# Modeling and Inverse Problems in EEG/MEG

by Felix Lucka

## Introduction

The physical phenomena of electromagnetic fields produced by living cells, tissue or organisms is called bioelectromagnetism. This phenomena can be used to infer information about biological processes in the human body in a non-invasive way. The most common example of a such a diagnostic procedure is probably the visulization of the electrical activity of the human heart by*Electrocardiography*(

*ECG*) recordings.

*Electroencephalography*(

*EEG*) and

*magnetoencephalography*(

*MEG*) record the electromagnetic fields generated by ion currents in active brain cells

**Figure 1:** Confocal image of pyramidal cell in mouse cortex. Source: Wikimedia Commons, file: GFPneuron.png

*source reconstruction*). This task involves challenging mathematical problems in different areas of mathematical modeling and imaging.

The imaging workgroup cooperates with the workgroup "Methods in Bioelectromagnetism" by PD. Dr. Carsten Wolters located at the Institute for Biomagnetism and Biosignalanalysis of the Münster Faculty of Medicine on various topics:

## Head Modeling and Forward Computation in EEG/MEG

The first step for source reconstruction is to generate an individual model of the patient's head that captures all its relevant physical properties (*head modeling*). Based on this head model, the mathematical modeling of bioelectromagnetism (

*forward modeling*) and computational methods (

*forward computation*), one is able to simulate the electromagnetic field distribution on the head surface for a given current source in the brain.

**Figure 2:** *Left:*MEG measurement device. *Right:*EEG electrode cap; Source: Wikimedia Commons, file: EEG_cap.jpg.

**Figure 3:** Different MRI scans of the same subjekt. *Left*: T1-MRI image; *middle*: T2-MRI image; *right*: DW-MRI image (FA is visualized).

### Image Registration for EEG/MEG Head Modeling

To generate an individual model of the patient's head one has to get detailed information about the head's tissues and their electrical properties. Magnetic resonance imaging provides a powerful and non-invasive imaging technique for this purpose. Different MRI scan sequences can reveal different aspects of the head tissues or provide a different contrast between them (see Figure 3).The task of merging (or

*fusing*) all acquired information into a common reference frame is called

*image registration*and relies on sophisticated mathematical techniques. An introduction into that topic is given here.

#### Own Contributions:

- Ruthotto, L., Kugel, H., Olesch, J., Fischer, B., Modersitzki, J., Burger, M., Wolters, C.H. (2012). Diffeomorphic Susceptibility Artefact Correction of Diffusion-Weighted Magnetic Resonance Images. In Physics in Medicine and Biology, 57(18):5715-5731.
- Olesch, J., Ruthotto, L., Kugel, H., Skare, S., Fischer, B., and Wolters, C.H. (2010). A variational approach for the correction of field-inhomogeneities in EPI sequences. In Proc. SPIE 7623.
- Ruthotto, L. (2010). Mass-preserving Registration of Medical Images. German diploma thesis (mathematics), University of Muenster. Supervision by M. Burger and C.H. Wolters.

### Image Segementation for EEG/MEG Head Modeling

Once the MRI images have been co-registered, information about the different head tissues can be inferred. The task of partitioning a digital image into multiple classes (in our case the head tissues we want to differentiate in our model) is called*image segmentation*(see Figure 4). The accurate segmentation of head tissues for the purpose of EEG/MEG source reconstruction poses specific challenges, which necessitates research into this direction.

**Figure 4:** Segementation of the MRI data into different head tissues: Skin (green), eyes (yellow), skull compacta (blue), skull spongiosa (red), csf (turquoise), gray matter (black), white matter (white).

#### Own Contributions:

- Lanfer B, Scherg M, Dannhauer M, Knösche TR, Burger M, Wolters CH. (2012). Influences of skull segmentation inaccuracies on EEG source analysis. In Neuroimage, 62(1):418-31.

### Forward Computation in EEG/MEG

The comparison of simulated and measured data is a central ingredient of all source reconstruction approaches. The relation between the neural currents, the head's tissue and the induced electromagnetic fields is described by*Maxwell's equations*. Under some simplifying assumptions, a solution of these equations by different numerical approaches can be considered. If a realistic and individual modeling of the head's tissues is of highest importance, the

*finite element method (FEM)*is the method of choice as it can account for complex geometries and anisotropic conductivities. To apply FEM, a mesh has to be build from the segmentation (see Figure 8) and a mathematical model of the neural activity has to be formulated and treated within the FEM framework (see Figure 6).

**Figure 5:**A finite element head model which can be used for an EEG/MEG forward computation.

**Figure 6:** The resulting EEG potentials and MEG fields of a FEM-based forward computation for a single dipolar current source (green cone).

#### Own Contributions:

- Vorwerk J., Clerc M., Burger M., Wolters C.H. (2012). Comparison of Boundary Element and Finite Element Approaches to the EEG Forward Problem. In Biomed Tech 2012, 57 (Suppl. 1):795-798.
- Pursiainen, S and Lucka, F and Wolters, C.H. (2012). Complete electrode model in EEG: relationship and differences to the point electrode model. In Physics in Medicine and Biology, 57(4):999-1017.
- Vorwerk, J. (2011). Comparison of Numerical Approaches to the EEG Forward Problem. German diploma thesis (mathematics), University of Muenster. Supervision by M. Burger and C.H. Wolters.

## The Inverse Problem of EEG/MEG

The reconstruction of the so-called*primary*or

*impressed*currents is called the

*EEG/MEG inverse problem*. In its generic formulation, it lacks a unique solution, and infinitely many source configurations, often with extremely different properties, can explain the measured fields. All inverse methods rely on the use of

*a priori information*on the source activity to choose a particular solution from the set of likely solutions. This a priori information can reflect computational constraints as well as neurological considerations. Nevertheless, because the problem is heavily under-determined, the results from different methods for the same measurement data can still differ considerably (see Figure 7). Consequently, most methods work well for certain source configurations while failing to recover other configurations. Therefore, a careful examination of the performance of the methods for different source scenarios is mandatory.

**Figure 7:** Different source reconstructions (red to yellow cones) for simulated measurement data from a source configuration consisting of three sources (turquoise cones). *Left*: Minimum norm solution; *right*: MAP estimate for hierarchical Bayesian model.

#### Own Contributions:

- Lucka, F., Pursiainen, S., Burger, M., Wolters, C.H. (2012). Hierarchical Bayesian inference for the EEG inverse problem using realistic FE head models: Depth localization and source separation for focal primary currents. In
*NeuroImage*, 61(4):1364-1382. - Lucka, F. (2011). Hierarchical Bayesian Approaches to the Inverse Problem of EEG/MEG Current Density Reconstruction. German diploma thesis (mathematics), University of Muenster. Supervision by M. Burger and C.H. Wolters.
- Steinsträter, O. Sillekens, S., Junghofer, M., Burger, M., Wolters, C.H. (2010). Sensitivity of beamformer source analysis to deficiencies in forward modeling, In
*Human Brain Mapping*, 31(12):1907-1927.

## Simulation of tDCS and TMS

The head modeling techniques and forward approaches developed for source reconstruction can be used to simulate another medical application of bioelectromagnetism, which has attracted a lot of attention lately: The noninvasive stimulation of cortical brain regions by currents which are induced by external electric or magnetic fields (*neurostimulation*). In transcranial Magnetic Stimulation (TMS) magnetic fields are generated by a coil device, while in transcranial direct-current stimulation (tDCS), an electric potential is applied by attaching electrodes to the head surface. Simulating the induced currents by means of realistic head modeling and forward computation can help to better understand and use neurostimulation techniques.

**Figure 8:** Orientations and amplitudes of induced currents of a tDCS stimulation. *Left*: Results for a simplified head model; *right*: Results for a more realistic head model.

#### Own Contributions:

- Wagner, S. (2011). An adjoint FEM approach for the EEG forward problem. German diploma thesis (mathematics), University of Muenster. Supervision by M. Burger and C.H. Wolters.