PhD Position INRIA France

June 8, 2008

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PhD position: A Coupled FEM/BEM/lineic Method for Solving the M/EEG Forward Problem

Magneto- and electro-encephalography provide non-invasive measurements of the activity of the brain at millisecond time-resolution. They measure the electrical (EEG) or magnetic (MEG) fields that are created outside of the head by electrical sources situated in the cortex. From these measurements one can use an inverse problem to obtain source localisations within the brain. This inverse problem combines the measurements and a regularisation procedure with a model of the electric propagation in the head (called a forward model), to compute the best source distribution that explains the measures and that is optimal with respect to a chosen regularization.

The Inria Project Team Odyssée has developped several techniques, based on Finite Element or Boundary Elements to tackle the forward problem. Each of these techniques is based on a specific type of head model, with associated advantages and drawbacks.

The Boundary Element Method (BEM) assumes that head conductivity is homogeneous and constant in domains separated by surfaces. This model is rather simple to use as meshes are only required on the interfaces and it reduces the discretization errors within each domains, as analytical formulae are used within these domains. But this is at the price of the simplified conductivity model.

The Finite Element Method (FEM), a volumic approach, requires the whole head domain to be discretized. This also allows for more general conductivity models (ie anisotropic models such as those provided by a tensor). Since some compartments of the head (eg the skull) have such types of conductivities, the generality of this model is interesting to properly account for these tissues. The FEM approach also has a computational advantage as it tends to produce sparse symmetric matrices which are much easier to manipulate than the full ones generated by the BEM model. However, the FEM model is bigger and more complicated to build.

The skull conductivity has a rather simple anisotropy structure (different radial and tangential conductivities). The white matter of the brain also has an anisotropic conductivity, but with a more complicated structure. Basically, the white matter is constituted of fibres contained in a homogenous and isotropic domain. Unfortunately, FEM cannot handle such complicated conductivity domains easily. The best way to handle such domains would be to have a superposition of an isotropic domain and an ad hoc model for fibres (a set of lineic models).
There are thus various solvers for the forward problem, each with its strengths and weaknesses. The goal of this PhD thesis is to study an hybrid method that combines these different solvers into a single solver that uses the best available method depending on the tissue properties.

We suggest the following main stages for the PhD:
1. A bibliographic study of domain decomposition and method coupling to clarify the state of the art and how these techniques can be applied to the M/EEG problem.
2. To effectively couple the existing BEM and FEM codes. In a first step, it may be possible to assume that the meshes are conforming for both codes (ie the FEM mesh corresponds to the BEM mesh on the interfaces), but we also would like to explore the case where the meshes can be totally different. It will be interesting to evaluate the improvements brought by the coupled method.
3. To propose a lineic model of the fibre part of the white matter. The fibre geometry will be extracted from DT-MRI images. A litterature survey is needed to design a good conductivity model. The conductivity model may be a tensor. However,much is known about the propagation of currents along fibres, and it seems appropriate to introduce a model with lineic conductivities, the directions of which can be estimated by tractography of diffusion Magnetic Resonance Images. As said above, the white matter also has an isotropic component, which can be modeled with BEM or FEM. The coupling of these two contribution needs to be defined and implemented.
4. Finally, to couple all these models into a single head model that uses the most appropriate model for each tissue of the head.

Prerequisites
1. Knowledge of C++ since the above-mentioned codes are written in this langage,
2. An ability for programming and understanding preexisting code,
3. A background in numerical analysis.

Research environment
The PhD thesis will take place at INRIA Sophia Antipolis, in close partnership with the Magnetoencephalography center of Marseille at La Timone. It is partly funded by the Regional Council of Provence Alpes Côte d’Azur.
For further information, please contact Maureen.Clerc@sophia.inria.fr (advisor) or Theodore.Papadopoulo@sophia.inria.fr (co-advisor).

Deadline: July 31, 2008.

Scholarship Tags: France, INRIA, PhD, Position