# American Institute of Mathematical Sciences

November  2013, 7(4): 1251-1270. doi: 10.3934/ipi.2013.7.1251

## Analytic sensing for multi-layer spherical models with application to EEG source imaging

 1 École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, Switzerland 2 Chinese University of Hong Kong, Shatin, Hong Kong, China

Received  June 2011 Revised  March 2013 Published  November 2013

Source imaging maps back boundary measurements to underlying generators within the domain; e.g., retrieving the parameters of the generating dipoles from electrical potential measurements on the scalp such as in electroencephalography (EEG). Fitting such a parametric source model is non-linear in the positions of the sources and renewed interest in mathematical imaging has led to several promising approaches.
One important step in these methods is the application of a sensing principle that links the boundary measurements to volumetric information about the sources. This principle is based on the divergence theorem and a mathematical test function that needs to be an homogeneous solution of the governing equations (i.e., Poisson's equation). For a specific choice of the test function, we have devised an algebraic non-iterative source localization technique for which we have coined the term analytic sensing''.
Until now, this sensing principle has been applied to homogeneous-conductivity spherical models only. Here, we extend it for multi-layer spherical models that are commonly applied in EEG. We obtain a closed-form expression for the test function that can then be applied for subsequent localization. A simulation study show the feasibility of the proposed approach.
Citation: Djano Kandaswamy, Thierry Blu, Dimitri Van De Ville. Analytic sensing for multi-layer spherical models with application to EEG source imaging. Inverse Problems & Imaging, 2013, 7 (4) : 1251-1270. doi: 10.3934/ipi.2013.7.1251
##### References:
 [1] S. Andrieux, T. N. Baranger and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional,, Inverse Problems, 22 (2006), 115. doi: 10.1088/0266-5611/22/1/007. Google Scholar [2] S. Andrieux and A. Ben Abda, The reciprocity gap: A general concept for flaws identification problems,, Mechanical Research Communications, 20 (1993), 415. doi: 10.1016/0093-6413(93)90032-J. Google Scholar [3] S. Andrieux, A. Ben Abda and J. Mohamed, On the inverse emergent plane crack problem,, Mathematical Methods in the Applied Sciences, 21 (1998), 895. Google Scholar [4] J. P. Ary, S. A. Klein and D. H. Fender, Location of sources of evoked scalp potentials: Corrections for skull and scalp thicknesses,, IEEE Transactions on Biomedical Engineering, BME-28 (1981), 447. doi: 10.1109/TBME.1981.324817. Google Scholar [5] K. A. Awada, D. R. Jackson, S. B. Baumann, B. Stephen, J. T. Williams, D. R. Wilton, P. Fink and B. Prasky, Effect of conductivity uncertainties and modeling errors on EEG source localization using a 2-D model,, IEEE Transactions on Biomedical Engineering, 45 (1998), 1135. doi: 10.1109/10.709557. Google Scholar [6] S. Baillet, J. C. Mosher and R. M. Leahy, Electromagnetic brain mapping,, IEEE Signal Processing Magazine, 18 (2001), 14. doi: 10.1109/79.962275. Google Scholar [7] L. Baratchart, A. Ben Abda, F. Ben Hassen and J. Leblond, Recovery of pointwise sources or small inclusions in 2D domains and rational approximation,, Inverse Problems, 21 (2005), 51. doi: 10.1088/0266-5611/21/1/005. Google Scholar [8] L. Baratchart, J. Leblond and J. P. Marmorat, Inverse sources problem in a 3D ball from best meromorphic approximation on 2D slices,, Electronic Transactions on Numerical Analysis, 25 (2006), 41. Google Scholar [9] G. R. Barnes and A. Hillebrand, Statistical flattening of MEG beamformer images,, Human Brain Mapping, 18 (2003), 1. doi: 10.1002/hbm.10072. Google Scholar [10] G. Birot, L. Albera, F. Wendling and I. Merlet, Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach,, NeuroImage, (2011). Google Scholar [11] T. Blu, P.-L. Dragotti, M. Vetterli, P. Marziliano and L. Coulot, Sparse sampling of signal innovations,, IEEE Signal Processing Magazine, 25 (2008), 31. doi: 10.1109/MSP.2007.914998. Google Scholar [12] M. Clerc and J. Kybic, Cortical mapping by Laplace-Cauchy transmission using a boundary element method,, Inverse Problems, 23 (2007), 2589. doi: 10.1088/0266-5611/23/6/020. Google Scholar [13] B. N. Cuffin, Effects of head shape on EEG's and MEG's,, IEEE Transactions On Biomedical Engineering, 37 (1990), 44. Google Scholar [14] A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis,, Inverse Problems, 16 (2000), 651. doi: 10.1088/0266-5611/16/3/308. Google Scholar [15] G. E. Fasshauerand, Mathematical Methods For Curves And Surfaces II,, Vanderbilt University Press, (1998). Google Scholar [16] D. B. Geselowitz, On bioelectric potentials in an inhomogeneous volume conductor,, Biophysical Journal, 7 (1967), 1. doi: 10.1016/S0006-3495(67)86571-8. Google Scholar [17] D. Gutirrez and A. Nehorai, Estimating brain conductivities and dipole source signals with EEG arrays,, IEEE Transactions On Biomedical Engineering, 51 (2004), 2113. doi: 10.1109/TBME.2004.836507. Google Scholar [18] H. L. F. Helmholtz, Über Einige Gesetze der Vertheilung Elektrischer Ströme in Köperlichen Leitern mit Anwendung auf die Thierisch-Elektrischen Versuche,, Annalen der Physik, 9 (1853), 211. Google Scholar [19] V. Isakov, Inverse Source Problems,, 34 of Mathematical Surveys and Monographs Series. AMS, (1990). Google Scholar [20] D. Kandaswamy, Analytic Sensing: Sparse Source Recovery From Boundary Measurements Using An Extension Of Prony's Method For The Poisson Equation,, PhD thesis, (2011). Google Scholar [21] D. Kandaswamy, T. Blu and D. Van De Ville, Analytic sensing: Noniterative retrieval of point sources from boundary measurements,, SIAM Journal on Scientific Computing, 31 (2009), 3179. doi: 10.1137/080712829. Google Scholar [22] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations,, Comput. Math. Phys., 31 (1991), 45. Google Scholar [23] J. Kybic, M. Clerc, T. Abboud, O. Faugeras, R. Keriven and T. Papadopoulo, A common formalism for the integral formulations of the forward EEG problem,, IEEE Transactions on Medical Imaging, 24 (2005), 12. doi: 10.1109/TMI.2004.837363. Google Scholar [24] J. Kybic, M. Clerc, O. Faugeras, R. Keriven and T. Papadopoulo, Fast multipole acceleration of the MEG/EEG boundary element method,, Physics in Medicine and Biology, 50 (2005), 4695. doi: 10.1088/0031-9155/50/19/018. Google Scholar [25] J. Kybic, M. Clerc, O. Faugeras, R. Keriven and T. Papadopoulo, Generalized head models for MEG/EEG: Boundary element method beyond nested volumes,, Physics in Medicine and Biology, 51 (2006), 13333. doi: 10.1088/0031-9155/51/5/021. Google Scholar [26] C. M. Michel, G. Lantz, L. Spinelli, R. Grave de Peralta, T. Landis and M. Seeck, 128-channel EEG source imaging in epilepsy: Clinical yield and localization precision,, J. Clin. Neurosphysiol, 21 (2004), 71. doi: 10.1097/00004691-200403000-00001. Google Scholar [27] C. M. Michel, M. M. Murray, G. Lantz, S. Gonzalez, L. Spinelli and R. Grave de Peralta, EEG source imaging,, Clin. Neurophysiol, 115 (2004), 2195. doi: 10.1016/j.clinph.2004.06.001. Google Scholar [28] K. Miller, Stabilized numerical prolongation with poles,, SIAM J. Appl. Math., 18 (1970), 346. doi: 10.1137/0118029. Google Scholar [29] S. Mingui, An efficient algorithm for computing multishell spherical volume conductor models in EEG dipole source localization,, IEEE Transactions On Biomedical Engineering, 44 (1997), 1243. Google Scholar [30] J. C. Mosher, P. S. Lewis and R. M. Leahy, Multiple dipole modeling and localization from spatio-temporal MEG data,, IEEE Transactions on Biomedical Engineering, 39 (1992), 541. doi: 10.1109/10.141192. Google Scholar [31] T. Nara and S. Ando, A projective method for an inverse source problem of the Poisson equation,, Inverse Problems, 19 (2003), 355. doi: 10.1088/0266-5611/19/2/307. Google Scholar [32] M. Scherg and D. von Cramon, Two bilateral sources of the late AEP as identified by a spatio-temporal dipole model,, Electroenceph Clinic Neurophysiol, 62 (1985), 32. doi: 10.1016/0168-5597(85)90033-4. Google Scholar [33] D. M. Schmidt, J. S. George and C. C. Wood, Bayesian inference applied to the electromagnetic inverse problem,, Human Brain Mapping, 7 (1999), 195. Google Scholar [34] L. Spinelli, S. G. Andino, G. Lantz, M. Seeck and C. M. Michel, Electromagnetic inverse solutions in anatomically constrained spherical head models,, Brain Topography, 13 (2000). Google Scholar [35] V. Srinivasan, C. Eswaran and N. Sriraam, Approximate entropy-based epileptic EEG detection using artificial neural networks,, IEEE Transactions On Information Technology In Biomedicine, 11 (2007), 288. doi: 10.1109/TITB.2006.884369. Google Scholar [36] O. Steinstrter, S. Sillekens, M. Junghoefer, M. Burger and C. H. Wolters, Sensitivity of beamformer source analysis to deficiencies in forward modeling,, Human Brain Mapping, 31 (2010), 1907. doi: 10.1002/hbm.20986. Google Scholar [37] A. N. Tikhonov, On the stability of inverse problems,, Dokl. Akad. Nauk SSSR, 39 (1943), 176. Google Scholar [38] S. Vallaghe and M. Clerc, A global sensitivity analysis of three- and four-layer EEG conductivity models,, IEEE Transactions on Biomedical Engineering, 56 (2009), 998. doi: 10.1109/TBME.2008.2009315. Google Scholar [39] B. Vanrumste, G. Van Hoey, R. Van de Walle, M. D'Have, I. Limahieu and P. Boon, Dipole location errors in electroencephalogram source analysis due to volume conductor model errors,, Medical & Biological Engineering & Computing, 38 (2000), 528. doi: 10.1007/BF02345748. Google Scholar [40] M. Vetterli, P. Marzilliano and T. Blu, Sampling signals with finite rate of innovation,, IEEE Transactions on Signal Processing, 50 (2002), 1417. doi: 10.1109/TSP.2002.1003065. Google Scholar [41] D. P. Wipf, J. P. Owena, H. T. Attiasb, K. Sekiharac and S. S. Nagarajana, Robust Bayesian estimation of the location, orientation, and time course of multiple correlated neural sources using MEG,, NeuroImage, 49 (2010), 641. doi: 10.1016/j.neuroimage.2009.06.083. Google Scholar

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##### References:
 [1] S. Andrieux, T. N. Baranger and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional,, Inverse Problems, 22 (2006), 115. doi: 10.1088/0266-5611/22/1/007. Google Scholar [2] S. Andrieux and A. Ben Abda, The reciprocity gap: A general concept for flaws identification problems,, Mechanical Research Communications, 20 (1993), 415. doi: 10.1016/0093-6413(93)90032-J. Google Scholar [3] S. Andrieux, A. Ben Abda and J. Mohamed, On the inverse emergent plane crack problem,, Mathematical Methods in the Applied Sciences, 21 (1998), 895. Google Scholar [4] J. P. Ary, S. A. Klein and D. H. Fender, Location of sources of evoked scalp potentials: Corrections for skull and scalp thicknesses,, IEEE Transactions on Biomedical Engineering, BME-28 (1981), 447. doi: 10.1109/TBME.1981.324817. Google Scholar [5] K. A. Awada, D. R. Jackson, S. B. Baumann, B. Stephen, J. T. Williams, D. R. Wilton, P. Fink and B. Prasky, Effect of conductivity uncertainties and modeling errors on EEG source localization using a 2-D model,, IEEE Transactions on Biomedical Engineering, 45 (1998), 1135. doi: 10.1109/10.709557. Google Scholar [6] S. Baillet, J. C. Mosher and R. M. Leahy, Electromagnetic brain mapping,, IEEE Signal Processing Magazine, 18 (2001), 14. doi: 10.1109/79.962275. Google Scholar [7] L. Baratchart, A. Ben Abda, F. Ben Hassen and J. Leblond, Recovery of pointwise sources or small inclusions in 2D domains and rational approximation,, Inverse Problems, 21 (2005), 51. doi: 10.1088/0266-5611/21/1/005. Google Scholar [8] L. Baratchart, J. Leblond and J. P. Marmorat, Inverse sources problem in a 3D ball from best meromorphic approximation on 2D slices,, Electronic Transactions on Numerical Analysis, 25 (2006), 41. Google Scholar [9] G. R. Barnes and A. Hillebrand, Statistical flattening of MEG beamformer images,, Human Brain Mapping, 18 (2003), 1. doi: 10.1002/hbm.10072. Google Scholar [10] G. Birot, L. Albera, F. Wendling and I. Merlet, Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach,, NeuroImage, (2011). Google Scholar [11] T. Blu, P.-L. Dragotti, M. Vetterli, P. Marziliano and L. Coulot, Sparse sampling of signal innovations,, IEEE Signal Processing Magazine, 25 (2008), 31. doi: 10.1109/MSP.2007.914998. Google Scholar [12] M. Clerc and J. Kybic, Cortical mapping by Laplace-Cauchy transmission using a boundary element method,, Inverse Problems, 23 (2007), 2589. doi: 10.1088/0266-5611/23/6/020. Google Scholar [13] B. N. Cuffin, Effects of head shape on EEG's and MEG's,, IEEE Transactions On Biomedical Engineering, 37 (1990), 44. Google Scholar [14] A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis,, Inverse Problems, 16 (2000), 651. doi: 10.1088/0266-5611/16/3/308. Google Scholar [15] G. E. Fasshauerand, Mathematical Methods For Curves And Surfaces II,, Vanderbilt University Press, (1998). Google Scholar [16] D. B. Geselowitz, On bioelectric potentials in an inhomogeneous volume conductor,, Biophysical Journal, 7 (1967), 1. doi: 10.1016/S0006-3495(67)86571-8. Google Scholar [17] D. Gutirrez and A. Nehorai, Estimating brain conductivities and dipole source signals with EEG arrays,, IEEE Transactions On Biomedical Engineering, 51 (2004), 2113. doi: 10.1109/TBME.2004.836507. Google Scholar [18] H. L. F. Helmholtz, Über Einige Gesetze der Vertheilung Elektrischer Ströme in Köperlichen Leitern mit Anwendung auf die Thierisch-Elektrischen Versuche,, Annalen der Physik, 9 (1853), 211. Google Scholar [19] V. Isakov, Inverse Source Problems,, 34 of Mathematical Surveys and Monographs Series. AMS, (1990). Google Scholar [20] D. Kandaswamy, Analytic Sensing: Sparse Source Recovery From Boundary Measurements Using An Extension Of Prony's Method For The Poisson Equation,, PhD thesis, (2011). Google Scholar [21] D. Kandaswamy, T. Blu and D. Van De Ville, Analytic sensing: Noniterative retrieval of point sources from boundary measurements,, SIAM Journal on Scientific Computing, 31 (2009), 3179. doi: 10.1137/080712829. Google Scholar [22] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations,, Comput. Math. Phys., 31 (1991), 45. Google Scholar [23] J. Kybic, M. Clerc, T. Abboud, O. Faugeras, R. Keriven and T. Papadopoulo, A common formalism for the integral formulations of the forward EEG problem,, IEEE Transactions on Medical Imaging, 24 (2005), 12. doi: 10.1109/TMI.2004.837363. Google Scholar [24] J. Kybic, M. Clerc, O. Faugeras, R. Keriven and T. Papadopoulo, Fast multipole acceleration of the MEG/EEG boundary element method,, Physics in Medicine and Biology, 50 (2005), 4695. doi: 10.1088/0031-9155/50/19/018. Google Scholar [25] J. Kybic, M. Clerc, O. Faugeras, R. Keriven and T. Papadopoulo, Generalized head models for MEG/EEG: Boundary element method beyond nested volumes,, Physics in Medicine and Biology, 51 (2006), 13333. doi: 10.1088/0031-9155/51/5/021. Google Scholar [26] C. M. Michel, G. Lantz, L. Spinelli, R. Grave de Peralta, T. Landis and M. Seeck, 128-channel EEG source imaging in epilepsy: Clinical yield and localization precision,, J. Clin. Neurosphysiol, 21 (2004), 71. doi: 10.1097/00004691-200403000-00001. Google Scholar [27] C. M. Michel, M. M. Murray, G. Lantz, S. Gonzalez, L. Spinelli and R. Grave de Peralta, EEG source imaging,, Clin. Neurophysiol, 115 (2004), 2195. doi: 10.1016/j.clinph.2004.06.001. Google Scholar [28] K. Miller, Stabilized numerical prolongation with poles,, SIAM J. Appl. Math., 18 (1970), 346. doi: 10.1137/0118029. Google Scholar [29] S. Mingui, An efficient algorithm for computing multishell spherical volume conductor models in EEG dipole source localization,, IEEE Transactions On Biomedical Engineering, 44 (1997), 1243. Google Scholar [30] J. C. Mosher, P. S. Lewis and R. M. Leahy, Multiple dipole modeling and localization from spatio-temporal MEG data,, IEEE Transactions on Biomedical Engineering, 39 (1992), 541. doi: 10.1109/10.141192. Google Scholar [31] T. Nara and S. Ando, A projective method for an inverse source problem of the Poisson equation,, Inverse Problems, 19 (2003), 355. doi: 10.1088/0266-5611/19/2/307. Google Scholar [32] M. Scherg and D. von Cramon, Two bilateral sources of the late AEP as identified by a spatio-temporal dipole model,, Electroenceph Clinic Neurophysiol, 62 (1985), 32. doi: 10.1016/0168-5597(85)90033-4. Google Scholar [33] D. M. Schmidt, J. S. George and C. C. Wood, Bayesian inference applied to the electromagnetic inverse problem,, Human Brain Mapping, 7 (1999), 195. Google Scholar [34] L. Spinelli, S. G. Andino, G. Lantz, M. Seeck and C. M. Michel, Electromagnetic inverse solutions in anatomically constrained spherical head models,, Brain Topography, 13 (2000). Google Scholar [35] V. Srinivasan, C. Eswaran and N. Sriraam, Approximate entropy-based epileptic EEG detection using artificial neural networks,, IEEE Transactions On Information Technology In Biomedicine, 11 (2007), 288. doi: 10.1109/TITB.2006.884369. Google Scholar [36] O. Steinstrter, S. Sillekens, M. Junghoefer, M. Burger and C. H. Wolters, Sensitivity of beamformer source analysis to deficiencies in forward modeling,, Human Brain Mapping, 31 (2010), 1907. doi: 10.1002/hbm.20986. Google Scholar [37] A. N. Tikhonov, On the stability of inverse problems,, Dokl. Akad. Nauk SSSR, 39 (1943), 176. Google Scholar [38] S. Vallaghe and M. Clerc, A global sensitivity analysis of three- and four-layer EEG conductivity models,, IEEE Transactions on Biomedical Engineering, 56 (2009), 998. doi: 10.1109/TBME.2008.2009315. Google Scholar [39] B. Vanrumste, G. Van Hoey, R. Van de Walle, M. D'Have, I. Limahieu and P. Boon, Dipole location errors in electroencephalogram source analysis due to volume conductor model errors,, Medical & Biological Engineering & Computing, 38 (2000), 528. doi: 10.1007/BF02345748. Google Scholar [40] M. Vetterli, P. Marzilliano and T. Blu, Sampling signals with finite rate of innovation,, IEEE Transactions on Signal Processing, 50 (2002), 1417. doi: 10.1109/TSP.2002.1003065. Google Scholar [41] D. P. Wipf, J. P. Owena, H. T. Attiasb, K. Sekiharac and S. S. Nagarajana, Robust Bayesian estimation of the location, orientation, and time course of multiple correlated neural sources using MEG,, NeuroImage, 49 (2010), 641. doi: 10.1016/j.neuroimage.2009.06.083. Google Scholar
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