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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[40]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

[40]

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

[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.

[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.

[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.

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