4 S/m) or that of the dura (0.02–0.1 S/m) (Weerasuriya et al., 1984). In the experimental setting of Lau et al (2014) (skull thickness = 2.1 mm and skull conductivity = 0.1 S/m), the attenuation should have been 4–6% when the conductivity of the opening is 0.3 S/m. These values are consistent with the values in our present study for a skull thickness of 2 mm.

5. Conclusions

The presence of a fontanel and more generally unfused cranial bones can affect MEG signals in the pediatric population. The effects based on three indices (D₁, MAG_rel and RDM*) indicate that the fontanel effects are relatively small for a head with a hole in the skull mimicking the anterior fontanel. The sum of the maximum difference in magnetic field between the fontanel head model and the reference model without a fontanel (D₁), for example, was <6% across a whole-head MEG sensor array, consistent with the long-held notion that MEG signals are relatively unaffected by unfused cranial bones. However, the fontanel effect was seen extensively for deeper primary sources and sources outside the fontanel region for larger fontanels. The effects for radial sources were comparable or even larger to those for tangential sources. Skull thickness significantly increased the fontanel effect, while skull conductivity had minor effects. Thus, the influence of the fontanel and other openings in infants and young children may be important, especially for quantitative analysis. The concept of secondary sources helps to quantitatively explain all of the present and some of the earlier results for effects of skull thickness and conductivity. Our analysis demonstrates that this approach may provide a simple, intuitive useful account of the effects in MEG signals due to unfused cranial bones or openings in the skull due to other causes.

Highlights.

• We evaluated alterations of MEG signals by unfused cranial bones in infants in a simulation study.

• Effects of the fontanel were small (<6%), but could be significant for radial sources, larger for thicker skulls and extensive for larger fontanels.

• The minor influence of unfused cranial bones simplifies MEG analysis, but it should be considered for quantitative analysis.

Abbreviations:

MEG

Magnetoencephalography

EEG

Electroencephalography

RDM*

Relative difference measure

MAG_rel

Relative magnitude difference

FEM

Finite element method