Keywords

1 Introduction

The synergetic effects connecting spatial and functional neuroimage techniques allows reduction of the weakness for single method analysis. Specifically, EEG Source Imaging (ESI) relating structural head models and distributed source localization techniques improves the time and spatial resolution of single MRI or EEG analysis [3]. ESI information is used for diagnosis and preoperative stages of brain surgery being, in most cases, the only suitable analysis tools because of the high risk of surgical interventions [5].

ESI techniques allow the estimation of neuronal activity from electrical potentials measured over the scalp (EEG). In particular, ESI solution needs real EEG signals, a method for mapping of the measured activity from electrodes to the sources (EEG inverse problem solution), and a correct modeling of the potentials conduction and morphology of the head, meaning, a forward solution. In this regard, the accuracy of ESI solutions directly depends on the capabilities of the forward model to adequately describe the information from sources to sensors [12].

In this regard, a more realistic representation of head volumes may be of benefit. In practice, a realistic head volume can be obtained from neuroimages such as MRI or CT containing a large number of slices in a series of two-dimensional images. Every slice must be registered in the same coordinate system to obtain a coherent three-dimensional volume. After the registration stage, the volume contains the information of head tissues codified in intensity values, that can be segmented to generate a labeling map holding compartments for specific tissues. In particular, the scalp (where the EEG electrodes are placed), the skull, the cerebrum spinal fluid, the gray matter, and the white matter are the most commonly considered tissues in the forward modeling. Due to the direct impact of the forward modeling on EEG source localization, we build a patient-specific and realistic head model holding five tissues, and anisotropic skull and white matter modeling in a 1 mm\(^3\) volumetric segmentation. Further, we use the FDM technique to calculate patient-specific head models, analyzing model complexity (number of tissue compartments) and anisotropic modeling [2] for two different ESI prior techniques, namely, Loreta-like priors (LOR) and empirical bayesian beamformer priors (EBB). Finally, we use Bayesian model selection for group studies to analyze the influence of the forward modeling in the considered distributed ESI solutions.

2 Methods

2.1 EEG Forward Problem

The EEG forward problem involves the calculation of potentials \(\phi ({\varvec{r}})\) induced by a primary current density \({\varvec{J}}({\varvec{r}})\) in a head volume \(\varOmega \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^3\) with \(\partial {\varOmega } \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^2\) boundary, holding inhomogeneous and anisotropic conductivity \({\varvec{\varSigma }} ({\varvec{r}}) \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{3 \times 3}\). The quasi-static approximation of Maxwell’s equations can be formulated, leading to the Poisson’s equation as follows [7]:

$$\begin{aligned} \nabla \negthickspace \cdot \negthickspace \left( {{\varvec{\varSigma }} ({\varvec{r}})\nabla {\varvec{\phi }} ({\varvec{r}})} \right)&= - \nabla \negthickspace \cdot \negthickspace {\varvec{J}}({\varvec{r}}), \,\,\, \forall {\varvec{r}} \in {\varOmega } \end{aligned}$$
(1a)
$$\begin{aligned} \left. {\varvec{\phi }} ({\varvec{r}}) \right| _{\varGamma _l }^+&= \left. {\varvec{\phi }} ({\varvec{r}}) \right| _{\varGamma _l }^- \,\,\, \text {on} \, {\varGamma }_l, \,\,\, \forall l = 1, \ldots ,N\end{aligned}$$
(1b)
$$\begin{aligned} \left. \left( {{\varvec{\varSigma }} ({\varvec{r}})\nabla {\varvec{\phi }} ({\varvec{r}})} \right) \negthickspace \cdot \negthickspace {\varvec{\hat{n}}}({\varvec{r}}) \right| _{\varGamma _l }^+&= \left. \left( {{\varvec{\varSigma }} ({\varvec{r}})\nabla {\varvec{\phi }} ({\varvec{r}})} \right) \negthickspace \cdot \negthickspace {\varvec{\hat{n}}}({\varvec{r}}) \right| _{\varGamma _l }^-, \,\,\, \text {on} \, {\varGamma }_l\end{aligned}$$
(1c)
$$\begin{aligned} \left. \left( {{\varvec{\varSigma }} ({\varvec{r}})\nabla {\varvec{\phi }} ({\varvec{r}})} \right) \negthickspace \cdot \negthickspace {\varvec{\hat{n}}}({\varvec{r}}) \right| _{\partial {\varOmega }}&= 0, \,\,\, \text {on boundary}\, \partial {\varOmega } \end{aligned}$$
(1d)

where \({\varvec{r}}\) is a specific head volume position, N is the number of interfaces \({\varGamma }_l\) (i.e., head layers), \({\varvec{\hat{n}}}({\varvec{r}}) \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^3\) is a unit vector normal to \({\varGamma }_l\) at \({\varvec{r}}\), and \(\left. g({\varvec{r}}) \right| _{\varGamma _l }^ \pm \) stands for the trace of function \(g({\varvec{r}})\) from both sides of the l-th interface \({\varGamma }_l\). Furthermore, the solution of Eq. (1a) requires establishing proper boundary conditions between adjacent compartments having different conductivities. Thus, Eqs. (1b) and (1c) stand for the Dirichlet and Neumann flux conditions respectively, while Eq. (1d) (or non-flux homogeneous Neumann condition) implies that no current can flow out through the human head interface \(\partial {\varOmega }\) into the air.

Fig. 1.
figure 1

Realistic head modeling methodology pipeline.

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Figure 1 shows the realistic head modeling methodology pipeline, for which we define 5 different tissue conductivities for an MRI based tissue-labeling segmentation in a 1 mm\(^3\) space as in [11]. The segmentation allows a voxelwise conductivity distribution for the FDM taking known conductivity parameters from the literature. In addition we define a reciprocity solution for the EEG sensors space, that is solved using the FDM technique.

2.2 Distributed Inverse Solutions

For a EEG dataset \({\varvec{Y}} \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{N_C \times T}\) of \(N_C\) sensors, T time samples, and a given lead-field matrix \({\varvec{L}}_m \in \mathbb {R}^{N_C \times N_D}\), the magnitude of the neural activity \({\varvec{J}} \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{N_D \times T}\) for \(N_D\) current dipoles distributed in the GM, is generally represented by the general linear model [3]:

$$\begin{aligned} {\varvec{Y}} = {\varvec{L}}_m{\varvec{J}} + {\varvec{\varXi }} \end{aligned}$$
(2)

where \({\varvec{\varXi }} \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{N_C \times T},\) is an additive white noise matrix with covariance \(\text {cov}({\varvec{\varXi }}) = \exp (\lambda _{0}){\varvec{I}}_C,\) where \({\varvec{I}}_{N_C} \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{N_C \times N_C}\) is an identity matrix, and \(\lambda _{0} \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{+}\) an hyperparameter modulating the sensor noise variance. Further, solving Eq. (2) to estimate \({\varvec{J}}\) becomes a optimization problem of the following form:

$$\begin{aligned} \tilde{{\varvec{J}}} = {\varvec{Q}}{\varvec{L}}_m^{\top }({\varvec{Q}}_{\varXi } + {\varvec{L}}_m{\varvec{Q}}{\varvec{L}}_m^{\top })^{-1}{\varvec{Y}} \end{aligned}$$
(3)

requiring prior information about the source covariance matrix \({\varvec{Q}}\).

Loreta-Like (LOR) Priors: Considering that sources vary smoothly over space using a Green’s function \({\varvec{Q}}_G = \exp (\sigma {\varvec{G}}_M),\) with \({\varvec{Q}}_G \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{N_D \times N_D,}\) where \({\varvec{G}}_M \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{N_D \times N_D}\) is a graph Laplacian that comprises inter-dipole connectivity information about all neighboring dipoles, and \(\sigma \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{+}\) rules the spatial expansion of the activated areas. Consequently, the source prior is computed as:

$$\begin{aligned} {\varvec{Q}} = \exp (\lambda _1) {\varvec{Q}}_G \end{aligned}$$
(4)

with \(\lambda _1 \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{+}\) and hyperparameter to be estimated.

Empirical Bayesian Beamformer (EBB) Priors: Assuming one global prior for the source covariance main diagonal, where the off-diagonal elements are zeros, i.e., no correlations assumed. Thus, the d-th position of the source covariance matrix main diagonal is calculated as [1]:

$$\begin{aligned} {\varvec{Q}} = diag\left( \exp (\lambda _1)({{\varvec{l}}_m}_d^{\top } {\varvec{C}}_{Y}^{-1}{{\varvec{l}}_m}_d)\right) \end{aligned}$$
(5)

where \({{\varvec{l}}_m}_d \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{N_D \times 1}\) is the d-th column of the lead field matrix, and \({\varvec{C}}_Y \,{{\,\mathrm{\negthinspace \in \negthinspace }\,}}\, \mathbb {R}^{N_C \times N_C}\) is the EEG data covariance matrix.

Further, to estimate the hyperparameter set \(\{\lambda _{{\varvec{\Xi }}},\lambda _{{\varvec{P}}}\}\), we use the verisimilitude function known as free-energy [14]. In this regard, for a given EEG recording and a certain forward model m, the free energy can be expressed as [6]:

$$\begin{aligned} F(m) = accuracy(\lambda ) - complexity(\lambda ), \end{aligned}$$
(6)

Free Energy can be maximized using standard variational schemes [14]. To perform this optimization scheme, we use a greedy search (GS) algorithm. Further, the set of GS hyperparameters were tuned through the Restricted Maximum Likelihood (ReML) algorithm as given in [1].

3 Experimental Setup of the FDM-Based Forward Solution

The solution of the Poisson Eq. (1a) for realistic patient-specific head volumes is only possible using numerical approximations. In particular, individual magnetic resonance (MR) and/or computed tomography (CT) images can be segmented into different tissue types, such as white and grey matter (WM/GM), cerebrospinal fluid (CSF), skull, skin, among others.

We build a realistic, high-resolution 1 mm\(^3\), patient-specific volume conductor model from neuroimages, including anisotropic skull and white matter modeling and considering five different tissue compartment segmentation as in [11], considering the following isotropic conductivity values: 0.33 (scalp), 0.0105 (skull), 1.79 (CSF), 0.33 (GM), and 0.14 (white matter) as in [12]. Further, we use T1, IDEAL T2 and diffusion-weighted imaging (DWI) MR scans acquired from a healthy 32-years-old male in the Rey Juan Carlos University, Medicine Faculty, Medical Image Analysis and Biometry Lab, Madrid, Spain. Further, based on the proposed methodology, we define three different forward model setups with increasing complexity (number of tissues). Moreover, We build three isotropic models, beginning with the simplest model M1 including only three tissues, namely skin, skull, and brain, then, for the model M2, we add the CSF, and for the model M3, we divide the brain area into GM and WM.

Furthermore, we use the multi-subject, multi-modal human neuroimaging dataset including visual event-related potentials (ERP’s) (among other neuroimaging data) [13], selecting 15 patients, 8 males, and 7 females, with an age range 23–37 years, all Caucasian with European origins. An evoked potential visual experiment is carried out using face images stimuli, including two sets of 300 greyscale photographs, half of famous people and the other half of nonfamous people (unknown to participants). In the data set, half of the faces were male, half female, leaving 3 trial-types (conditions): Familiar Faces (Famous), Unfamiliar Faces (Nonfamous) and Scrambled Faces. Stimuli were projected onto a screen approximately 1.3 mts in front of the participant, and visual markers were projected to synchronize the stimuli apparition. A 70 channel Easycap EEG cap was used to record the EEG data simultaneously, with electrode layout conforming to the extended 10–10 system. EEG data were acquired at an 1100 Hz sampling rate with a low-pass filter at 350 Hz and no high-pass filter, including processing stages for automatic detection of wrong channels throughout the run, notch-filtering of the 50 Hz line-noise and its harmonics and a trial rejecting stage. Finally, an averaging the remaining trials for each of the three conditions was made to calculate ERP’s for 1 s time windows.

Finally, we used Bayesian model selection based on free-energy in order to study the head model influence in the studied group [8]. To this end, we calculate free-energy factors to the full ERP time window (1 s) using the LOR and EBB techniques for the 15 considered patients, the three different visual stimuli and the 3 proposed head models for a total of 900 test. Then we group the free-energy of three reconstructions over stimulus conditions for each ESI technique to apply a random effects analysis at the group level, where the log group Bayes factor can be obtained summing over the 15 subjects [9].

4 Results

The Fig. 2 show the expected posterior model frequency for both ESI considered techniques, EBB (right) and LOR (left) and the three considered isotropic head models. Moreover, we include separate results for the different visual stimulus, Famous (blue), Unfamiliar (green), and Scrambled (red), showing the Bayesian omnibus risk (BOR) values in the button part of the charts. The results show strong evidence in favor of the most complex model M3 for both considerer inverse priors, namely, LOR and EBB, with an appreciable increment of the exceedance probability between the most straightforward model M1 and the model M2. All test have low BOR values indicating strong evidence in favor of the obtained results.

Fig. 2.
figure 2

Random fixed analysis showing expected posterior model frequency, and Bayesian omnibus risk (BOR) for the considered stimuli. (Color figure online)

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Fig. 3.
figure 3

Model complexity.

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4.1 Complexity Modeling of ESI Tasks

We analyze the model complexity influence in the source localization using both, the LOR and the EBB technique for the three considered models. This test shows the effect of increasing the number of tissues (M1 to M2) and also the effect of include anisotropy in both, WM and skull areas (M2 to M3). Figure 3 shows a normalized maximum intensity projection for the considered models, where it can be appreciated that the energy is more spread in model M1 compared to model M2. Moreover, the inclusion of anisotropic skull and WM for the model M3 show not only a concentrated activation area, but also possible spatial separation for individual activations in the visual cortex that appears as a mixed and single activation in the models M1 and M2. We used red squares to show the source activation zone for the model M1, and green squares in the concentrated area of activation in the model M3, additionally, we used blue squares to show the source energy separation between two different sources that are very near one to each other. Moreover, the energy separation for more complex head models is consistent with similar analysis results [4].

4.2 Results for Visual Stimulus

Finally, we analyze the stimulus-response for the most complex model M3 using EBB and LOR source priors for a single subject (S9). Figure 4 shows the maximum intensity projection maps for the three considered stimuli namely, Famous, Unfamiliar and Scrambled, illustrating appreciable differences in the energy distribution for the different considered stimulus, with less intensity in the Thalamus area for the Unfamiliar stimulus compared to the Famous stimulus.

Fig. 4.
figure 4

Stimulus response.

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5 Concluding Remark

We analyze the forward modeling complexity influence in the ESI task using a patient-specific realistic head model and real EEG data. We calculate FDM solutions in a reciprocity setup for the 70 electrodes configuration given in the multi-modal, multi-subject database [13]. Bayesian model selection for group studies using random effect analysis results, Fig. 2 shows concluding evidence in favor of most complex head models, with a high posterior model frequency probability for the two considered prior methods and visual stimulus. Results are similar to the reported in [10], where only analyze CSF inclusion. Based on our results, we suggest that at least 5-layer tissue compartments segmentation are needed to use head models for distributed ESI techniques; this result is similar to the suggested by [10].