IEEE TRANSACTIONS ON MEDICAL IMAGING 1 A Mixed Finite … · 2017-08-25 · IEEE TRANSACTIONS ON MEDICAL IMAGING 1 A Mixed Finite Element Method to Solve the EEG Forward Problem J

IEEE TRANSACTIONS ON MEDICAL IMAGING 1

A Mixed Finite Element Methodto Solve the EEG Forward Problem

J Vorwerk C Engwer S Pursiainen and CH Wolters

copy2016 IEEE Personal use of this material is permitted Permission from IEEE must be obtained for all other uses in any current or future media includingreprintingrepublishing this material for advertising or promotional purposes creating new collective works for resale or redistribution to servers or lists orreuse of any copyrighted component of this work in other works DOI 101109TMI20162624634

AbstractmdashFinite element methods have been shown to achievehigh accuracies in numerically solving the EEG forward problemand they enable the realistic modeling of complex geometries andimportant conductive features such as anisotropic conductivitiesTo date most of the presented approaches rely on the sameunderlying formulation the continuous Galerkin (CG)-FEMIn this article a novel approach to solve the EEG forwardproblem based on a mixed finite element method (Mixed-FEM)is introduced To obtain the Mixed-FEM formulation the electriccurrent is introduced as an additional unknown besides theelectric potential As a consequence of this derivation the Mixed-FEM is by construction current preserving in contrast to theCG-FEM Consequently a higher simulation accuracy can beachieved in certain scenarios eg when the diameter of thininsulating structures such as the skull is in the range of themesh resolution

A theoretical derivation of the Mixed-FEM approach for EEGforward simulations is presented and the algorithms imple-mented for solving the resulting equation systems are describedSubsequently first evaluations in both sphere and realistic headmodels are presented and the results are compared to previouslyintroduced CG-FEM approaches Additional visualizations areshown to illustrate the current preserving property of the Mixed-FEM

Based on these results it is concluded that the newly presentedMixed-FEM can at least complement and in some scenarioseven outperform the established CG-FEM approaches whichmotivates a further evaluation of the Mixed-FEM for applicationsin bioelectromagnetism

Index TermsmdashEEG forward problem source analysis mixedfinite element method realistic head modeling

Copyright (c) 2016 IEEE Personal use of this material is permittedHowever permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissionsieeeorg

Asterisk indicates corresponding authorJV is with the Institute for Biomagnetism and Biosignalanalysis University

of Munster Germany and the Scientific Computing and Imaging (SCI) Insti-tute University of Utah Salt Lake City USA e-mail jvorwerksciutahedu

CE is with the Institute for Applied Mathematics University of MunsterGermany and the Cluster of Excellence EXC 1003 Cells in Motion CiMMunster Germany

SP is with the Department of Mathematics Tampere University of Tech-nology Finland and the Department of Mathematics and System AnalysisAalto University Helsinki Finland

CHW is with the Institute for Biomagnetism and BiosignalanalysisUniversity of Munster Germany

JV and CHW were supported by the Priority Program 1665 of theDeutsche Forschungsgemeinschaft (DFG) (WO14255-2 WO14257-1) andthe EU project ChildBrain (Marie Curie Innovative Training Networks grantagreement no 641652) CE was supported by the Cluster of Excellence 1003of the Deutsche Forschungsgemeinschaft (DFG EXC 1003 Cells in Motion)SP was supported by the Academy of Finland (Centre of Excellence inInverse Problems Research and Key Project number 305055)

I INTRODUCTION

THE EEG forward problem is to simulate the electricpotential on the head surface that is generated by a

minimal patch of active brain tissue Its accurate solution isfundamental for precise EEG source analysis An accuratesolution can be achieved via numerical methods that allow totake the realistic head geometry into account In this contextfinite element methods (FEM) achieve high numerical accura-cies and enable to realistically model tissue boundaries withcomplicated shapes such as the gray matterCSF interface andto incorporate tissue conductivity anisotropy The importanceof incorporating these model features for the computationof accurate forward solutions and in consequence also forprecise source analysis has been shown in multiple studies[1]ndash[3]

Different FEM approaches to solve the EEG forward prob-lem have been proposed eg St Venant partial integrationWhitney or subtraction approaches [4]ndash[9] These approachesdiffer in the way the dipole source is modeled but theunderlying discretization of the continuous partial differentialequation (PDE) is the same a conforming Galerkin-FEM (CG-FEM) with most often linear Ansatz-functions The necessarydiscretization of the head volume can be achieved using eithertetrahedral or hexahedral head models Hexahedral modelshave the advantage that they can be directly generated fromvoxel-based magnetic resonance images (MRI) whereas thegeneration of surface-based tetrahedral meshes can be com-plicated Therefore hexahedral meshes are more and morefrequently used in praxis [10] [11] and have furthermorerecently been positively validated in an animal study [12]

In this article a mixed finite element method (Mixed-FEM) to solve the EEG forward problem is introducedCompared to the CG-FEM it has the advantage that thecurrent source can be represented in a direct way whereaseither an approximation using electrical monopoles has to bederived or the subtraction approach has to be applied whenusing the CG-FEM Furthermore the Mixed-FEM is currentpreserving and thereby prevents the effects of the (local)current leakages through the skull that might occur for the CG-FEM [13] [14] Mixed- and CG-FEM are compared in sucha leakage scenario in Section IV-C An accurate simulation ofthe currents penetrating the skull is important as the influenceof an accurate representation of the skull for accurate forwardsimulations has been shown [15]ndash[17] The accuracy of theMixed-FEM in comparison to CG-FEM approaches and arecently presented approach based on a discontinuous Galerkin(DG) FEM formulation [14] is evaluated in sphere and realistic

head models It is shown that the Mixed-FEM achieves higheraccuracies in solving the EEG forward problem than the CG-FEM for highly eccentric sources in sphere models and thanboth CG- and DG-FEM in realistic head models

II THEORY

APPLYING the quasistatic approximation of Maxwellrsquosequations [18] [19] the forward problem of EEG is com-

monly formulated as a second-order PDE with homogeneousNeumann boundary condition

nabla middot (σnablau) = nabla middot jp in Ω (1a)σpartnu = 0 on partΩ = Γ (1b)

Here u denotes the electric potential jp the source currentand σ the conductivity distribution in Ω In (1) the electriccurrent j is already eliminated as an unknown For our purposewe start at the previous step in the derivation of the quasistaticapproximation and keep the electric current as an unknownThus our starting point is the system of first-order PDEs

j + σnablau = jp (2a)nabla middot j = 0 in Ω (2b)〈jn〉 = 〈jpn〉 on partΩ = Γ (2c)

Since the source current jp in general fulfills 〈jpn〉 = 0 onΓ as supp jp sub Ω for physiological reasons (there are nosources in the skin) (2c) can be simplified to 〈jn〉 = 0 on ΓThe Mixed-FEM formulation for the EEG forward problem isnow derived from (2) instead of discretizing (1) as would bedone for the CG-FEM

A A (Mixed) Weak Formulation of the EEG Forward Problem

Due to the vector-valued equation (2a) it is necessary tointroduce a space of vector-valued test functions to be able toderive a weak formulation of (2) A natural function space forthe current in the mixed formulation is H(div Ω)

H(div Ω) =q isin L2(Ω)3 nabla middot q isin L2(Ω)

Akin to the scalar-valued Sobolev spaces Hk(Ω) this spacebecomes a Hilbert space with the norm

qH(divΩ) =(q2L2(Ω)3 + nabla middot q2L2(Ω)

We introduce a subspace H0(divΩ) of H(div Ω) in whichthe boundary condition 〈jn〉 = 0 on partΩ = Γ is fulfilled byconstruction

H0(divΩ) = q isin H(div Ω) 〈qn〉 = 0 on partΩ (5)

For the scalar-valued equation (2b) one can simply choose thespace of square-integrable functions L2(Ω) as the test space

Now we can introduce a weak formulation of (2)intΩ

〈σminus1jq〉dx minusint

nabla middot qudx =intΩ

〈σminus1jpq〉dx for all q isin H0(divΩ) (6a)

nabla middot jv dx = 0 for all v isin L2(Ω) (6b)

This is the so-called dual mixed formulation [20]ndash[22] TheNeumann boundary condition (2c)(5) is an essential boundarycondition in the (dual) mixed formulation and has to beimposed explicitly in solving the discrete problem We definethe bilinear forms

a(pq) = (σminus1pq)L2(Ω)3 (7a)b(p v) = (nabla middot p v)L2(Ω) (7b)

and the functional

l(q) = (σminus1jpq)L2(Ω)3 (7c)

for pq isin H0(divΩ) v isin L2(Ω) jp isin L2(Ω)3 σ isin Linfin(Ω)σ gt 0 Therefore to solve (6) is to

find (u j) isin L2(Ω)timesH0(divΩ) such that

a(jq) + b(q u) = l(q) for all q isin H0(divΩ) (8a)

b(j v) = 0 for all v isin L2(Ω) (8b)

In this notation the saddle point structure of problem (8) andthus also (6) is recognizable As a consequence the existenceand uniqueness of a solution cannot be shown using theLemma of Lax-Milgram

Instead it can be shown that a solution to (8) exists ifthe operator a is H0(divΩ)-elliptic on the kernel of b andb fulfills an inf-sup condition which in this case is alsocalled the LBB condition named after the mathematiciansLadyzhenskaya Babuska and Brezzi At this point we shallnote only that these conditions are fulfilled by a and bdefined as in (7) and thereby the existence of a solution(u j) isin L2(Ω) times H0(divΩ) is given While the solutionfor j is unique u is defined up to an element of ker b(j v)v isin L2(Ω) [20] [23] Uniqueness for u can be obtained byintroducing an additional condition such as fixing the valueofint

Ωu or

intpartΩu For a detailed proof and discussion we refer

the reader to eg [20] [22] [24]

B Mixed Finite Element Method

Obtaining a numerical solution for (6)(8) necessitateschoosing suitable discrete approximations for the test functionspaces H0(divΩ) and L2(Ω) Utilizing a Galerkin approachthese are also the spaces in which the discrete solution (uh jh)lies

In order to construct the discrete subspaces the volume Ωis subdivided and approximated by a set of simple geometricalobjects In three dimensions these objects are usually tetrahe-dra or hexahedra For the sake of simplicity any subdivisionof Ω into either tetrahedra or hexahedra is henceforth referredto as triangulation T = T1 T2 T3 Tm In this paper wefollow the definition of a triangulation according to [24] Wefurther assume that the triangulation T is admissible [24] andwrite Th instead of T if each element T isin T has a diameterof maximally 2h

We can now choose the space P0 of piecewise constantfunctions on each element as a discrete subspace of L2(Ω)

P0(Th) =v isin L2(Ω) v|T equiv cT cT isin R for all T isin Th

A basis of this space is given by the set of characteristicfunctions 1T isin L2(Ω) for each element T isin Th We denotethis set of P0 basis functions by SP0

h = 1T T isin ThFor H0(divΩ) we start by defining the space RT0 of

the lowest-order Raviart-Thomas elements on a single regularhexahedron T [25] [26]

RT0(T ) =P100(T )times P010(T )times P001(T )

where Pijk(T ) denotes the set of polynomial functionsdefined on T of degrees i j and k in x1 x2 and x3 Weexpand this definition to a discrete subspace of H(div Ω)

RT0(Th) =q isin L2(Ω)3 q|T isin RT0(T ) and [q]partT = 0

for all T isin Th

= q isin H(div Ω) q|T isin RT0(T ) for all T isin Th

[middot]γ indicates the jump of the normal component at aboundary γ

Using Fortinrsquos criterion [22] [24] it can be shown thatthe existence and uniqueness of a solution to (8) ndash as notedin Section II-A ndash are conserved when replacing L2(Ω) andH0(divΩ) by their discrete approximations P0 and RT0 Fordetails we refer the reader to [22]

A basis of the space RT0 can be defined for both tetrahedraland hexahedral elements We explicitly note only the hexahe-dral case which is also used in the numerical evaluations For

Fig 1 Zeroth-order Raviart-Thomas basis function supported on two hexa-hedra Q1 and Q2

a regular hexahedral mesh with edge length h a RT0 basisfunction wk is supported on the two hexahedra Q1 Q2 isin Thsharing the face fk = Q1 cap Q2 with normal vector nk andcentroid xk It can be defined via

wk(x) =

1minus |〈xminus xknk〉|h

)nk if x isin Q1 cup Q2

0 otherwise(9)

This definition can be transferred to nondegenerated paral-lelepipeds using a Piola transformation to preserve the normalcomponents [22] [25] [27] We denote the set of Raviart-Thomas basis functions wk by SRT0

h The discrete approximation of (8) can now be written as a

matrix equation

)︸︷︷︸

〈σminus1wiwj〉dx Bkj =

vk(nabla middotwj) dx

〈σminus1jpwi〉dx (12)

for vk isin SP0

h wiwj isin SRT0

For the submatrices A and B we have mA = nA = facesand mB = elements nB = faces respectively and thusthe dimension of K is mK = nK = faces + elementsUsing SRT0

h for the matrix setup in (10) we did not enforcethe Neumann boundary condition (2c) in the discrete equationsystem so far This has to be done explicitly when solving (10)by eliminating the respective degrees of freedom

C Comparison to Other FE Methods for Solving the EEGForward Problem

The state-of-the-art FE method to solve the EEG forwardproblem is the CG-FEM for which a variety of differentsource models has been derived [4]ndash[9] In addition in [14]a discontinuous Galerkin (DG)-FEM for the EEG forwardproblem has been proposed The DG-FEM like the Mixed-FEM is current preserving and was derived to prevent skullleakages and to obtain more accurate and reliable results How-ever whereas the Mixed-FEM actually preserves the physicalcurrent jh = σnablauh the DG-FEM preserves jh = σnablauhminusησγ

hγ[uh]n at each element boundary which converges to the

physical current for h rarr 0 Here middot and [middot] indicate theaverage and jump of the limit values from both sides at an(element) boundary γ η is a regularization parameter and σγand hγ are local definitions of electric conductivity and meshwidth at the surface γ [8] [14] [28] [29]

For sufficiently regular solutions all three methods are con-sistent with the strong problem and show optimal convergencerates ie O(h2) in the L2-norm and O(h) in the energynorm for CG- and DG-FEM and O(h) in the L2-norm forthe Mixed-FEM Furthermore the Mixed-FEM and the DG-FEM are locally charge preserving For details we refer thereader to [24] for the CG-FEM to [30] for the DG-FEM andto [22] for the Mixed-FEM

Remark 1 The above-mentioned a priori convergence re-sults will in general not apply in our case as the dipoleon the right-hand side is not in L2(Ω) For classical globalconvergence results for the CG-FEM and singular right-handsides we refer the reader to [31] [32]

CG- and DG-FEM will be used to evaluate the numericalaccuracy of the approaches based on the Mixed-FEM in thenumerical evaluations in Section IV

D Solving the Linear Equation System (10)

Due to the size of the matrix K in (10) the applicationof direct solvers is not feasible Since the matrix K has alarge 0-block Krylov subspace algorithms such as variants ofthe conjugate gradient (CG) or generalized minimal residual(GMRES) method are also not as efficient as for manyother problems since the commonly used methods for pre-conditioning fail [23] Nevertheless much research has beenperformed to find preconditioning techniques that enable asolution using CG-solvers [33] [34] A further approach tosolve (10) was proposed based on the idea of introducingLagrangian multipliers to achieve the interelement continuityof the RT0-basis functions instead of including this conditionby construction [35] This approach has the advantage thatthe resulting equation system has only faces unknowns butthe derivation is rather technical [22] [35] For our firstevaluation of Mixed-FEM to solve the EEG forward problemwe therefore chose to apply a more direct approach thatmakes use of the fact that A is ndash unlike K ndash positive (semi-) definite The chosen approach follows the ideas of [23]and is based on a modification of the frequently describedUzawa-iteration [24] [36] It was shown that this approach iscompetitive with regard to computation time when comparedto the approach based on Lagrangian multipliers called mixed-hybrid formulation in [23] and a (preconditioned) AugmentedLagrangian approach [23] [37] in a similar scenario as theone considered here The origin of the derivation is identicalto that of the Uzawa-iteration

If we write (10) as a system of two equations

Aj +BTu = b (13a)Bj = 0 (13b)

we can left-multiply Aminus1 to (13a) and solve for j ie j =Aminus1(bminusBTu) Substituting this representation of j into (13b)leads to

Bj = BAminus1(bminusBTu) = 0

hArr BAminus1BTu = BAminus1b(14)

S = BAminus1BT is the so-called Schur complement mS =ns = elements S is positive semidefinite (if ker(B) =0 positive definite) and since A is symmetric also S issymmetric [22] Thus with h = BAminus1b solving (10) isreduced to solving

Su = h (15)

(15) could now be solved using the (conjugated) Uzawa-iteration [22] [24] [36]

However Aminus1 is a dense matrix so that an explicit compu-tation of Aminus1 (and S) is not efficient considering the matrixdimensions occurring in our scenario Instead we access Aminus1

on-the-fly by solving an additional linear equation system foreach iteration ie instead of calculating x = Aminus1y we solveAx = y This equation system can for example be solvedefficiently using preconditioned CG-solvers With the obtainedimplicit representation of S common solver schemes such asthe gradient descent or CG method can be applied to (15)

When solving (15) via the CG algorithm with the implicitrepresentation of S preconditioning is advisable as S has a

large condition number [23] Since S is not directly accessibleit is necessary to use an approximation of S for precondition-ing The use of BBT is proposed in [38] but is efficientonly in the case of constant conductivities [23] AlthoughBBT approximates the pattern of S well enough it does notprovide a reasonable approximation of the matrix entries ofS Instead it is suggested to choose a diagonal matrix D thatin some sense approximates A and to use BDminus1BT as inputto the preconditioner [23] It is further proposed to chooseDii = l2(Ai) = (

sumj A

12 Indeed this approximationled to the best results when it was compared to the choicesDii = Aii Dii =

sumj Aij and Dii = l1(Ai) =

sumj |Aij |

[8]Since all considered choices for D are diagonal the struc-

ture of the matrix P = BDminus1BT is identical to the structureof BBT and cannot be easily inverted Also due to thestructure of P commonly chosen preconditioners such as theincomplete LU-factorization (ILU) cannot be expected to beefficient [39 p 330] We found that approximating Pminus1 usingan algebraic multigrid (AMG) method leads to a performancethat is sufficient for our first evaluations [8]

Besides preconditioning of the ldquoouter iterationrdquo a furtherspeed-up of the solver could be achieved by reducing theaccuracy with which the inner equation Ax = y is solvedThis approach can be interpreted to be similar to inexactUzawa-algorithms as they are proposed in the literature [40]Since reducing the number of iterations for solving the innerequation did not result in an increase in the number of outeriterations that is necessary to reach the desired solution accu-racy performing only one iteration led to the fastest solvingspeed Using this approach solving the equation system (15)took less than two minutes for the finest used spherical modelwith 1 mm mesh resolution (model seg 1 res 1 in Table II) [8]

Through the integration of algebraic multigrid precondition-ers to the Uzawa-like method proposed in [23] our solutionalgorithm has similarities to the combined conjugate gradient-multigrid algorithm proposed in [41] However in [41] nopreconditioning of the outer iteration is performed

E Modeling of a Dipole Source

This section focuses on the exact choice of the source dis-tribution jp In principle arbitrary distributions jp isin L2(Ω)3supp jp sub Ω can be modeled The common choice in EEGforward modeling is jp = mδx0

where δx0is the Dirac

delta distribution and m the dipole moment Since maximallyδ isin Hminus32minusε the assumption jp isin L2(Ω)3 is violatedThe authors are not aware of any literature investigating theinfluence of singular right-hand sides jp for the Mixed-FEMHowever in the case of the CG-FEM it was shown that sucha singular right-hand side does not affect the existence anduniqueness of a solution in general but leads to a lowerregularity of the solution and in consequence to worseglobal a priori error estimates [31] [32] (Quasi-) optimalconvergence for the CG-FEM can be shown in seminormsthat exclude the locations of the singularities [42]

As (1) is represented by a system of two PDEs now thereare two options to model the dipole source The dipole can be

modeled either in the ldquocurrent spacerdquo (6a) or in the ldquopotentialspacerdquo (6b) (sometimes also called ldquopressure spacerdquo due to theorigin of Mixed-FEM in reservoir simulations [38]) The firstoption corresponds to an evaluation of the functional l in thediscrete space RT0 as it was defined in (12) For jp = mδx0

ie a current dipole with moment m at position x0 we have

bi = bcuri =

〈σminus1mδx0 wi〉dx

〈σminus1mwi(x0)〉 if x0 isin suppwi

0 otherwise(16)

This approach will be called the direct approach with h =hdirect = BAminus1bcur

A representation of the dipole in the potential space hence-forth called the projected approach can be obtained using thematrix B which can be interpreted as a mapping betweenthe current and the potential space Figuratively the (source)current is mapped to the distribution of sinks and sourcesgenerating this current The projected approach is similar tothe Whitney approach that was introduced for the CG-FEM[6] [43] except for using the scalar space P0 instead of P1 Inboth approaches a current source represented by RT0 basisfunctions is mapped to the potential space To achieve thisrepresentation for the Mixed-FEM we redefine b to be theapproximation of jp in the space RT0

bpoti =

〈mδx0wi〉dx

〈mwi(x0)〉 if x0 isin suppwi

0 otherwise(17)

bpot is then projected to the space P0 using B We obtainh = hproj = Bbpot the dipole is represented by a source anda sink in the potential space in this case (Figure 2 top)

Remark 2 If a single RT0 function is chosen as the sourcedistribution and a hexahedral mesh is used ie the sourceis positioned on the face fi and the direction is nfi onlyone entry of b is nonzero (cf (16) (17)) When applying theprojection to the potential space using the matrix B which hasonly two nonzero entries per column (cf (11)) the right-handside vector which is given by h = hproj = Bbpot also hasonly two nonzero entries (Figure 2 top) In contrast the right-hand side hdirect = BAminus1bcur causes a blurring of the currentsource when interpreting it as a monopole distribution andvisualizing it in the pressure space It leads to nonzero right-hand side entries hi assigned to all elements that are ldquoin thesource directionrdquo (cf Figure 2 middle and bottom Figure 2bottom shows the sign function of all elements correspondingto nonzero right-hand side entries through red-blue coloring)However most of these values are small

This structure of b transforms accordingly to the case ofarbitrarily positioned and oriented sources as the right-handside vectors b ndash and thereby also h ndash are linear combinationsof the solutions for dipoles oriented in the directions of themesh basis vectors in this case The accuracies of the differentrepresentations are evaluated in Section IV

Fig 2 Visualization of h = hproj = Bbpot (top) h = hdirect =BAminus1bcur (middle) and full view of the support of hdirect throughvisualizing sign(hdirect) (bottom) for a source positioned in the center of aface fi and direction nfi (green cone) The slice is taken at the dipole positionin the y-plane The coloring indicates the values for the P0 basis functioncorresponding to the respective element red is positive blue is negative

III METHODS

A Implementation

FOR this study both the direct (ie h = hdirect =BAminus1bcur) and the projected (h = hproj = Bbpot)

Mixed-FEM approaches were implemented in the DUNEframework [44] [45] using the DUNE-PDELab toolbox [46]In addition a solver corresponding to a conjugate Uzawa-iteration with additional preconditioning and implicit represen-tation of Aminus1 as derived in Section II-D was implementedusing the CG-solver template from the DUNE module iterativesolvers template library (DUNE-ISTL) in combination withthe AMG preconditioner [47]

B EvaluationIn order to evaluate the accuracy of the Mixed-FEM

different comparisons both in hexahedral four-layer spheremodels and in realistic head models were performed As iscommon for the evaluation of EEG forward approaches theerror measures RDM (minimal error 0 maximal error 2) andlnMAG (minimal error 0 maximal error plusmninfin) were used [48][49]

RDM(unum uref ) =

∥∥∥∥ unum

unum2minus uref

∥∥∥∥2

lnMAG(unum uref ) = ln

(unum2uref2

) (18)

In the sphere models the solution was evaluated on the wholeouter boundary instead of using single electrode positionsso that the results are independent of the choice of sensorpositions For the realistic head model the sensor positions ofa realistic 80-electrode EEG cap were used [3] [8]

TABLE IFOUR-LAYER SPHERE MODELS (COMPARTMENTS FROM IN- TO OUTSIDE)

Compartment Outer Radius σ Reference

Brain 78 mm 033 Sm [50]CSF 80 mm 179 Sm [51]Skull 86 mm 001 Sm [17]Skin 92 mm 043 Sm [17] [50]

Besides the two Mixed-FEM approaches the Whitney CG-FEM was included in our sphere model comparisons as itrelies on the same approximation of the dipole source [6] [43]By including the Whitney CG-FEM the differences betweenMixed- and CG-FEM can be directly evaluated Two four-layerhexahedral sphere models seg 1 res 1 and seg 2 res 2 with amesh resolution of 1 and 2 mm respectively were generated(Tables I II) Sources were placed at 10 different radii andfor each radius 10 sources were randomly distributed Thisdistribution of the test sources allows us to gain a statisticaloverview of the range of the numerical accuracy at eacheccentricity Since the numerical errors increase along withthe eccentricity ie the quotient of source radius and radiusof the innermost compartment boundary the radii of thesource positions were chosen so that the distances to the nextconductivity jump (brainCSF boundary) were logarithmicallydistributed The most exterior eccentricity 0993 correspondsto a distance of only asymp 05 mm to the conductivity jump Inpraxis (and for the realistic head model used in this study)sources are usually placed so that at least one layer of elementsis between the source element and the conductivity jumpwhich is fulfilled for the considered eccentricities le 0987 inthe 1 mm model and the eccentricities le 0964 in the 2 mmmodel The reference solutions uref were computed using aquasianalytical solution for sphere models [52]

In the first study for each model the sources were placed onthe closest face center and the source directions were chosenaccording to the face normals so that only one basis function

TABLE IISPHERE MODEL PARAMETERS

Mesh width (h) vertices elements faces

seg 1 res 1 1 mm 3342701 3262312 9866772seg 2 res 2 2 mm 428185 407907 1243716

TABLE IIIREALISTIC HEAD MODEL PARAMETERS

6C hex 1mm 1 mm 3965968 3871029 117074016C hex 2mm 2 mm 508412 484532 14771646C tet hr ndash 2242186 14223508 27314610

Fig 3 Visualization of realistic six-compartment hexahedral (6C hex 2mmleft) and high-resolution reference head model (6C tet hr right)

contributes to the right-hand side vectors b (cf (11) (16))Therefore the results are not influenced by the interpolationthat is needed for arbitrary source directions and positions Forthe Whitney approach it was shown that it has the highestaccuracy of all CG-FEM approaches in this scenario [43]Next the three approaches were compared in the same modelsusing the initially generated random source positions andradial source directions so that neither positions nor directionswere adjusted to the mesh We limit our investigations to radialsources as eccentric radial sources were shown to lead tohigher numerical errors than tangential sources in previousstudies [53] Finally the projected Mixed-FEM and WhitneyCG-FEM were evaluated in combination with the modelsseg 2 res 2 r82 seg 2 res 2 r83 and seg 2 res 2 r84 generatedfrom model seg 2 res 2 but with an especially thin skull layeragain with random positions and radial source directions TableIV indicates the outer skull radii of the different models andthe resulting number of leakages ie the number of nodes inwhich elements of skin and CSF compartment touch

Mixed-FEM CG-FEM and DG-FEM were further eval-uated in a more realistic scenario Two realistic six-compartment hexahedral head models with mesh widths of1 mm 6C hex 1mm and 2 mm 6C hex 2mm were createdresulting in 3965968 vertices and 3871029 elements and508412 vertices and 484532 elements respectively (TableIII Figure 3) As the model with a mesh width of 2 mmwas not corrected for leakages 1164 vertices belonging toboth CSF and skin elements were found mainly located at thetemporal bone The conductivities were chosen according to[3] Of 18893 source positions placed in the gray matter with anormal constraint those not fully contained in the gray mattercompartment (ie where the source was placed in an elementat a compartment boundary) were excluded In consequence17870 source positions remained for the 1 mm model and17843 source positions for the 2 mm model As sensorconfiguration an 80 channel realistic EEG cap was chosen Theinvestigated approaches were projected Mixed-FEM WhitneyCG-FEM St Venant CG-FEM [4] and Partial Integration DG-

TABLE IVMODEL LEAKS

Model Outer Skull Radius leaksseg 2 res 2 r82 82 mm 10080seg 2 res 2 r83 83 mm 1344seg 2 res 2 r84 84 mm 0

FEM [8] [14] St Venant CG-FEM and Partial IntegrationDG-FEM were additionally included since they were shownto achieve the highest accuracies of the different CG- and DG-FEM approaches respectively when choosing arbitrary sourcedirections and positions [14] [43] Solutions for all methodswere computed in the 2 mm model and a solution in the1 mm model was calculated using the St Venant CG-FEMIn the realistic scenario RDM and lnMAG were evaluatedin comparison to a reference solution that was computedusing the St Venant method in a high-resolution tetrahedralmodel 6C tet hr based on the same segmentation (Table III2242186 vertices 14223508 elements) For details of thismodel we refer the reader to [3] [8]

IV RESULTS

In this paper a new finite element method to solve theEEG forward problem is introduced It is expected that itshould be preferrable compared to the commonly used CG-FEM approaches especially in leakage and realistic scenariosThe goal of Sections IV-A and IV-B is to show that this newmethod performs appropriately when compared to the estab-lished CG-FEM in common sphere models and in SectionsIV-C and IV-D the accuracy in leakage and realistic scenariosis evaluated

A Comparison of Whitney CG-FEM and Mixed-FEM forOptimal Source Positions

COMPARING the three approaches with regard to theRDM in model seg 1 res 1 (Figure 4) no remarkable

differences are found up to an eccentricity of 0964 (distancefrom next conductivity jump ge 28 mm) with maximal errorsbelow 005 for all approaches (Figure 4 top row) At aneccentricity of 0979 (dist asymp 16 mm) the maximal errorsfor the Mixed-FEM slightly increase However the maximalerrors remain clearly below 01 Also the Whitney CG-FEMhas a maximal error below 01 at this eccentricity and theupper quartile and median are lower than for the Mixed-FEMFor the highest three eccentricities the RDM clearly increasesfor all considered approaches The variance especially for thehighest eccentricities is lowest for projected Mixed-FEM andWhitney CG-FEM In the coarser model seg 2 res 2 direct andprojected Mixed-FEM perform similar up to eccentricities of0933 or 0964 (dist ge 28 mm) whereas the errors for theWhitney CG-FEM are lower and have less variance For highereccentricities a rating of the accuracies is hardly possible dueto the higher variance

With regard to the lnMAG (Figure 4 bottom row) onlyminor differences are recognizable for model seg 1 res 1 Inmodel seg 2 res 2 it is notable that the direct Mixed-FEMleads to very high maximal errors for eccentricities of 0987whereas Whitney CG-FEM and projected Mixed-FEM per-form similar with a tendency of the Whitney CG-FEM towardlower errors

B Comparison of Whitney CG-FEM and Mixed-FEM forRandom Source Positions

The next comparison expands the previous results to randomsource positions and radial source orientations When compar-

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

seg_1_res_1 Mixed-FEM

seg_1_res_1 proj Mixed-FEM

seg_1_res_1 Whitney CG-FEM

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

Fig 4 Comparison of direct and projected Mixed-FEM and Whitney CG-FEM in meshes seg 1 res 1 and seg 2 res 2 Results for optimized dipolepositions Visualized boxplots of RDM (top row) and lnMAG (bottom row)Dipole positions outside the brain compartment in the discretized models aremarked as dots Note the logarithmic scaling of the x-axes

ing the two Mixed-FEM approaches with regard to the RDM(Figure 5 top row) both models show no major differencesup to an eccentricity of 0964 (dist ge 28 mm) but theWhitney CG-FEM leads to lower errors especially in modelseg 2 res 2 For model seg 1 res 1 the RDM is constantlybelow 005 at low eccentricities (up to eccentricity le 0964ie dist ge 28 mm) With increasing eccentricity the RDMfor the projected Mixed-FEM and Whitney CG-FEM mainlyremains below 01 whereas the maximal RDM is at nearly03 for the direct approach and the median is above 01 Alsoin model seg 2 res 2 the projected approach outperforms thedirect approach with regard to the RDM The less accurateapproximation of the geometry leads to higher errors in thesemodels eg the minimal RDM at an eccentricity of 0964(dist ge 28 mm) is already at nearly 01 for both approachesin model seg 2 res 2 The Whitney CG-FEM performs clearlybetter than both Mixed-FEM approaches in this model withmaximal errors below 013 at this eccentricity For moreeccentric sources the projected approach again performsbetter than the direct approach Nevertheless the errors forthe Whitney CG-FEM remain at a lower level

The results for the lnMAG (Figure 5 bottom row) do notshow remarkable differences for all models up to an eccentric-

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

Fig 5 Comparison of direct and projected Mixed-FEM and WhitneyCG-FEM in meshes seg 1 res 1 and seg 2 res 2 Results for random dipolepositions Visualized boxplots of RDM (top row) and lnMAG (bottom row)Dipole positions outside the brain compartment in the discretized models aremarked as dots Note the logarithmic scaling of the x-axes

ity of 0964 In model seg 1 res 1 the projected Mixed-FEMleads to the lowest spread for the three highest eccentricitiesHowever the lnMAG decreases from positive values for allsource positions at low eccentricities to completely negativevalues at the highest eccentricity This effect is even strongerfor the Whitney CG-FEM In contrast the median of thedirect Mixed-FEM remains close to constant up to the highesteccentricity but with a higher spread The same behavior ofthe three approaches just at a generally higher error level isfound for model seg 2 res 2

C Comparison of Mixed-FEM Approaches in Leaky SphereModels

The results of Sections IV-A and IV-B suggest that theprojected Mixed-FEM is superior to the direct Mixed-FEM Tokeep the presentation concise we from here on compare onlythe projected Mixed-FEM with the Whitney CG-FEM Theresults for model seg 2 res 2 r84 (Table IV) which does notcontain any skull leakages mainly resemble those for modelseg 2 res 2 for both RDM and lnMAG (Figure 6)

In models seg 2 res 2 r82 and seg 2 res 2 r83 the effectsof the leakages become apparent With regard to the RDM(Figure 6 top row) the projected Mixed-FEM leads to lower

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

seg_2_res_2_r84 proj Mixed-FEM

seg_2_res_2_r84 Whitney CG-FEM

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

Fig 6 Comparison of projected Mixed-FEM and Whitney CG-FEM inmeshes with thin skull compartment Results for random dipole positionsVisualized boxplots of RDM (top row) and lnMAG (bottom row) Dipolepositions outside the brain compartment in the discretized models are markedas dots Note the logarithmic scaling of the x-axes

errors in both models In model seg 2 res 2 r83 the differencesbetween the two approaches are still moderate However espe-cially up to an eccentricity of 0964 (dist ge 28 mm) a higheraccuracy for the projected Mixed-FEM is clearly observableThe increased number of leakages in seg 2 res 2 r82 intensifiesthe difference between the approaches The errors for theWhitney CG-FEM are clearly higher than for the Mixed-FEMhere with maximal errors larger than 05 at eccentricitiesabove 0964 (dist le 16 mm)

Also with regard to the lnMAG (Figure 6 bottom row)the influence of the skull leakages is apparent In modelsseg 2 res 2 r82 and seg 2 res 2 r83 the lnMAG increases upto an eccentricity of 0964 and only decreases for highereccentricities This effect is clearly stronger for the WhitneyCG-FEM than for the Mixed-FEM In contrast the lnMAG forthe Whitney CG-FEM decreases clearly stronger than for theMixed-FEM in model seg 2 res 2 r84 with increasing eccen-tricity leading to a switch from about 02 for eccentricitiesbelow 0964 to values lower than 02 at an eccentricity of0993 Especially in model seg 2 res 2 r83 the Whitney CG-FEM also leads to a higher variance of the lnMAG but thisvariance is less distinct in the other models

For a single exemplary dipole the distribution of the

Fig 7 Geometry of leaky four-layer sphere model (left compartments from in- to outsidebottom left to top right are brain CSF skull skin and air) andvisualization of strength (only skull and skin in microAmm2) and direction of volume currents for CG-FEM (middle) and Mixed-FEM simulation (right)

volume currents in skull and skin in model seg 2 res 2 r82simulated with the Whitney CG- and projected Mixed-FEMis visualized in Figure 7 The leakage effect for the CG-FEM (Figure 7 middle) is obvious While the Mixed-FEM(Figure 7 right) leads to a smooth current distribution andthe highest current strengths among skull and skin elementsare found in the skull compartment (up to asymp 13 microAmm2) thecurrent strength peaks in the skin compartment for the WhitneyCG-FEM (maximum asymp 144 microAmm2) and is increased by afactor of more than 11 compared to the Mixed-FEM (note thedifferent scaling of the colorbars) Compared to the maximalcurrent strength in the skin compartment the current strengthin the skull is very low here showing the leakage of thevolume currents through the nodes shared between CSF andthe skin

D Realistic Head Model Study

The cumulative relative frequencies of RDM and lnMAGare displayed in Figure 8 Due to the rough approximationof the smooth surfaces all models consisting of regularhexahedra (especially at the mesh width of 2 mm) lead to rel-atively high topography and magnitude errors when comparedto the surface-based tetrahedral reference model Comparingthe results in model 6C hex 2mm with regard to the RDM(Figure 8 top) the projected Mixed-FEM performs best withroughly 95 of the errors below 031 (95 indicated by upperhorizontal bar in Figure 8 top) Therefore the result is nearlyas good as that achieved with the St Venant approach in the 1mm model 6C hex 1mm where 95 of the errors are below028 The partial integration DG-FEM performs nearly equallywell to the Mixed-FEM with 95 of the errors reached atabout 036 Whitney and St Venant CG-FEM perform nearlyidentically and for these approaches the 95th percentile isreached at an RDM of nearly 04

With regard to the lnMAG the differences between theresults obtained using the mesh resolutions of 1 and 2 mm andalso between Mixed- DG- and the two CG-FEM approachesare larger than for the RDM (Figure 8 bottom) The projectedMixed-FEM performs best for model 6C hex 2mm with 90of the errors in the range from -015 and 035 (interval betweenlower and upper horizontal lines in Figure 8) The partialintegration DG-FEM performs only slightly worse with 90of the errors in the range from -015 and 04 Again Whitney

0 01 02 03 04 05

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

-02 -01 0 01 02 03 04 05 06

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

Fig 8 Cumulative relative errors of RDM (top) and lnMAG (bottom) forEEG in realistic six-layer head model The horizontal lines indicate the 5thand 95th percentile (lower and upper lines respectively)

and St Venant CG-FEM lead to nearly identical accuraciesand show the highest errors for the model 6C hex 2mm bothwith regard to absolute values and spread (90 of the errors inthe range from -01 to 054) The increase in accuracy whenusing model 6C hex 1mm instead of model 6C hex 2mm isclearer for the lnMAG than for the RDM For the St VenantCG-FEM 90 of the lnMAG-errors are in the range from -02to 025 thus showing both a smaller spread than the resultsin the model 6C hex 2mm and also lower absolute values

V DISCUSSION AND CONCLUSION

THIS study introduced the Mixed-FEM approach for theEEG forward problem Two approaches to model the

dipole source were derived the direct and the projectedNumerical results for sphere and realistic head models werepresented and compared to different established numericalmethods

The results suggest that the Mixed-FEM achieves an appro-priate accuracy for common sphere models especially the pro-jected approach The comparison with the Whitney CG-FEMapproach with optimized positions and orientations shows thatthe Mixed-FEM leads to comparable accuracies (Figure 4) Forboth optimized and arbitrary source positions the projectedapproach achieved a superior accuracy compared to the directapproach Previous publications concentrated on evaluating theWhitney CG-FEM in tetrahedral models [43] In these studiesthe accuracy of the Whitney approach deteriorated when usingarbitrary source positions and orientations potentially dueto the interpolation necessary to represent arbitrary sourcepositions and orientations with the Whitney approach Thiseffect is not found in the hexahedral models used here anda high accuracy is achieved (Figure 4) These results shouldbe investigated in more depth in further studies In the leakymodels seg 2 res 2 r82 and seg 2 res 2 r83 the Mixed-FEMperforms better than the Whitney CG-FEM (Figure 6) Thishigher accuracy was expected from the Mixed-FEM basedon theoretical considerations since the Mixed-FEM is byconstruction charge preserving which should prevent currentleakages [54]

For EEG forward modeling the Mixed-FEM approachesshare this current preserving property with the recently pro-posed approaches based on the DG-FEM [14] Both thedirect Mixed-FEM and the partial integration DG-FEM wereevaluated against CG-FEM approaches in the realistic six-compartment head model 6C hex 2mm In this head modelboth Mixed- and DG-FEM were advantageous in comparisonto the CG-FEM (Figure 8) The projected Mixed-FEM clearlyoutperforms both Whitney and St Venant CG-FEM in thisscenario and achieves a slightly higher accuracy than thepartial integration DG-FEM Since only a few skull leakagesoccurred in this model and as these were concentrated inthe area of the temporal bone leakage effects do not sufficeto explain the higher accuracy of Mixed- and DG-FEM Anoverall higher accuracy of these approaches in this kind ofmodel ie regular hexahedral with a mesh resolution of 2mm can be assumed The relatively high level of errors isa consequence of the coarse regular hexahedral meshes thatwere used whereas the reference solution was computed ina highly resolved tetrahedral model The result for the StVenant CG-FEM in the model with a mesh resolution of1 mm 6C hex 1mm helps to estimate the relation betweenthe influence of the different numerical approaches and theaccuracy of the approximation of the geometry It is shown thatthe difference between projected Mixed-FEM and Whitneyand St Venant CG-FEM in model 6C hex 2mm is nearly asbig as the difference between using models 6C hex 1mm and6C hex 2mm for the St Venant CG-FEM

Realizing these differences in accuracy directly leads tothe three main sources of error in these evaluations Besidesthe previously discussed leakage effects these are inaccuraterepresentation of the geometry and numerical inaccuracies Amajor source of error is the representation of the geometrySince regular hexahedral meshes were used the influence ofgeometry errors is significant especially for coarse mesheswith resolutions of 2 mm or higher No explicit convergencestudy comparing the results in models with increasing meshresolution but a constant representation of the geometry wasperformed However it can be assumed from the resultsof previous studies that the geometry error dominates thenumerical errors due to lower mesh resolutions [8] [14]

In order to reduce the geometry error the use of geometry-adapted meshes was considered for the CG-FEM Suchmeshes have been shown to clearly improve the represen-tation of the geometry in previous studies [53] [55] [56]Although the use of nondegenerated parallelepipeds is un-critical for the Mixed-FEM ldquosome complications may arisefor general elementsrdquo [22] However it was shown that theH(div Ω)-convergence is preserved on shape-regular asymp-totically parallelepiped hexahedral meshes [27] and for thetwo-dimensional case error estimates for general quadrilat-eral grids can be obtained when modifying the lowest-orderRaviart-Thomas elements [57] [58] and for convex quadri-laterals even superconvergence was shown [54] The use ofgeometry-adapted hexahedral meshes in combination with theMixed-FEM should therefore be evaluated in future studies

Regarding the numerical inaccuracy due to the discretizationof the equations and the source singularity the Mixed-FEMallows to increase the regularity of the right-hand side byone degree As a consequence of the first-order formulation(6) applying the derivative to the delta distribution includedin the primary current jp can be circumvented The resultsobtained show high numerical accuracies especially at thehighest eccentricities and particularly for the projected Mixed-FEM This increase in accuracy comes at the cost of a highernumber of degrees of freedom than that of the CG-FEM asthe current j is also considered as an unknown now meaningthat it has to be discretized Furthermore the discrete problemhas a saddle point structure (10) and cannot be efficientlysolved with AMG-CG solvers without further modificationsAlthough the number of unknowns is clearly increased com-pared to the CG-FEM eg in model seg 2 res 2 we haveDOFM = 1 243 716 + 407 904 and DOFCG = 428 185(cf Table II) by introducing an algorithm based on the ideaof the conjugated Uzawa-iteration (Section II-D) the solvingtime even in the finest model seg 1 res 1 was reduced to lessthan two minutes This solving time is only a few secondsslower than that for the CG-FEM Furthermore as the equationsystem (10) is symmetric the transfer matrix approach [59][60] can be applied for the Mixed-FEM to reduce the numberof equation systems that have to be solved to equal the numberof sensors

As an alternative to the straightforward approach presentedhere for solving the linear equation system (13) using theSchur complement an approach based on the method ofLagrange multipliers has been proposed [35] In this approach

the continuity of the vector-valued basis functions is no longerenforced by the definition of the basis functions but by intro-ducing interelement Lagrange multipliers This approach leadsto a linear equation system having as many unknowns as thenumber of faces in the case of lowest-order Raviart-Thomaselements This equation system is symmetric positive definiteand sparse Although this approach does not necessarily leadto a decrease of the solving time [23] [41] a higher orderof convergence is predicted in theory when employing theinformation contained in the Lagrangian multipliers [22] [61]Therefore it is desirable to evaluate this solution approach insubsequent studies

The lowest-order Raviart-Thomas elements used in thisstudy are the most classical but only one of many dif-ferent elements that have been developed to approximateH(div Ω) Further element types are eg Brezzi-Douglas-Marini (BDM) [62] [63] and Brezzi-Douglas-Fortin-Marini(BDFM) [64] elements To overcome known limitations ofthese classical element types further elements to approximateH(div Ω) were developed more recently [65] [66] Due todifferent approximation properties of the element types theevaluation of further element types for solving the EEG for-ward problem using the Mixed-FEM in future studies might beworthwhile Also the use of higher-order Raviart-Thomas ele-ments eg RT1 elements in combination with discontinuouslinear Ansatz-functions for the potential should be consideredas the theoretically predicted convergence rates improve forhigher element orders For an overview of the most commonfinite element spaces to approximate H(div Ω) includinghigher-order elements and their convergence properties werefer the reader to [22] However the use of higher orderelements comes at the cost of an increased number of degreesof freedom Thus the use of higher mesh resolutions shouldalways be considered as an alternative to the use of higher-order elements

As mentioned the Mixed-FEM guarantees the conserva-tion of charge by construction In consequence especiallyin models with thin insulating compartments and at highesteccentricities it still leads to high accuracies which alsoencourages the use of the Mixed-FEM in related applicationsthat depend on an accurate simulation of the electric currentsuch as the magnetoencephalography (MEG) forward problemtranscranial direct current stimulation (tDCS) or deep brainstimulation (DBS) simulations

Overall we conclude that the Mixed-FEM is an interestingnew approach that can at least complement and in somescenarios even outperform standard continuous Galerkin FEMapproaches for simulation studies in bioelectromagnetism Theuse of different element types and solving algorithms shouldbe investigated in further studies

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their valuable comments and suggestions to improve thequality of the paper We are also grateful to Prof Dr SteffenBorm for proof-reading and his advice with regard to theTheory section

REFERENCES

[1] Z Akalin-Acar and S Makeig ldquoEffects of forward model errors onEEG source localizationrdquo Brain Topography vol 26 no 3 pp 378ndash396 2013

[2] J-H Cho J Vorwerk C H Wolters and T R Knosche ldquoInfluenceof the head model on EEG and MEG source connectivity analysesrdquoNeuroImage vol 110 pp 60ndash77 2015

[3] J Vorwerk J-H Cho S Rampp H Hamer T R Knosche and C HWolters ldquoA guideline for head volume conductor modeling in EEG andMEGrdquo NeuroImage vol 100 pp 590ndash607 2014

[4] H Buchner G Knoll M Fuchs A Rienacker R Beckmann M Wag-ner J Silny and J Pesch ldquoInverse localization of electric dipole currentsources in finite element models of the human headrdquo Electroencephalog-raphy and Clinical Neurophysiology vol 102 pp 267ndash278 1997

[5] Y Yan P L Nunez and R T Hart ldquoFinite-element model of the humanhead Scalp potentials due to dipole sourcesrdquo Medical amp BiologicalEngineering amp Computing vol 29 pp 475ndash481 1991

[6] S Pursiainen A Sorrentino C Campi and M Piana ldquoForwardsimulation and inverse dipole localization with the lowest orderRaviart-Thomas elements for electroencephalographyrdquo Inverse Prob-lems vol 27 no 4 2011

[7] C H Wolters H Kostler C Moller J Hartlein L Grasedyck andW Hackbusch ldquoNumerical mathematics of the subtraction method forthe modeling of a current dipole in EEG source reconstruction usingfinite element head modelsrdquo SIAM Journal on Scientific Computingvol 30 no 1 pp 24ndash45 2007

[8] J Vorwerk ldquoNew finite element methods to solve the EEGMEGforward problemrdquo PhD thesis in Mathematics Westfalische Wilhelms-Universitat Munster February 2016 [Online] Available httpsciutahedusimwoltersPaperWolters2016Vorwerk Dissertation 2016pdfrdquo

[9] C H Wolters H Kostler C Moller J Hardtlein and A AnwanderldquoNumerical approaches for dipole modeling in finite element methodbased source analysisrdquo International Congress Series vol 1300 pp189ndash192 2007

[10] U Aydin J Vorwerk P Kupper M Heers H Kugel A GalkaL Hamid J Wellmer C Kellinghaus S Rampp and C H WoltersldquoCombining EEG and MEG for the reconstruction of epileptic activityusing a calibrated realistic volume conductor modelrdquo PLOS ONE vol 9no 3 p e93154 2014

[11] M Rullmann A Anwander M Dannhauer S Warfield F H Duffyand C H Wolters ldquoEEG source analysis of epileptiform activity usinga 1mm anisotropic hexahedra finite element head modelrdquo NeuroImagevol 44 no 2 pp 399ndash410 2009

[12] S Lau D Gullmar L Flemming D B Grayden M Cook C HWolters and J Haueisen ldquoSkull defects in finite element head modelsfor source reconstruction from magnetoencephalography signalsrdquo Fron-tiers in Neuroscience vol 10 no 141 2016

[13] H Sonntag J Vorwerk C H Wolters L Grasedyck J Haueisenand B Maess ldquoLeakage effect in hexagonal FEM meshes of the EEGforward problemrdquo in International Conference on Basic and ClinicalMultimodal Imaging (BaCI) 2013

[14] C Engwer J Vorwerk J Ludewig and C H Wolters ldquoA discontinuousGalerkin method for the EEG forward problemrdquo arXiv1511048922015

[15] V Montes-Restrepo P van Mierlo G Strobbe S Staelens S Van-denberghe and H Hallez ldquoInfluence of skull modeling approaches onEEG source localizationrdquo Brain Topography vol 27 no 1 pp 95ndash1112014

[16] B Lanfer M Scherg M Dannhauer T R Knosche M Burger andC H Wolters ldquoInfluences of skull segmentation inaccuracies on EEGsource analysisrdquo NeuroImage vol 62 no 1 pp 418ndash431 2012

[17] M Dannhauer B Lanfer C H Wolters and T R Knosche ldquoModelingof the human skull in EEG source analysisrdquo Human Brain Mappingvol 32 no 9 pp 1383ndash1399 2011

[18] M S Hamalainen R Hari R J Ilmoniemi J Knuutila and O VLounasmaa ldquoMagnetoencephalography ndash theory instrumentation andapplications to noninvasive studies of the working human brainrdquo Re-views of Modern Physics vol 65 no 2 pp 413ndash497 1993

[19] R Brette and A Destexhe Handbook of Neural Activity MeasurementCambridge University Press 2012 [Online] Available httpwwwdiensfrsimbretteHandbookMeasurement

[20] J Roberts and J-M Thomas ldquoMixed and hybrid methodsrdquo in FiniteElement Methods (Part 1) ser Handbook of Numerical Analysis P GCiarlet and J L Lions Eds Elsevier 1991 vol 2 pp 523 ndash 639

[21] D N Arnold ldquoMixed finite element methods for elliptic problemsrdquoComputer Methods in Applied Mechanics and Engineering vol 82no 1 pp 281ndash300 1990

[22] F Brezzi and M Fortin Mixed and hybrid finite element methodsSpringer 1991 vol 15

[23] L Bergamaschi S Mantica and F Saleri ldquoMixed finite elementapproximation of Darcyrsquos law in porous mediardquo Report CRS4 AppMath-94-20 CRS4 Cagliari Italy 1994

[24] D Braess Finite elements theory fast solvers and applications in solidmechanics Cambridge University Press 2007

[25] J-C Nedelec ldquoMixed finite elements in R3rdquo Numerische Mathematikvol 35 no 3 pp 315ndash341 1980

[26] P-A Raviart and J-M Thomas ldquoA mixed finite element method for 2-nd order elliptic problemsrdquo in Mathematical Aspects of Finite ElementMethods Springer 1977 pp 292ndash315

[27] A Bermudez P Gamallo M R Nogueiras and R Rodrıguez ldquoAp-proximation properties of lowest-order hexahedral raviartndashthomas finiteelementsrdquo Comptes Rendus Mathematique vol 340 no 9 pp 687ndash6922005

[28] A Ern A F Stephansen and P Zunino ldquoA discontinuous Galerkinmethod with weighted averages for advectionndashdiffusion equations withlocally small and anisotropic diffusivityrdquo IMA Journal of NumericalAnalysis vol 29 no 2 pp 235ndash256 2009

[29] S Giani and P Houston ldquoAnisotropic hp-adaptive discontinuousGalerkin finite element methods for compressible fluid flowsrdquo Inter-national Journal of Numerical Analysis and Modeling vol 9 no 4 pp928ndash949 2012

[30] A Ern A F Stephansen and P Zunino ldquoA discontinuous Galerkinmethod with weighted averages for advectionndashdiffusion equations withlocally small and anisotropic diffusivityrdquo IMA Journal of NumericalAnalysis 2008

[31] E Casas ldquoL2 estimates for the finite element method for the Dirichletproblem with singular datardquo Numerische Mathematik vol 47 no 4 pp627ndash632 1985

[32] R Scott ldquoFinite element convergence for singular datardquo NumerischeMathematik vol 21 no 4 pp 317ndash327 1973

[33] O Axelsson Iterative solution methods Cambridge University PressNew York 1994

[34] G H Golub and C F Van Loan Matrix computations The JohnHopkins University Press Baltimore and London 2nd edition 1989

[35] B F De Veubeke and G Sander ldquoAn equilibrium model for platebendingrdquo International Journal of Solids and Structures vol 4 no 4pp 447ndash468 1968

[36] K J Arrow L Hurwicz and H Uzawa Studies in linear and non-linearprogramming ser Stanford mathematical studies in the social sciencesStanford University Press 1972

[37] R Glowinski and P Le Tallec Augmented Lagrangian and operator-splitting methods in nonlinear mechanics SIAM 1989 vol 9

[38] E Ng B Nitrosso and B Peyton ldquoOn the solution of Stokesrsquos pressuresystem within N3S using supernodal Cholesky factorizationrdquo FiniteElements in Fluids New Trends and Applications 1993

[39] K Chen Matrix preconditioning techniques and applications Cam-bridge University Press 2005 no 19

[40] H C Elman and G H Golub ldquoInexact and preconditioned Uzawaalgorithms for saddle point problemsrdquo SIAM Journal on NumericalAnalysis vol 31 no 6 pp 1645ndash1661 1994

[41] R Verfurth ldquoA combined conjugate gradient-multi-grid algorithm for thenumerical solution of the Stokes problemrdquo IMA Journal of NumericalAnalysis vol 4 no 4 pp 441ndash455 1984

[42] T Koppl and B Wohlmuth ldquoOptimal a priori error estimates for anelliptic problem with dirac right-hand siderdquo SIAM Journal on NumericalAnalysis vol 52 no 4 pp 1753ndash1769 2014

[43] M Bauer S Pursiainen J Vorwerk H Kostler and C H WoltersldquoComparison study for Whitney (Raviart-Thomas)-type source models infinite element method based EEG forward modelingrdquo IEEE Transactionson Biomedical Engineering vol 62 no 11 pp 2648ndash2656 2015

[44] P Bastian M Blatt A Dedner C Engwer R Klofkorn M Ohlbergerand O Sander ldquoA generic grid interface for parallel and adaptivescientific computing Part I Abstract frameworkrdquo Computing vol 82no 2ndash3 pp 103ndash119 July 2008

[45] P Bastian M Blatt A Dedner C Engwer R Klofkorn R KornhuberM Ohlberger and O Sander ldquoA generic grid interface for paralleland adaptive scientific computing Part II Implementation and tests inDUNErdquo Computing vol 82 no 2ndash3 pp 121ndash138 July 2008

[46] P Bastian F Heimann and S Marnach ldquoGeneric implementationof finite element methods in the distributed and unified numericsenvironment (DUNE)rdquo Kybernetika vol 46 no 2 pp 294ndash315 2010

[47] M Blatt ldquoA parallel algebraic multigrid method for elliptic problemswith highly discontinuous coefficientsrdquo PhD thesis in MathematicsHeidelberg University 2010

[48] J W H Meijs O W Weier M J Peters and A van OosteromldquoOn the numerical accuracy of the boundary element methodrdquo IEEETransactions on Biomedical Engineering vol 36 pp 1038ndash1049 1989

[49] D Gullmar J Haueisen and J R Reichenbach ldquoInfluence ofanisotropic electrical conductivity in white matter tissue on theEEGMEG forward and inverse solution a high-resolution whole headsimulation studyrdquo NeuroImage 2010

[50] C Ramon P Schimpf J Haueisen M Holmes and A IshimaruldquoRole of soft bone CSF and gray matter in EEG simulationsrdquo BrainTopography vol 16 no 4 pp 245ndash248 2004

[51] S B Baumann D R Wozny S K Kelly and F M Meno ldquoThe elec-trical conductivity of human cerebrospinal fluid at body temperaturerdquoIEEE Transactions on Biomedical Engineering vol 44 no 3 pp 220ndash223 1997

[52] J C de Munck and M J Peters ldquoA fast method to compute thepotential in the multisphere modelrdquo IEEE Transactions on BiomedicalEngineering vol 40 no 11 pp 1166ndash1174 1993

[53] C H Wolters A Anwander G Berti and U Hartmann ldquoGeometry-adapted hexahedral meshes improve accuracy of finite element methodbased EEG source analysisrdquo IEEE Transactions on Biomedical Engi-neering vol 54 no 8 pp 1446ndash1453 2007

[54] R E Ewing M M Liu and J Wang ldquoSuperconvergence of mixedfinite element approximations over quadrilateralsrdquo SIAM Journal onNumerical Analysis vol 36 no 3 pp 772ndash787 1999

[55] D Camacho R Hopper G Lin and B Myers ldquoAn improved methodfor finite element mesh generation of geometrically complex structureswith application to the skullbaserdquo Journal of Biomechanics vol 30no 10 pp 1067ndash1070 1997

[56] S Wagner F Lucka J Vorwerk C S Herrmann G Nolte M Burgerand C H Wolters ldquoUsing reciprocity for relating the simulation of tran-scranial current stimulation to the EEG forward problemrdquo NeuroImage2016

[57] S H Chou D Y Kwak and K Y Kim ldquoFlux recovery from primalhybrid finite element methodsrdquo SIAM Journal on Numerical Analysisvol 40 no 2 pp 403ndash415 2002

[58] D Y Kwak and H C Pyo ldquoMixed finite element methods for generalquadrilateral gridsrdquo Applied Mathematics and Computation vol 217no 14 pp 6556ndash6565 2011

[59] D Weinstein L Zhukov and C Johnson ldquoLead-field bases for elec-troencephalography source imagingrdquo Annals of Biomedical Engineeringvol 28 no 9 pp 1059ndash1066 2000

[60] C H Wolters L Grasedyck and W Hackbusch ldquoEfficient computationof lead field bases and influence matrix for the FEM-based EEG andMEG inverse problemrdquo Inverse Problems vol 20 no 4 pp 1099ndash11162004

[61] D N Arnold and F Brezzi ldquoMixed and nonconforming finite elementmethods implementation postprocessing and error estimatesrdquo RAIRO-Modelisation mathematique et analyse numerique vol 19 no 1 pp7ndash32 1985

[62] F Brezzi J Douglas Jr and L D Marini ldquoTwo families of mixed finiteelements for second order elliptic problemsrdquo Numerische Mathematikvol 47 no 2 pp 217ndash235 1985

[63] F Brezzi J Douglas Jr R Duran and M Fortin ldquoMixed finite elementsfor second order elliptic problems in three variablesrdquo NumerischeMathematik vol 51 no 2 pp 237ndash250 1987

[64] F Brezzi J Douglas Jr M Fortin and L D Marini ldquoEfficientrectangular mixed finite elements in two and three space variablesrdquoRAIRO-Modelisation mathematique et analyse numerique vol 21 no 4pp 581ndash604 1987

[65] D N Arnold D Boffi and R S Falk ldquoQuadrilateral H(div) finiteelementsrdquo SIAM Journal on Numerical Analysis vol 42 no 6 pp2429ndash2451 2005

[66] R S Falk P Gatto and P Monk ldquoHexahedral H(div) and H(curl)finite elementsrdquo ESAIM Mathematical Modelling and Numerical Anal-ysis vol 45 no 1 pp 115ndash143 2011

I Introduction

II Theory

II-A A (Mixed) Weak Formulation of the EEG Forward Problem

II-B Mixed Finite Element Method

II-C Comparison to Other FE Methods for Solving the EEG Forward Problem

II-D Solving the Linear Equation System ()

II-E Modeling of a Dipole Source

III Methods

III-A Implementation

III-B Evaluation

IV Results

IV-A Comparison of Whitney CG-FEM and Mixed-FEM for Optimal Source Positions

IV-B Comparison of Whitney CG-FEM and Mixed-FEM for Random Source Positions

IV-C Comparison of Mixed-FEM Approaches in Leaky Sphere Models

IV-D Realistic Head Model Study

V Discussion and Conclusion

References

head models It is shown that the Mixed-FEM achieves higheraccuracies in solving the EEG forward problem than the CG-FEM for highly eccentric sources in sphere models and thanboth CG- and DG-FEM in realistic head models

II THEORY

APPLYING the quasistatic approximation of Maxwellrsquosequations [18] [19] the forward problem of EEG is com-

monly formulated as a second-order PDE with homogeneousNeumann boundary condition

nabla middot (σnablau) = nabla middot jp in Ω (1a)σpartnu = 0 on partΩ = Γ (1b)

Here u denotes the electric potential jp the source currentand σ the conductivity distribution in Ω In (1) the electriccurrent j is already eliminated as an unknown For our purposewe start at the previous step in the derivation of the quasistaticapproximation and keep the electric current as an unknownThus our starting point is the system of first-order PDEs

j + σnablau = jp (2a)nabla middot j = 0 in Ω (2b)〈jn〉 = 〈jpn〉 on partΩ = Γ (2c)

Since the source current jp in general fulfills 〈jpn〉 = 0 onΓ as supp jp sub Ω for physiological reasons (there are nosources in the skin) (2c) can be simplified to 〈jn〉 = 0 on ΓThe Mixed-FEM formulation for the EEG forward problem isnow derived from (2) instead of discretizing (1) as would bedone for the CG-FEM

A A (Mixed) Weak Formulation of the EEG Forward Problem

Due to the vector-valued equation (2a) it is necessary tointroduce a space of vector-valued test functions to be able toderive a weak formulation of (2) A natural function space forthe current in the mixed formulation is H(div Ω)

H(div Ω) =q isin L2(Ω)3 nabla middot q isin L2(Ω)

Akin to the scalar-valued Sobolev spaces Hk(Ω) this spacebecomes a Hilbert space with the norm

qH(divΩ) =(q2L2(Ω)3 + nabla middot q2L2(Ω)

We introduce a subspace H0(divΩ) of H(div Ω) in whichthe boundary condition 〈jn〉 = 0 on partΩ = Γ is fulfilled byconstruction

H0(divΩ) = q isin H(div Ω) 〈qn〉 = 0 on partΩ (5)

For the scalar-valued equation (2b) one can simply choose thespace of square-integrable functions L2(Ω) as the test space

Now we can introduce a weak formulation of (2)intΩ

〈σminus1jq〉dx minusint

nabla middot qudx =intΩ

〈σminus1jpq〉dx for all q isin H0(divΩ) (6a)

nabla middot jv dx = 0 for all v isin L2(Ω) (6b)

This is the so-called dual mixed formulation [20]ndash[22] TheNeumann boundary condition (2c)(5) is an essential boundarycondition in the (dual) mixed formulation and has to beimposed explicitly in solving the discrete problem We definethe bilinear forms

a(pq) = (σminus1pq)L2(Ω)3 (7a)b(p v) = (nabla middot p v)L2(Ω) (7b)

and the functional

l(q) = (σminus1jpq)L2(Ω)3 (7c)

for pq isin H0(divΩ) v isin L2(Ω) jp isin L2(Ω)3 σ isin Linfin(Ω)σ gt 0 Therefore to solve (6) is to

find (u j) isin L2(Ω)timesH0(divΩ) such that

a(jq) + b(q u) = l(q) for all q isin H0(divΩ) (8a)

b(j v) = 0 for all v isin L2(Ω) (8b)

In this notation the saddle point structure of problem (8) andthus also (6) is recognizable As a consequence the existenceand uniqueness of a solution cannot be shown using theLemma of Lax-Milgram

Instead it can be shown that a solution to (8) exists ifthe operator a is H0(divΩ)-elliptic on the kernel of b andb fulfills an inf-sup condition which in this case is alsocalled the LBB condition named after the mathematiciansLadyzhenskaya Babuska and Brezzi At this point we shallnote only that these conditions are fulfilled by a and bdefined as in (7) and thereby the existence of a solution(u j) isin L2(Ω) times H0(divΩ) is given While the solutionfor j is unique u is defined up to an element of ker b(j v)v isin L2(Ω) [20] [23] Uniqueness for u can be obtained byintroducing an additional condition such as fixing the valueofint

Ωu or

intpartΩu For a detailed proof and discussion we refer

the reader to eg [20] [22] [24]

B Mixed Finite Element Method

Obtaining a numerical solution for (6)(8) necessitateschoosing suitable discrete approximations for the test functionspaces H0(divΩ) and L2(Ω) Utilizing a Galerkin approachthese are also the spaces in which the discrete solution (uh jh)lies

In order to construct the discrete subspaces the volume Ωis subdivided and approximated by a set of simple geometricalobjects In three dimensions these objects are usually tetrahe-dra or hexahedra For the sake of simplicity any subdivisionof Ω into either tetrahedra or hexahedra is henceforth referredto as triangulation T = T1 T2 T3 Tm In this paper wefollow the definition of a triangulation according to [24] Wefurther assume that the triangulation T is admissible [24] andwrite Th instead of T if each element T isin T has a diameterof maximally 2h

for all T isin Th

wk(x) =

0 otherwise(9)

matrix equation

)︸︷︷︸

for vk isin SP0

h wiwj isin SRT0

Aj +BTu = b (13a)Bj = 0 (13b)

Su = h (15)

sumj A

sumj |Aij |

bi = bcuri =

0 otherwise(16)

bpoti =

〈mδx0wi〉dx

0 otherwise(17)

III METHODS

A Implementation

RDM(unum uref ) =

∥∥∥∥ unum

unum2minus uref

∥∥∥∥2

(unum2uref2

) (18)

TABLE IVMODEL LEAKS

IV RESULTS

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0 01 02 03 04 05

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

-02 -01 0 01 02 03 04 05 06

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

for all T isin Th

wk(x) =

0 otherwise(9)

matrix equation

)︸︷︷︸

for vk isin SP0

h wiwj isin SRT0

Aj +BTu = b (13a)Bj = 0 (13b)

Su = h (15)

sumj A

sumj |Aij |

bi = bcuri =

0 otherwise(16)

bpoti =

〈mδx0wi〉dx

0 otherwise(17)

III METHODS

A Implementation

RDM(unum uref ) =

∥∥∥∥ unum

unum2minus uref

∥∥∥∥2

(unum2uref2

) (18)

TABLE IVMODEL LEAKS

IV RESULTS

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0 01 02 03 04 05

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

-02 -01 0 01 02 03 04 05 06

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

Aj +BTu = b (13a)Bj = 0 (13b)

Su = h (15)

sumj A

sumj |Aij |

bi = bcuri =

0 otherwise(16)

bpoti =

〈mδx0wi〉dx

0 otherwise(17)

III METHODS

A Implementation

RDM(unum uref ) =

∥∥∥∥ unum

unum2minus uref

∥∥∥∥2

(unum2uref2

) (18)

TABLE IVMODEL LEAKS

IV RESULTS

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0 01 02 03 04 05

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

-02 -01 0 01 02 03 04 05 06

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

bi = bcuri =

0 otherwise(16)

bpoti =

〈mδx0wi〉dx

0 otherwise(17)

III METHODS

A Implementation

RDM(unum uref ) =

∥∥∥∥ unum

unum2minus uref

∥∥∥∥2

(unum2uref2

) (18)

TABLE IVMODEL LEAKS

IV RESULTS

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0 01 02 03 04 05

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

-02 -01 0 01 02 03 04 05 06

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

TABLE IVMODEL LEAKS

IV RESULTS

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0 01 02 03 04 05

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

-02 -01 0 01 02 03 04 05 06

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

IV RESULTS

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0 01 02 03 04 05

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

-02 -01 0 01 02 03 04 05 06

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0010 0502 0748 0871 0933 0964 0979 0987 0991 0993

Eccentricity

0 01 02 03 04 05

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

-02 -01 0 01 02 03 04 05 06

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

0 01 02 03 04 05

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

-02 -01 0 01 02 03 04 05 06

requen

proj M-FEM 2 mm

Whitney CG-FEM 2 mm

PI DG-FEM 2 mm

Venant CG-FEM 2 mm

Venant CG-FEM 1 mm

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

ACKNOWLEDGMENT

REFERENCES

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

I Introduction

II Theory






III Methods


III-B Evaluation

IV Results






References

Documents

IEEE TRANSACTIONS ON MEDICAL IMAGING 1 A Mixed Finite … · 2017-08-25 · IEEE TRANSACTIONS ON MEDICAL IMAGING 1 A Mixed Finite Element Method to Solve the EEG Forward Problem J