... | ... | @@ -7,15 +7,22 @@ An overview of the different parameters is given below. |
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# Partial Integration
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Direct application of the distributional derivative of the dipolar source term to the test function. The resulting right hand side will have as many non-zero entries as the element containing the dipole has associated degrees of freedom (e.g. nodes for linear basis functions). For tetrahedral meshes, the partial integration approach is constant within each element.
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For a mathematical description see, e.g.:
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Bauer M, Pursiainen S, Vorwerk J, Kostler H, Wolters CH. Comparison Study for Whitney (Raviart-Thomas)-Type Source Models in Finite-Element-Method-Based EEG Forward Modeling. IEEE Trans Biomed Eng. 2015 Nov;62(11):2648-56. [https://doi.org/10.1109/TBME.2015.2439282](https://doi.org/10.1109/TBME.2015.2439282)
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# St. Venant
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The idea of the approach following St. Venant is to replace the dipolar source by a distribution of electrical monopoles which best reproduces the source moment.
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The paper cited above (Bauer et al. 2015), also contains a short overview of the St. Venant method implemented.
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# Whitney
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For the Whitney source modeling approach, a source term with a higher regularity is
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chosen. The primary current $`j_p`$ is discretized in a vector-valued function space consisting of functions with square-integrable divergence.
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For a detailed description, please see
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For a detailed description, please see:
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Miinalainen, T, Rezaei, A, Us, D, Nüßing, A, Engwer, C, Wolters, CH & Pursiainen, S 2019, 'A realistic, accurate and fast source modeling approach for the EEG forward problem', NeuroImage, vol. 184, no. 1, pp. 56-67. [https://doi.org/10.1016/j.neuroimage.2018.08.054](https://doi.org/10.1016/j.neuroimage.2018.08.054)
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# Subtraction
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While $`u_\infty`$ can be computed analytically, the insertion of the above decomposition into the EEG forward problem results in a Poisson equation for the correction potential with a right hand side $`- \nabla \cdot (\sigma_{corr} \nabla u_\infty)`$ and inhomogeneous Neumann boundary conditions.
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For more details, see, e.g., the following publications:
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Drechsler F, Wolters CH, Dierkes T, Si H, Grasedyck L. A full subtraction approach for finite element method based source analysis using constrained Delaunay tetrahedralisation. Neuroimage. 2009 Jul 15;46(4):1055-65. doi: . [https://doi.org/10.1016/j.neuroimage.2009.02.024](https://doi.org/10.1016/j.neuroimage.2009.02.024)
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Wolters, CH, Köstler, H, Möller, C, Härdtlein, J, Grasedyck, L and Hackbusch, W. 2007. Numerical Mathematics of the Subtraction Method for the Modeling of a Current Dipole in EEG Source Reconstruction Using Finite Element Head Models. SIAM J. Sci. Comput. 30, 1 (November 2007), 24–45. [https://doi.org/10.1137/060659053](https://doi.org/10.1137/060659053). |