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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

G. R. Barnes and A. Hillebrand, Statistical flattening of MEG beamformer images,, Human Brain Mapping, 18 (2003), 1. doi: 10.1002/hbm.10072.

[10]

G. Birot, L. Albera, F. Wendling and I. Merlet, Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach,, NeuroImage, (2011).

[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.

[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.

[13]

B. N. Cuffin, Effects of head shape on EEG's and MEG's,, IEEE Transactions On Biomedical Engineering, 37 (1990), 44.

[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.

[15]

G. E. Fasshauerand, Mathematical Methods For Curves And Surfaces II,, Vanderbilt University Press, (1998).

[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.

[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.

[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.

[19]

V. Isakov, Inverse Source Problems,, 34 of Mathematical Surveys and Monographs Series. AMS, (1990).

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

K. Miller, Stabilized numerical prolongation with poles,, SIAM J. Appl. Math., 18 (1970), 346. doi: 10.1137/0118029.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[37]

A. N. Tikhonov, On the stability of inverse problems,, Dokl. Akad. Nauk SSSR, 39 (1943), 176.

[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.

[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.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

G. R. Barnes and A. Hillebrand, Statistical flattening of MEG beamformer images,, Human Brain Mapping, 18 (2003), 1. doi: 10.1002/hbm.10072.

[10]

G. Birot, L. Albera, F. Wendling and I. Merlet, Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach,, NeuroImage, (2011).

[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.

[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.

[13]

B. N. Cuffin, Effects of head shape on EEG's and MEG's,, IEEE Transactions On Biomedical Engineering, 37 (1990), 44.

[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.

[15]

G. E. Fasshauerand, Mathematical Methods For Curves And Surfaces II,, Vanderbilt University Press, (1998).

[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.

[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.

[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.

[19]

V. Isakov, Inverse Source Problems,, 34 of Mathematical Surveys and Monographs Series. AMS, (1990).

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

K. Miller, Stabilized numerical prolongation with poles,, SIAM J. Appl. Math., 18 (1970), 346. doi: 10.1137/0118029.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[37]

A. N. Tikhonov, On the stability of inverse problems,, Dokl. Akad. Nauk SSSR, 39 (1943), 176.

[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.

[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.

[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.

[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.

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