• Previous Article
    Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model
  • MBE Home
  • This Issue
  • Next Article
    The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics
August 2018, 15(4): 905-932. doi: 10.3934/mbe.2018041

EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity

1. 

GRAMFC INSERM U1105, Department of Medicine, Amiens University Hospital, 80000 Amiens, France

2. 

Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80039 Amiens, France

3. 

Laboratoire de Mathématiques Appliquées de Compiègne, Sorbonne Université, Université de Technologie de Compiègne, 60205 Compiègne, France

4. 

Laboratoire de Mathématiques de Reims, EA4535, Université de Reims Champagne-Ardenne, 51687 Reims cedex 2, France

* Corresponding authorr: marion.darbas@u-picardie.fr.

Received  July 20, 2017 Accepted  October 13, 2017 Published  March 2018

The paper is devoted to the analysis of electroencephalography (EEG) in neonates. The goal is to investigate the impact of fontanels on EEG measurements, i.e. on the values of the electric potential on the scalp. In order to answer this clinical issue, a complete mathematical study (modeling, existence and uniqueness result, realistic simulations) is carried out. A model for the forward problem in EEG source localization is proposed. The model is able to take into account the presence and ossification process of fontanels which are characterized by a variable conductivity. From a mathematical point of view, the model consists in solving an elliptic problem with a singular source term in an inhomogeneous medium. A subtraction approach is used to deal with the singularity in the source term, and existence and uniqueness results are proved for the continuous problem. Discretization is performed with 3D Finite Elements of type P1 and error estimates are proved in the energy norm ($H^1$-norm). Numerical simulations for a three-layer spherical model as well as for a realistic neonatal head model including or not the fontanels have been obtained and corroborate the theoretical results. A mathematical tool related to the concept of Gâteau derivatives is introduced which is able to measure the sensitivity of the electric potential with respect to small variations in the fontanel conductivity. This study attests that the presence of fontanels in neonates does have an impact on EEG measurements.

Citation: Hamed Azizollahi, Marion Darbas, Mohamadou M. Diallo, Abdellatif El Badia, Stephanie Lohrengel. EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity. Mathematical Biosciences & Engineering, 2018, 15 (4) : 905-932. doi: 10.3934/mbe.2018041
References:
[1]

Z. Akalin Acar and S. Makeig, Effects of Forward Model Errors on EEG Source Localization, Brain Topogrography, 26 (2013), 378-396.

[2]

A. Alonso-RodriguezJ. CamanoR. Rodriguez and A. Valli, Assessment of two approximation methods for the inverse problem of electroencephalography, Int. J. of Numerical Analysis and Modeling, 13 (2016), 587-609.

[3]

H. AzizollahiA. Aarabi and F. Wallois, Effects of uncertainty in head tissue conductivity and complexity on EEG forward modeling in neonates, Hum. Brain Ma, 37 (2016), 3604-3622. doi: 10.1002/hbm.23263.

[4]

H. T. BanksD. Rubio and N. Saintier, Optimal design for parameter estimation in EEG problems in a 3D multilayered domain, Mathematical Biosciences and Engineering, 12 (2015), 739-760. doi: 10.3934/mbe.2015.12.739.

[5]

M. BauerS. PursiainenJ. VorwerkH. Köstler and C. H. Wolters, Comparison Study for Whitney (Raviart-Thomas)-Type Source Models in Finite-Element-Method-Based EEG Forward Modeling, IEEE Trans. Biomed. Eng., 62 (2015), 2648-2656. doi: 10.1109/TBME.2015.2439282.

[6]

J. Borggaard and V. L. Nunes, Fréchet Sensitivity Analysis for Partial Differential Equations with Distributed Parameters, American Control Conference, San Francisco, 2011.

[7]

H. Brezis, Functional Analysis, Sobolev Spaces And Partial Differential Equations, Universitext. Springer, New York, 2011.

[8]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, New York, 1978.

[9]

M. Clerc and J. Kybic, Cortical mapping by Laplace-Cauchy transmission using a boundary element method, Journal on Inverse Problems, 23 (2007), 2589-2601. doi: 10.1088/0266-5611/23/6/020.

[10]

M. Clerc, J. Leblond, J. -P. Marmorat and T. Papadopoulo, Source localization using rational approximation on plane sections, Inverse Problems, 28 (2012), 055018, 24 pp.

[11]

M. Darbas, M. M. Diallo, A. El Badia and S. Lohrengel, An inverse dipole source problem in inhomogeneous media: application to the EEG source localization in neonates, in preparation.

[12]

A. El Badia and T. Ha Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663. doi: 10.1088/0266-5611/16/3/308.

[13]

A. El Badia and M. Farah, Identification of dipole sources in an elliptic equation from boundary measurements, J. Inv. Ill-Posed Problems, 14 (2006), 331-353. doi: 10.1515/156939406777571012.

[14]

A. El Badia and M. Farah, A stable recovering of dipole sources from partial boundary measurements, Inverse Problems, 26 (2010), 115006, 24pp.

[15]

Q. Fang and D. A. Boas, Tetrahedral mesh generation from volumetric binary and grayscale images, EEE International Symposium on Biomedical Imaging: From Nano to Macro, (2009), SBI? 09. Boston, Massachusetts, USA, 1142–1145.

[16]

M. Farah, Problémes Inverses de Sources et Lien avec l'Electro-encéphalo-graphie, Thése de doctorat, Université de Technologie de Compiégne, 2007.

[17]

O. Faugeras, F. Clément, R. Deriche, R. Keriven, T. Papadopoulo, J. Roberts, T. Viéville, F. Devernay, J. Gomes, G. Hermosillo, P. Kornprobst and D. Lingrand, The Inverse EEG and MEG Problems: The Adjoint State Approach I: The Continuous Case ,Rapport de recherche, 1999.

[18]

P. GargiuloP. BelfioreE. A. FriogeirssonS. Vanhalato and C. Ramon, The effect of fontanel on scalp EEG potentials in the neonate, Clin. Neurophysiol, 126 (2015), 1703-1710. doi: 10.1016/j.clinph.2014.12.002.

[19]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order Springer, Berlin, 1977.

[20]

R. Grech, T. Cassar, J. Muscat, K. P. Camilleri, S. G. Fabri, M. Zervakis, P. Xanthopoulos, V. Sakkalis and B. Vanrumste, Review on solving the inverse problem in EEG source analysis J. NeuorEng. Rehabil. , 5 (2008). doi: 10.1186/1743-0003-5-25.

[21]

H. Hallez, B. Vanrumste, R. Grech, J. Muscat, W. De Clercq, A. Vergult, Y. d'Asseler, K. P. Camilleri, S. G. Fabri, S. Van Huffel and I. Lemahieu, Review on solving the forward problem in EEG source analysis J. NeuorEng. Rehabil., 4 (2007). doi: 10.1186/1743-0003-4-46.

[22]

M. HämäläinenR. HariJ. IlmoniemiJ. Knuutila and O. V. Lounasmaa, Magnetoencephalography-theory, instrumentation, and applications to noninvasive studies of the working human brain, Rev. Mod. Phys., 65 (1993), 413-497.

[23]

F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuka, FreeFem++ Manual, 2014.

[24]

E. Hernández and R. Rodríguez, Finite element Approximation of Spectral Problems with Neumann Boundary Conditions on curved domains, Math. Comp., 72 (2002), 1099-1115.

[25]

R. Kress, Linear Integral Equations, Second edition, Applied Mathematical Sciences 82, Spinger-Verlag, 1999.

[26]

J. KybicM. ClercT. AbboudO. FaugerasR. Keriven and T. Papadopoulo, A common formalism for the integral formulations of the forward EEG Problem, IEEE Transactions on Medical Imaging, 24 (2005), 12-28. doi: 10.1109/TMI.2004.837363.

[27]

J. KybicM. ClercO. FaugerasR. Keriven and T. Papadopoulo, Fast multipole acceleration of the MEG/EEG boundary element method, Physics in Medicine and Biology, 50 (2005), 4695-4710. doi: 10.1088/0031-9155/50/19/018.

[28]

J. KybicM. ClercT. AbboudO. FaugerasR. Keriven and T. Papadopoulo, Generalized head models for MEG/EEG: Boundary element method beyond nested volumes, Phys. Med. Biol., 51 (2006), 1333-1346. doi: 10.1088/0031-9155/51/5/021.

[29]

J. Leblond, Identifiability properties for inverse problems in EEG data processing and medical engineering, with observability and optimization issues, Acta Applicandae Mathematicae, 135 (2015), 175-190. doi: 10.1007/s10440-014-9951-7.

[30]

S. LewD. D. SilvaM. ChoeP. Ellen GrantY. OkadaC. H. Wolters and M. S. Hämäläinen, Effects of sutures and fontanels on MEG and EEG source analysis in a realistic infant head model, Neuroimage, 76 (2013), 282-293. doi: 10.1016/j.neuroimage.2013.03.017.

[31]

T. MedaniD. LautruD. SchwartzZ. Ren and G. Sou, FEM method for the EEG forward problem and improvement based on modification of the saint venant's method, Progress In Electromagnetic Research, 153 (2015), 11-22.

[32]

J. C. de Munck and M. J. Peters, A fast method to compute the potential in the multisphere model, IEEE Trans. Biomed. Eng., 40 (1993), 1166-1174.

[33]

OdabaeeA. TokarievS. LayeghyM. MesbahP. B. ColditzC. Ramon and S. Vanhatalo, Neonatal EEG at scalp is focal and implies high skull conductivity in realistic neonatal head models, NeuroImage, 96 (2014), 73-80.

[34]

P. A. Raviart and J. M. Thomas, Introduction á l'Analyse Numérique des Equations aux Dérivées Partielles, Masson, Paris, 1983.

[35]

N. Roche-LabarbeA. AarabiG. KongoloC. Gondry-JouetM. DümpelmannR. Grebe and F. Wallois, High-resolution electroencephalography and source localization in neonates, Human Brain Mapping, 29 (2008), 167-76. doi: 10.1002/hbm.20376.

[36]

C. RordenL. BonilhaJ. FridrikssonB. Bender and H. O. Karnath, Age-specific CT and MRI templates for spatial normalization, NeuroImage, 61 (2012), 957-965. doi: 10.1016/j.neuroimage.2012.03.020.

[37]

M. Schneider, A multistage process for computing virtual dipole sources of EEG discharges from surface information, IEEE Trans. on Biomed. Eng., 19, 1-19.

[38]

M. I. Troparevsky, D. Rubio and N. Saintier, Sensitivity analysis for the EEG forward problem Frontiers in Computational Neuroscience, 4 (2010), p138. doi: 10.3389/fncom.2010.00138.

[39]

J. VorwerkJ. H. ChoS. RamppH. HamerT. T. Knösche and C. H. Wolters, A guideline for head volume conductor modeling in EEG and MEG, NeuroImage, 100 (2014), 590-607. doi: 10.1016/j.neuroimage.2014.06.040.

[40]

C. H. WoltersH. KöstlerC. MöllerJ. HärdtleinL. Grasedyck and W. Hackbusch, Numerical mathematics of the subtraction approach for the modeling of a current dipole in EEG source reconstruction using finite element head models, SIAM J. Sci. Comput., 30 (2007), 24-45.

[41]

C. H. WoltersH. KöstlerC. MöllerJ. Härdtlein and A. Anwander, Numerical approaches for dipole modeling in finite element method based source analysis, Int. Congress Ser., 1300 (2007), 189-192. doi: 10.1016/j.ics.2007.02.014.

[42]

Z. Zhang, A fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres, Phys. Med. Biol., 40 (1995), 335-349. doi: 10.1088/0031-9155/40/3/001.

show all references

References:
[1]

Z. Akalin Acar and S. Makeig, Effects of Forward Model Errors on EEG Source Localization, Brain Topogrography, 26 (2013), 378-396.

[2]

A. Alonso-RodriguezJ. CamanoR. Rodriguez and A. Valli, Assessment of two approximation methods for the inverse problem of electroencephalography, Int. J. of Numerical Analysis and Modeling, 13 (2016), 587-609.

[3]

H. AzizollahiA. Aarabi and F. Wallois, Effects of uncertainty in head tissue conductivity and complexity on EEG forward modeling in neonates, Hum. Brain Ma, 37 (2016), 3604-3622. doi: 10.1002/hbm.23263.

[4]

H. T. BanksD. Rubio and N. Saintier, Optimal design for parameter estimation in EEG problems in a 3D multilayered domain, Mathematical Biosciences and Engineering, 12 (2015), 739-760. doi: 10.3934/mbe.2015.12.739.

[5]

M. BauerS. PursiainenJ. VorwerkH. Köstler and C. H. Wolters, Comparison Study for Whitney (Raviart-Thomas)-Type Source Models in Finite-Element-Method-Based EEG Forward Modeling, IEEE Trans. Biomed. Eng., 62 (2015), 2648-2656. doi: 10.1109/TBME.2015.2439282.

[6]

J. Borggaard and V. L. Nunes, Fréchet Sensitivity Analysis for Partial Differential Equations with Distributed Parameters, American Control Conference, San Francisco, 2011.

[7]

H. Brezis, Functional Analysis, Sobolev Spaces And Partial Differential Equations, Universitext. Springer, New York, 2011.

[8]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, New York, 1978.

[9]

M. Clerc and J. Kybic, Cortical mapping by Laplace-Cauchy transmission using a boundary element method, Journal on Inverse Problems, 23 (2007), 2589-2601. doi: 10.1088/0266-5611/23/6/020.

[10]

M. Clerc, J. Leblond, J. -P. Marmorat and T. Papadopoulo, Source localization using rational approximation on plane sections, Inverse Problems, 28 (2012), 055018, 24 pp.

[11]

M. Darbas, M. M. Diallo, A. El Badia and S. Lohrengel, An inverse dipole source problem in inhomogeneous media: application to the EEG source localization in neonates, in preparation.

[12]

A. El Badia and T. Ha Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663. doi: 10.1088/0266-5611/16/3/308.

[13]

A. El Badia and M. Farah, Identification of dipole sources in an elliptic equation from boundary measurements, J. Inv. Ill-Posed Problems, 14 (2006), 331-353. doi: 10.1515/156939406777571012.

[14]

A. El Badia and M. Farah, A stable recovering of dipole sources from partial boundary measurements, Inverse Problems, 26 (2010), 115006, 24pp.

[15]

Q. Fang and D. A. Boas, Tetrahedral mesh generation from volumetric binary and grayscale images, EEE International Symposium on Biomedical Imaging: From Nano to Macro, (2009), SBI? 09. Boston, Massachusetts, USA, 1142–1145.

[16]

M. Farah, Problémes Inverses de Sources et Lien avec l'Electro-encéphalo-graphie, Thése de doctorat, Université de Technologie de Compiégne, 2007.

[17]

O. Faugeras, F. Clément, R. Deriche, R. Keriven, T. Papadopoulo, J. Roberts, T. Viéville, F. Devernay, J. Gomes, G. Hermosillo, P. Kornprobst and D. Lingrand, The Inverse EEG and MEG Problems: The Adjoint State Approach I: The Continuous Case ,Rapport de recherche, 1999.

[18]

P. GargiuloP. BelfioreE. A. FriogeirssonS. Vanhalato and C. Ramon, The effect of fontanel on scalp EEG potentials in the neonate, Clin. Neurophysiol, 126 (2015), 1703-1710. doi: 10.1016/j.clinph.2014.12.002.

[19]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order Springer, Berlin, 1977.

[20]

R. Grech, T. Cassar, J. Muscat, K. P. Camilleri, S. G. Fabri, M. Zervakis, P. Xanthopoulos, V. Sakkalis and B. Vanrumste, Review on solving the inverse problem in EEG source analysis J. NeuorEng. Rehabil. , 5 (2008). doi: 10.1186/1743-0003-5-25.

[21]

H. Hallez, B. Vanrumste, R. Grech, J. Muscat, W. De Clercq, A. Vergult, Y. d'Asseler, K. P. Camilleri, S. G. Fabri, S. Van Huffel and I. Lemahieu, Review on solving the forward problem in EEG source analysis J. NeuorEng. Rehabil., 4 (2007). doi: 10.1186/1743-0003-4-46.

[22]

M. HämäläinenR. HariJ. IlmoniemiJ. Knuutila and O. V. Lounasmaa, Magnetoencephalography-theory, instrumentation, and applications to noninvasive studies of the working human brain, Rev. Mod. Phys., 65 (1993), 413-497.

[23]

F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuka, FreeFem++ Manual, 2014.

[24]

E. Hernández and R. Rodríguez, Finite element Approximation of Spectral Problems with Neumann Boundary Conditions on curved domains, Math. Comp., 72 (2002), 1099-1115.

[25]

R. Kress, Linear Integral Equations, Second edition, Applied Mathematical Sciences 82, Spinger-Verlag, 1999.

[26]

J. KybicM. ClercT. AbboudO. FaugerasR. Keriven and T. Papadopoulo, A common formalism for the integral formulations of the forward EEG Problem, IEEE Transactions on Medical Imaging, 24 (2005), 12-28. doi: 10.1109/TMI.2004.837363.

[27]

J. KybicM. ClercO. FaugerasR. Keriven and T. Papadopoulo, Fast multipole acceleration of the MEG/EEG boundary element method, Physics in Medicine and Biology, 50 (2005), 4695-4710. doi: 10.1088/0031-9155/50/19/018.

[28]

J. KybicM. ClercT. AbboudO. FaugerasR. Keriven and T. Papadopoulo, Generalized head models for MEG/EEG: Boundary element method beyond nested volumes, Phys. Med. Biol., 51 (2006), 1333-1346. doi: 10.1088/0031-9155/51/5/021.

[29]

J. Leblond, Identifiability properties for inverse problems in EEG data processing and medical engineering, with observability and optimization issues, Acta Applicandae Mathematicae, 135 (2015), 175-190. doi: 10.1007/s10440-014-9951-7.

[30]

S. LewD. D. SilvaM. ChoeP. Ellen GrantY. OkadaC. H. Wolters and M. S. Hämäläinen, Effects of sutures and fontanels on MEG and EEG source analysis in a realistic infant head model, Neuroimage, 76 (2013), 282-293. doi: 10.1016/j.neuroimage.2013.03.017.

[31]

T. MedaniD. LautruD. SchwartzZ. Ren and G. Sou, FEM method for the EEG forward problem and improvement based on modification of the saint venant's method, Progress In Electromagnetic Research, 153 (2015), 11-22.

[32]

J. C. de Munck and M. J. Peters, A fast method to compute the potential in the multisphere model, IEEE Trans. Biomed. Eng., 40 (1993), 1166-1174.

[33]

OdabaeeA. TokarievS. LayeghyM. MesbahP. B. ColditzC. Ramon and S. Vanhatalo, Neonatal EEG at scalp is focal and implies high skull conductivity in realistic neonatal head models, NeuroImage, 96 (2014), 73-80.

[34]

P. A. Raviart and J. M. Thomas, Introduction á l'Analyse Numérique des Equations aux Dérivées Partielles, Masson, Paris, 1983.

[35]

N. Roche-LabarbeA. AarabiG. KongoloC. Gondry-JouetM. DümpelmannR. Grebe and F. Wallois, High-resolution electroencephalography and source localization in neonates, Human Brain Mapping, 29 (2008), 167-76. doi: 10.1002/hbm.20376.

[36]

C. RordenL. BonilhaJ. FridrikssonB. Bender and H. O. Karnath, Age-specific CT and MRI templates for spatial normalization, NeuroImage, 61 (2012), 957-965. doi: 10.1016/j.neuroimage.2012.03.020.

[37]

M. Schneider, A multistage process for computing virtual dipole sources of EEG discharges from surface information, IEEE Trans. on Biomed. Eng., 19, 1-19.

[38]

M. I. Troparevsky, D. Rubio and N. Saintier, Sensitivity analysis for the EEG forward problem Frontiers in Computational Neuroscience, 4 (2010), p138. doi: 10.3389/fncom.2010.00138.

[39]

J. VorwerkJ. H. ChoS. RamppH. HamerT. T. Knösche and C. H. Wolters, A guideline for head volume conductor modeling in EEG and MEG, NeuroImage, 100 (2014), 590-607. doi: 10.1016/j.neuroimage.2014.06.040.

[40]

C. H. WoltersH. KöstlerC. MöllerJ. HärdtleinL. Grasedyck and W. Hackbusch, Numerical mathematics of the subtraction approach for the modeling of a current dipole in EEG source reconstruction using finite element head models, SIAM J. Sci. Comput., 30 (2007), 24-45.

[41]

C. H. WoltersH. KöstlerC. MöllerJ. Härdtlein and A. Anwander, Numerical approaches for dipole modeling in finite element method based source analysis, Int. Congress Ser., 1300 (2007), 189-192. doi: 10.1016/j.ics.2007.02.014.

[42]

Z. Zhang, A fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres, Phys. Med. Biol., 40 (1995), 335-349. doi: 10.1088/0031-9155/40/3/001.

Figure 1.1.  Fontanels and skull of a neonate.
Figure 2.1.  Three-layer head model.
Figure 5.1.  Behavior of factors RDM and MAG with respect to the eccentricity of the dipole. Different mesh sizes (finest mesh $M_3$). Neonatal three-layer spherical head model without fontanels. Exact reference solution.
Figure 5.2.  A spherical head model with the main fontanel.
Figure 5.3.  Errors in $H^1$-norm with respect to the mesh size $h$ in logarithm scale. Three-layer spherical head model with the anterior fontanel (Gaussian behavior for the fontanel conductivity). Numerical reference solution computed on $M_{\tiny{\mbox{ref}}}$. Left: one single source $S = (0, 0, 40mm)$, $\mathbf{q} = (0, 0, J)$. Right: two sources $S_1 = (0, 0, 10mm)$, $S_2 = (0, 10mm, 0)$ with moments $\mathbf{q}_1 = (0, 0, J)$, $\mathbf{q}_2 = (0, J, 0)$. Intensity $J = 10^{-6} A.m^{-2}$.
Figure 5.4.  Behavior of factors RDM and MAG with respect to the eccentricity dipole position for different meshes. Three-layer spherical model with the anterior fontanel (Gaussian behavior for the fontanel conductivity). Numerical reference solution computed on $M_{\tiny{\mbox{ref}}}$.
Figure 6.1.  Realistic head model of a neonate. Left: skull and fontanels. Right: mesh of the fontanels.
Figure 6.2.  The coronal, sagittal and axial plane of the head model and its 3D reconstruction.
Figure 6.3.  Variations of factors RDM and MAG with respect to different conductivities $(\sigma_{\!f}, \sigma_{skull})$. Four-layer realistic head model. Reference solution computed with the model without fontanels.
Figure 6.4.  Sensitivity of the electric potential on the scalp with respect to eccentricity. Distance source-interface brain/CSF $\approx 5$mm (left) and $\approx 15$mm (right).
Figure 6.5.  Sensitivity of the electric potential on the scalp with respect to orientation. Distance source-interface brain/CSF $\approx 15$mm. Left: moment $\mathbf{q} = (0, J, J)$. Right: moment $\mathbf{q} = (J, J, 0)$.
Figure 6.6.  Sensitivity of the electric potential on the scalp for a deep source.
Table 1.  Definition of meshes (neonatal three-layer spherical head model).
Mesh Nodes Tetrahedra Boundary nodes $h_{min}$ [m] $h_{max}$ [m]
$M_1$ $102 540$ $594 907$ $16 936$ $8.16 10^{-4}$ $4.81 10^{-3}$
$M_2$ $302 140$ $1\ 855 005$ $23 339$ $6.35 10^{-4}$ $3.07 10^{-3}$
$M_3$ $596 197$ $3 632 996$ $54 290$ $4.1 10^{-4}$ $2.46 10^{-3}$
$M_{\rm ref}$ $2 754 393$ $17 263 316$ $124 847$ $2.5 10^{-4}$ $1.51 10^{-3}$
Mesh Nodes Tetrahedra Boundary nodes $h_{min}$ [m] $h_{max}$ [m]
$M_1$ $102 540$ $594 907$ $16 936$ $8.16 10^{-4}$ $4.81 10^{-3}$
$M_2$ $302 140$ $1\ 855 005$ $23 339$ $6.35 10^{-4}$ $3.07 10^{-3}$
$M_3$ $596 197$ $3 632 996$ $54 290$ $4.1 10^{-4}$ $2.46 10^{-3}$
$M_{\rm ref}$ $2 754 393$ $17 263 316$ $124 847$ $2.5 10^{-4}$ $1.51 10^{-3}$
Table 2.  Four-layer realistic head model
Mesh Nodes Tetrahedra Boundary faces $h_{min}$ [m] $h_{max}$ [m]
$M_{real}$ 108 669 590 878 55 660 $3.4\ 10^{-4}$ $14\ 10^{-3}$
Mesh Nodes Tetrahedra Boundary faces $h_{min}$ [m] $h_{max}$ [m]
$M_{real}$ 108 669 590 878 55 660 $3.4\ 10^{-4}$ $14\ 10^{-3}$
[1]

Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051

[2]

Edward Della Torre, Lawrence H. Bennett. Analysis and simulations of magnetic materials. Conference Publications, 2005, 2005 (Special) : 854-861. doi: 10.3934/proc.2005.2005.854

[3]

Peter Monk, Jiguang Sun. Inverse scattering using finite elements and gap reciprocity. Inverse Problems & Imaging, 2007, 1 (4) : 643-660. doi: 10.3934/ipi.2007.1.643

[4]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631

[5]

Eric Dubach, Robert Luce, Jean-Marie Thomas. Pseudo-Conform Polynomial Lagrange Finite Elements on Quadrilaterals and Hexahedra. Communications on Pure & Applied Analysis, 2009, 8 (1) : 237-254. doi: 10.3934/cpaa.2009.8.237

[6]

Zhangxin Chen, Qiaoyuan Jiang, Yanli Cui. Locking-free nonconforming finite elements for planar linear elasticity. Conference Publications, 2005, 2005 (Special) : 181-189. doi: 10.3934/proc.2005.2005.181

[7]

Philip Gerlee, Alexander R. A. Anderson. Diffusion-limited tumour growth: Simulations and analysis. Mathematical Biosciences & Engineering, 2010, 7 (2) : 385-400. doi: 10.3934/mbe.2010.7.385

[8]

Jaroslaw Smieja, Malgorzata Kardynska, Arkadiusz Jamroz. The meaning of sensitivity functions in signaling pathways analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2697-2707. doi: 10.3934/dcdsb.2014.19.2697

[9]

Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241

[10]

Daniel Peterseim. Robustness of finite element simulations in densely packed random particle composites. Networks & Heterogeneous Media, 2012, 7 (1) : 113-126. doi: 10.3934/nhm.2012.7.113

[11]

P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692

[12]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[13]

Martin Burger, Peter Alexander Markowich, Jan-Frederik Pietschmann. Continuous limit of a crowd motion and herding model: Analysis and numerical simulations. Kinetic & Related Models, 2011, 4 (4) : 1025-1047. doi: 10.3934/krm.2011.4.1025

[14]

Marcin Studniarski. Finding all minimal elements of a finite partially ordered set by genetic algorithm with a prescribed probability. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 389-398. doi: 10.3934/naco.2011.1.389

[15]

Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216

[16]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[17]

Lorena Bociu, Jean-Paul Zolésio. Sensitivity analysis for a free boundary fluid-elasticity interaction. Evolution Equations & Control Theory, 2013, 2 (1) : 55-79. doi: 10.3934/eect.2013.2.55

[18]

Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381

[19]

Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial & Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1

[20]

Krzysztof Fujarewicz, Krzysztof Łakomiec. Parameter estimation of systems with delays via structural sensitivity analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2521-2533. doi: 10.3934/dcdsb.2014.19.2521

2017 Impact Factor: 1.23

Metrics

  • PDF downloads (47)
  • HTML views (315)
  • Cited by (0)

[Back to Top]