2015, 12(4): 739-760. doi: 10.3934/mbe.2015.12.739

Optimal design for parameter estimation in EEG problems in a 3D multilayered domain

1. 

Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212

2. 

Centro de Matemática Aplicada, Universidad de San Martín, Buenos Aires, Argentina

3. 

Instituto de Ciencias, Universidad Nacional Gral. Sarmiento, Buenos Aires, Argentina

4. 

Dep. de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires, Argentina

Received  April 2014 Revised  December 2014 Published  April 2015

The fundamental problem of collecting data in the ``best way'' in order to assure statistically efficient estimation of parameters is known as Optimal Experimental Design. Many inverse problems consist in selecting best parameter values of a given mathematical model based on fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data. We consider an electromagnetic interrogation problem, specifically one arising in an electroencephalography (EEG) problem, of finding optimal number and locations for sensors for source identification in a 3D unit sphere from data on its boundary. In this effort we compare the use of the classical $D$-optimal criterion for observation points as opposed to that for a uniform observation mesh. We consider location and best number of sensors and report results based on statistical uncertainty analysis of the resulting estimated parameters.
Citation: H. T. Banks, D. Rubio, N. Saintier, M. I. Troparevsky. Optimal design for parameter estimation in EEG problems in a 3D multilayered domain. Mathematical Biosciences & Engineering, 2015, 12 (4) : 739-760. doi: 10.3934/mbe.2015.12.739
References:
[1]

I. Akduman and R. Kress, Electrostatic imaging via conformal mapping,, Inverse Problems, 18 (2002), 1659.  doi: 10.1088/0266-5611/18/6/315.  Google Scholar

[2]

K. Adoteye, H. T. Banks and K. B. Flores, Optimal design of non-equilibrium experiments for genetic network interrogation,, Applied Math Letters, 40 (2015), 84.  doi: 10.1016/j.aml.2014.09.013.  Google Scholar

[3]

R. A. Albanese, R. L. Medina and J. W. Penn, Mathematics, medicine and microwaves,, Inverse Problems, 10 (1994), 995.  doi: 10.1088/0266-5611/10/5/001.  Google Scholar

[4]

R. A. Albanese, J. W. Penn and R. L. Medina, Short-rise-time microwave pulse propagation through dispersive biological media,, J. Optical Society of America A, 6 (1989), 1441.  doi: 10.1364/JOSAA.6.001441.  Google Scholar

[5]

N. D. Aparicio and M. K. Pidcock, The boundary inverse problem for the Laplace equation in two dimensions,, Inverse Problems, 12 (1996), 565.  doi: 10.1088/0266-5611/12/5/003.  Google Scholar

[6]

H. T. Banks, R. Baraldi, K. Cross, K. B. Flores, C. McChesney, L. Poag and E. Thorpe, Uncertainty quantification in modeling HIV viral mechanics., CRSC-TR13-16, (2013), 13.   Google Scholar

[7]

H. T. Banks, J. E. Banks, K. Link, J. A. Rosenheim, C. Ross and K. A. Tillman, Model comparison tests to determine data information content,, Applied Math Letters, 43 (2015), 10.  doi: 10.1016/j.aml.2014.11.002.  Google Scholar

[8]

H. T. Banks, M. W. Buksas and T. Lin, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts,, Frontiers in Applied Mathematics, (2000).  doi: 10.1137/1.9780898719871.  Google Scholar

[9]

H. T. Banks, S. Dediu and S. L. Ernstberger, Sensitivity functions and their uses in inverse problems,, J. Inverse and Ill-posed Problems, 15 (2007), 683.  doi: 10.1515/jiip.2007.038.  Google Scholar

[10]

H. T. Banks, S. Dediu, S. L. Ernstberger and F. Kappel, A new optimal approach to optimal design problem,, J. Inverse and Ill-posed Problems, 18 (2010), 25.  doi: 10.1515/JIIP.2010.002.  Google Scholar

[11]

H. T. Banks, M. Doumic, C. Kruse, S. Prigent and H. Rezaei, Information content in data sets for a nucleated-polymerization model,, CRSC-TR14-15, (2014), 14.   Google Scholar

[12]

H. T. Banks, K. Holm and F. Kappel, Comparison of optimal design methods in inverse problems,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/7/075002.  Google Scholar

[13]

H. T. Banks, S. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty,, CRC Press, (2014).   Google Scholar

[14]

H. T. Banks and F. Kojima, Boundary shape identification in two-dimensional electrostatic problems using SQUIDs,, J. Inverse and Ill-Posed Problems, 8 (2000), 487.  doi: 10.1515/jiip.2000.8.5.487.  Google Scholar

[15]

H. T. Banks and K. L. Rehm, Experimental design for vector output systems,, Inverse Problems in Sci. and Engr., 22 (2014), 557.  doi: 10.1016/j.aml.2012.08.003.  Google Scholar

[16]

H. T. Banks and K. L. Rehm, Experimental design for distributed parameter vector systems,, Applied Mathematics Letters, 26 (2013), 10.  doi: 10.1016/j.aml.2012.08.003.  Google Scholar

[17]

H. T. Banks, D. Rubio, N. Saintier and M. I. Troparevsky, Optimal design techniques for distributed parameter systems,, CRSC-TR13-01, (2013), 13.   Google Scholar

[18]

H. T. Banks, D. Rubio, N. Saintier and M. I. Troparevsky, Optimal electrode positions for the inverse problem of EEG in a simplified model in 3D,, MACI, 4 (2013), 521.   Google Scholar

[19]

R. Baraldi, K. Cross, C. McChesney, L. Poag, E. Thorpe, K. B. Flores and H. T. Banks, Uncertainty quantification for a model of HIV-1 patient response to antiretroviral therapy interruptions,, Proceedings of the American Control Conference, (2014), 2753.  doi: 10.1109/ACC.2014.6858714.  Google Scholar

[20]

F. Ben Hassen, Y. Boukari and H. Haddar, Inverse impedance boundary problem via the conformal mapping method: the case of small impedances,, Revue ARIMA, 13 (2010), 47.   Google Scholar

[21]

M. Clerc, J. Leblond, J.-P. Marmorat and T. Papadopoulo, Source localization using rational approximation on plane sections,, Inverse Problems, 28 (2012), 1.  doi: 10.1088/0266-5611/28/5/055018.  Google Scholar

[22]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer Applied Mathematical Sciences, (2013).  doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[23]

D. Colton, R. Kress and P. Monk, A new algorithm in electromagnetic inverse scattering theory with an application to medical imaging,, Math Methods Applied Science, 20 (1997), 385.  doi: 10.1002/(SICI)1099-1476(19970325)20:5<385::AID-MMA815>3.0.CO;2-Y.  Google Scholar

[24]

J. C. de Munck, The potential distribution in a layered anisotropic spheroidal volume conductor,, J. Appl. Phys., 64 (1988), 464.   Google Scholar

[25]

J. C. de Munck, H. Huizenga, L. J. Waldrop and R. M. Heethaar, Estimating stationary dipoles from MEG/EEG data contaminated with spatially and temporal correlated background noise,, IEEE Trans. On Signal Processing, 50 (2002), 1565.   Google Scholar

[26]

A. El Badia, A inverse source problem in an anisotropic medium by boundary measurements,, Inverse Problems, 16 (2000), 651.  doi: 10.1088/0266-5611/16/3/308.  Google Scholar

[27]

A. El Badia, Summary of some results on an EEG inverse problem,, Neurology and Clinical Neurophysiology, 2004 (2004).   Google Scholar

[28]

A. El Badia and M. Farah, Identification of dipole sources in an elliptic equation from boundary measurements: Application to the inverse EEG problem,, J. Inv. Ill-Posed Problems, 14 (2006), 331.  doi: 10.1515/156939406777571012.  Google Scholar

[29]

C. Gabriel, S. Gabriel and E. Corthout, The dielectric properties of biological tissues: I. Literature survey,, Phys. Med. Biol., 41 (1996), 2231.  doi: 10.1088/0031-9155/41/11/001.  Google Scholar

[30]

S. Gabriel, R. W. Lau and C. Gabriel, The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,, Phys. Med. Biol., 41 (1996), 2251.  doi: 10.1088/0031-9155/41/11/002.  Google Scholar

[31]

S. Gabriel, R. W. Lau and C. Gabriel, The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues,, Phys. Med. Biol., 41 (1996), 2271.  doi: 10.1088/0031-9155/41/11/003.  Google Scholar

[32]

M. Hamalainen, R. Hari, R.J. Ilmoniemi, J. Knuutila and O. Lounasmaa, Magnetoencephalography, theory, instrumentation and applications to noninvasive studies of the working human brain,, Reviews of Modern Physics, 65 (1993), 414.   Google Scholar

[33]

H. Huizenga, J. C. de Munck, L. J. Waldrop and R. P. Grasman, Spatiotemporal EEG/MEG source analysis based on a parametric noise covariance model,, IEEE Trans. Biomedical Engineering, 49 (2002), 533.  doi: 10.1109/TBME.2002.1001967.  Google Scholar

[34]

R. Kress, Inverse Dirichlet problem and conformal mapping,, Mathematics and Computers in Simulation, 66 (2004), 255.  doi: 10.1016/j.matcom.2004.02.006.  Google Scholar

[35]

R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem,, Inverse Problems, 21 (2005), 1207.  doi: 10.1088/0266-5611/21/4/002.  Google Scholar

[36]

J. C. Mosher, R. M. Leahy and P. S. Lewis, EEG and MEG: Forward solutions for inverse methods,, Trans. Biomedical Engineering, 46 (1999), 245.  doi: 10.1109/10.748978.  Google Scholar

[37]

D. Rubio and M. I. Troparevsky, The EEG forward problem: Theoretical and numerical aspects,, Latin American Applied Research, 36 (2006), 87.   Google Scholar

[38]

J. Sarvas, Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem,, Phy. Med Biol., 32 (1987), 11.  doi: 10.1088/0031-9155/32/1/004.  Google Scholar

[39]

P. H. Schimpf, C. Ramon and J. Haneisen, Dipole models for EEG and MEG,, IEEE Trans. Biomedical Engineering, 49 (2002), 409.  doi: 10.1109/10.995679.  Google Scholar

[40]

G. A. F. Seber and C. J. Wild, Nonlinear Regression,, John Wiley & Sons, (1989).  doi: 10.1002/0471725315.  Google Scholar

[41]

M. I. Troparevsky and D. Rubio, On the weak solutions of the forward problem in EEG,, J. of Applied Mathematics, 12 (2003), 647.  doi: 10.1155/S1110757X03305030.  Google Scholar

[42]

M. I. Troparevsky and D. Rubio, Weak solutions of the forward problem in EEG for different conductivity values,, Mathematical and Computer Modeling, 41 (2005), 1437.  doi: 10.1016/j.mcm.2004.02.037.  Google Scholar

[43]

I. S. Yetik, A. Nehorai, C. H. Muravchik and J. Haueisen, Line-source modeling and estimation with magnetoencephalography,, IEEE Trans. Biomed. Eng., 2 (2004), 1339.  doi: 10.1109/ISBI.2004.1398794.  Google Scholar

show all references

References:
[1]

I. Akduman and R. Kress, Electrostatic imaging via conformal mapping,, Inverse Problems, 18 (2002), 1659.  doi: 10.1088/0266-5611/18/6/315.  Google Scholar

[2]

K. Adoteye, H. T. Banks and K. B. Flores, Optimal design of non-equilibrium experiments for genetic network interrogation,, Applied Math Letters, 40 (2015), 84.  doi: 10.1016/j.aml.2014.09.013.  Google Scholar

[3]

R. A. Albanese, R. L. Medina and J. W. Penn, Mathematics, medicine and microwaves,, Inverse Problems, 10 (1994), 995.  doi: 10.1088/0266-5611/10/5/001.  Google Scholar

[4]

R. A. Albanese, J. W. Penn and R. L. Medina, Short-rise-time microwave pulse propagation through dispersive biological media,, J. Optical Society of America A, 6 (1989), 1441.  doi: 10.1364/JOSAA.6.001441.  Google Scholar

[5]

N. D. Aparicio and M. K. Pidcock, The boundary inverse problem for the Laplace equation in two dimensions,, Inverse Problems, 12 (1996), 565.  doi: 10.1088/0266-5611/12/5/003.  Google Scholar

[6]

H. T. Banks, R. Baraldi, K. Cross, K. B. Flores, C. McChesney, L. Poag and E. Thorpe, Uncertainty quantification in modeling HIV viral mechanics., CRSC-TR13-16, (2013), 13.   Google Scholar

[7]

H. T. Banks, J. E. Banks, K. Link, J. A. Rosenheim, C. Ross and K. A. Tillman, Model comparison tests to determine data information content,, Applied Math Letters, 43 (2015), 10.  doi: 10.1016/j.aml.2014.11.002.  Google Scholar

[8]

H. T. Banks, M. W. Buksas and T. Lin, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts,, Frontiers in Applied Mathematics, (2000).  doi: 10.1137/1.9780898719871.  Google Scholar

[9]

H. T. Banks, S. Dediu and S. L. Ernstberger, Sensitivity functions and their uses in inverse problems,, J. Inverse and Ill-posed Problems, 15 (2007), 683.  doi: 10.1515/jiip.2007.038.  Google Scholar

[10]

H. T. Banks, S. Dediu, S. L. Ernstberger and F. Kappel, A new optimal approach to optimal design problem,, J. Inverse and Ill-posed Problems, 18 (2010), 25.  doi: 10.1515/JIIP.2010.002.  Google Scholar

[11]

H. T. Banks, M. Doumic, C. Kruse, S. Prigent and H. Rezaei, Information content in data sets for a nucleated-polymerization model,, CRSC-TR14-15, (2014), 14.   Google Scholar

[12]

H. T. Banks, K. Holm and F. Kappel, Comparison of optimal design methods in inverse problems,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/7/075002.  Google Scholar

[13]

H. T. Banks, S. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty,, CRC Press, (2014).   Google Scholar

[14]

H. T. Banks and F. Kojima, Boundary shape identification in two-dimensional electrostatic problems using SQUIDs,, J. Inverse and Ill-Posed Problems, 8 (2000), 487.  doi: 10.1515/jiip.2000.8.5.487.  Google Scholar

[15]

H. T. Banks and K. L. Rehm, Experimental design for vector output systems,, Inverse Problems in Sci. and Engr., 22 (2014), 557.  doi: 10.1016/j.aml.2012.08.003.  Google Scholar

[16]

H. T. Banks and K. L. Rehm, Experimental design for distributed parameter vector systems,, Applied Mathematics Letters, 26 (2013), 10.  doi: 10.1016/j.aml.2012.08.003.  Google Scholar

[17]

H. T. Banks, D. Rubio, N. Saintier and M. I. Troparevsky, Optimal design techniques for distributed parameter systems,, CRSC-TR13-01, (2013), 13.   Google Scholar

[18]

H. T. Banks, D. Rubio, N. Saintier and M. I. Troparevsky, Optimal electrode positions for the inverse problem of EEG in a simplified model in 3D,, MACI, 4 (2013), 521.   Google Scholar

[19]

R. Baraldi, K. Cross, C. McChesney, L. Poag, E. Thorpe, K. B. Flores and H. T. Banks, Uncertainty quantification for a model of HIV-1 patient response to antiretroviral therapy interruptions,, Proceedings of the American Control Conference, (2014), 2753.  doi: 10.1109/ACC.2014.6858714.  Google Scholar

[20]

F. Ben Hassen, Y. Boukari and H. Haddar, Inverse impedance boundary problem via the conformal mapping method: the case of small impedances,, Revue ARIMA, 13 (2010), 47.   Google Scholar

[21]

M. Clerc, J. Leblond, J.-P. Marmorat and T. Papadopoulo, Source localization using rational approximation on plane sections,, Inverse Problems, 28 (2012), 1.  doi: 10.1088/0266-5611/28/5/055018.  Google Scholar

[22]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer Applied Mathematical Sciences, (2013).  doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[23]

D. Colton, R. Kress and P. Monk, A new algorithm in electromagnetic inverse scattering theory with an application to medical imaging,, Math Methods Applied Science, 20 (1997), 385.  doi: 10.1002/(SICI)1099-1476(19970325)20:5<385::AID-MMA815>3.0.CO;2-Y.  Google Scholar

[24]

J. C. de Munck, The potential distribution in a layered anisotropic spheroidal volume conductor,, J. Appl. Phys., 64 (1988), 464.   Google Scholar

[25]

J. C. de Munck, H. Huizenga, L. J. Waldrop and R. M. Heethaar, Estimating stationary dipoles from MEG/EEG data contaminated with spatially and temporal correlated background noise,, IEEE Trans. On Signal Processing, 50 (2002), 1565.   Google Scholar

[26]

A. El Badia, A inverse source problem in an anisotropic medium by boundary measurements,, Inverse Problems, 16 (2000), 651.  doi: 10.1088/0266-5611/16/3/308.  Google Scholar

[27]

A. El Badia, Summary of some results on an EEG inverse problem,, Neurology and Clinical Neurophysiology, 2004 (2004).   Google Scholar

[28]

A. El Badia and M. Farah, Identification of dipole sources in an elliptic equation from boundary measurements: Application to the inverse EEG problem,, J. Inv. Ill-Posed Problems, 14 (2006), 331.  doi: 10.1515/156939406777571012.  Google Scholar

[29]

C. Gabriel, S. Gabriel and E. Corthout, The dielectric properties of biological tissues: I. Literature survey,, Phys. Med. Biol., 41 (1996), 2231.  doi: 10.1088/0031-9155/41/11/001.  Google Scholar

[30]

S. Gabriel, R. W. Lau and C. Gabriel, The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,, Phys. Med. Biol., 41 (1996), 2251.  doi: 10.1088/0031-9155/41/11/002.  Google Scholar

[31]

S. Gabriel, R. W. Lau and C. Gabriel, The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues,, Phys. Med. Biol., 41 (1996), 2271.  doi: 10.1088/0031-9155/41/11/003.  Google Scholar

[32]

M. Hamalainen, R. Hari, R.J. Ilmoniemi, J. Knuutila and O. Lounasmaa, Magnetoencephalography, theory, instrumentation and applications to noninvasive studies of the working human brain,, Reviews of Modern Physics, 65 (1993), 414.   Google Scholar

[33]

H. Huizenga, J. C. de Munck, L. J. Waldrop and R. P. Grasman, Spatiotemporal EEG/MEG source analysis based on a parametric noise covariance model,, IEEE Trans. Biomedical Engineering, 49 (2002), 533.  doi: 10.1109/TBME.2002.1001967.  Google Scholar

[34]

R. Kress, Inverse Dirichlet problem and conformal mapping,, Mathematics and Computers in Simulation, 66 (2004), 255.  doi: 10.1016/j.matcom.2004.02.006.  Google Scholar

[35]

R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem,, Inverse Problems, 21 (2005), 1207.  doi: 10.1088/0266-5611/21/4/002.  Google Scholar

[36]

J. C. Mosher, R. M. Leahy and P. S. Lewis, EEG and MEG: Forward solutions for inverse methods,, Trans. Biomedical Engineering, 46 (1999), 245.  doi: 10.1109/10.748978.  Google Scholar

[37]

D. Rubio and M. I. Troparevsky, The EEG forward problem: Theoretical and numerical aspects,, Latin American Applied Research, 36 (2006), 87.   Google Scholar

[38]

J. Sarvas, Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem,, Phy. Med Biol., 32 (1987), 11.  doi: 10.1088/0031-9155/32/1/004.  Google Scholar

[39]

P. H. Schimpf, C. Ramon and J. Haneisen, Dipole models for EEG and MEG,, IEEE Trans. Biomedical Engineering, 49 (2002), 409.  doi: 10.1109/10.995679.  Google Scholar

[40]

G. A. F. Seber and C. J. Wild, Nonlinear Regression,, John Wiley & Sons, (1989).  doi: 10.1002/0471725315.  Google Scholar

[41]

M. I. Troparevsky and D. Rubio, On the weak solutions of the forward problem in EEG,, J. of Applied Mathematics, 12 (2003), 647.  doi: 10.1155/S1110757X03305030.  Google Scholar

[42]

M. I. Troparevsky and D. Rubio, Weak solutions of the forward problem in EEG for different conductivity values,, Mathematical and Computer Modeling, 41 (2005), 1437.  doi: 10.1016/j.mcm.2004.02.037.  Google Scholar

[43]

I. S. Yetik, A. Nehorai, C. H. Muravchik and J. Haueisen, Line-source modeling and estimation with magnetoencephalography,, IEEE Trans. Biomed. Eng., 2 (2004), 1339.  doi: 10.1109/ISBI.2004.1398794.  Google Scholar

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