the method of fundamental solutions in solving coupled boundary value problems...

The Method of Fundamental Solutions in SolvingCoupled Boundary Value Problems for EEG/MEG

Salvatore [email protected]

DEIM, University of Palermo, Italy

Joint work with G. Ala, G. Fasshauer, E. Francomano and M. McCourt

MAIA 2013Erice, 25-30 September, 2013

S. Ganci The Method of Fundamental Solutions in Solving Coupled Boundary Value Problems for EEG/MEG 0 / 21

Outline

1 Problem Formulation

2 State of the Art and Motivation

3 Methodology

4 Numerical Results

5 Conclusions


Problem Formulation

Background

What are EEG and MEG?EEG and MEG are two electromagnetic techniques for brain activityinvestigation, i.e. to locate active neural sources

How they work?Neural sources (location and amplitude) are reconstructed starting frommeasurements of electric potential on the scalp (EEG) or magnetic fieldnear the head (MEG). This is a typical inverse problem.

What is needed to perform them?

1 A data set (measurements)2 An inverse algorithm3 An efficient and accurate forward solver


Problem Formulation

Model for the Head

Brain (Ω1) Skull (Ω2)

Scalp (Ω3)

Figure 1 : Compartment modelfor the head

The head can be modeled as a linear,piecewise homogeneous, volume con-ductor domain Ω ⊂ R3 formed by Lnested layers.Let p be a point in Ω.

A model with three layers (L = 3) iscommon: brain, skull and scalp.

Let Ω` and ∂Ω` be the `-th layer in thedomain Ω, with known conductivity σ`,and its boundary, respectively.

The medium surrounding the head is theair and it can be considered as an un-bounded region of null electrical conduc-tivity.


Problem Formulation

Electromagnetic Modeling of the Brain Activity

The forward problem for the electric potential φ(p) can be formulatedas the following BVP:

σ`(p)∇2φ(p) = S`(p), p ∈ Ω`

φ(p−) = φ(p+), p ∈ ∂Ω` ∩ ∂Ω`+1

σ`n(p) · ∇φ(p−) = σ`+1n(p) · ∇φ(p+), p ∈ ∂Ω` ∩ ∂Ω`+1

where:

S`(p) =

∇ · (Qδ(p− p′)) neural source in p′ ∈ Ω`

0 otherwise

n(p) is the outward unit vector normal to the interface ∂Ω` ∩ ∂Ω`+1

at p

p− and p+ are limit points for two spatial sequences converging to pfrom inside and from outside the interface, respectively


Problem Formulation

Electromagnetic Modeling of the Brain Activity

The forward problem for the magnetic field can be formulated startingfrom the forward problem for the electric potential.

In fact, the Maxwell’s equations yield:

∇2B(p) = −µ∇× J(p) (1)

where B(p) is the magnetic induction, µ is the permeability of the mediumand the current density J(p) is known once φ(p) is known.

The solution of (1) with condition of null magnetic field at infinite distancefrom sources, is known as Ampere-Laplace law [Sarvas (1987)]:

B(p) = − µ

4π

∫Ωσ(p∗)∇φ(p∗)× p− p∗

‖p− p∗‖3dΩ∗


State of the Art and Motivation

State of the Art

Finite Element Method

Domain method → 3D meshes

Very costly

Boundary Element Method

Boundary method → 2D meshes

Comparable to FEM in accuracy [Adde et al. (2003)]

Implemented in popular toolboxes for EEG/MEGanalysis, e.g. FieldTrip [Oostenveld et al. (2011)],Brainstorm [Tadel et al. (2011)].



Motivation

Drawbacks of the state of the art solvers:

1 High quality meshes are needed to avoid mesh-related artifacts inreconstructed neural activation patterns

2 Mesh generation is a complex and time consuming pre-processingtask, even with automatic algorithms

3 Numerical integration is required and turns out to be the dominatingcomputational task in the process

4 Complex computer codes (not flexible).

What could be done to overcome these difficulties?



Motivation

The Method of Particular Solutions (MPS) allows for the application of theMethod of Fundamental Solutions (MFS)

1 Boundary-type method, like BEM

2 No meshing is required: ability to handle complex geometries in aneasy way

3 No numerical integration is required

4 Accuracy: potential for exponential convergence with smooth data anddomains [Cheng (1987); Katsurada (1994); Katsurada and Okamoto(1996)]

5 Flexibility: easy implementation


Methodology

The underlying idea

The MFS is a kernel-based method, introduced during 60’s [Kupradze andAleksidze (1964a,b); Kupradze (1967)]Let’s consider a homogeneous elliptic PDE of the form:

Lu(p) = 0, p ∈ Ω ⊆ R3 (2)

Like BEM, MFS is applicable when a fundamental solution of the PDE isknown.

Definition – Fundamental solution

A fundamental solution of the PDE (2) a function K(p, q) such thatLK(p, q) = −δ(p− q), p, q ∈ R3

q is called the singularity point (or source point) of the fundamental solutionsince K is defined everywhere except there, where it is singular.


Methodology

The underlying idea

The idea of the MFS is to estimate the solution by means of a linear com-bination of fundamental solutions of the governing PDE:

u(p) ≈ u(p) =

#Ξ∑j=1

cjK(p, ξj), p ∈ Ω, ξj ∈ Ξ (3)

were Ξ is a set of source points placed on a fictitious boundary outsidethe physical domain.The coefficients cj have to be determined by imposing (3) to satisfy theboundary conditions:

T u(p) = f∂Ω(p), p ∈ ∂Ω

at a set P of collocation points:

#Ξ∑j=1

cjTK(pi, ξj) = f∂Ω(pi) pi ∈ P, ξj ∈ Ξ (4)


Methodology

Inhomogeneous problems

Let’s consider an inhomogeneous BVP of the form:Lu(p) = fΩ(p), p ∈ Ω ⊆ R3

T u(p) = f∂Ω(p), p ∈ ∂Ω(5)

It can be reduced to a homogeneous problem by the Method of Particular Solutions(MPS), i.e. by splitting u into a particular solution up and its associated homogeneoussolution uh:

u = uh + up

Definition – Particular solution

A particular solution of the BVP (5) is a function up on Ω ∪ ∂Ω which satisfies theinhomogeneous PDE but not necessarily the boundary conditions.

Then we get the homogenous BVP:Luh(p) = fΩ(p)− Lup(p) = 0, p ∈ Ω

T uh(p) = f∂Ω(p)− T up(p), p ∈ ∂Ω


Methodology

Application to the EEG potential problem

Let’s apply the MFS, via MPS, to the EEG potential problem.

1 The fundamental solution for the Laplace equation in 3D is:

K(p, q) =1

4π‖p− q‖

2 An analytical expression for a function φp(p) that satisfies the equation

σ∇2φ(p) = ∇ · (Qδ(p− p′))

in an unbounded domain is known [Sarvas (1987)]:

φp(p) =1

4πσ

p− p′

‖p− p′‖3·Q


Methodology


Brain (Ω1) Skull (Ω2)

Scalp (Ω3)

The EEG potential problem in Ω can be addressed byconsidering a number L of coupled BVPs interactingthrough the boundary conditions.By introducing the parameter

α` =

1, neural source in Ω`

0, otherwise

the potential function in each layer can be expressed as:

φ`(p) = φh,`(p) + α`φp,`(p)

φh,` is given by the solution of the following homogeneous BVP:∇2φh,`(p) = 0, p ∈ Ω`

φh,`(p)− φh,`+1(p) = α`+1φp,`+1(p)− α`φp,`(p), p ∈ ∂Ω` ∩ ∂Ω`+1

σ`n(p) · ∇φh,`(p)− σ`+1n(p) · ∇φh,`+1(p) =

= α`+1σ`+1n(p) · ∇φp,`+1(p)− α`σ`n(p) · ∇φp,`(p) p ∈ ∂Ω` ∩ ∂Ω`+1


Methodology


The homogeneous solution is approximated by:

φh,`(p) =

#Ξ∑j=1

c`jK(p, ξj), p ∈ Ω`, ξj ∈ Ξ` (6)

where Ξ` is the set of source points relative to the layer Ω`.

In order to estimate the L sets of coefficients c`L`=1, the collocationhas to be performed on each interface.

Let PD`,`+1 and PN`,`+1 be the sets of collocation points on the interfacebetween the layer ` and the layer `+ 1 where Dirichlet conditions andNeumann conditions, respectively, are intended to be imposed.


Methodology

Application to the EEG potential problemThe collocation yields the following coupled linear system:

D11,2 D2

1,2

N11,2 N2

1,2

D22,3 D3

2,3

N22,3 N3

2,3

. . .

D``,`+1 D`+1

`,`+1

N``,`+1 N`+1

`,`+1

. . .

DL−1L−1,L DL

L−1,L

NL−1L−1,L NL

L−1,L

NLL,L+1

c1

c2

.

.

.c`

.

.

.cL−1

cL

=

bD1,2bN1,2bD2,3bN2,3...

bD`,`+1

bN`,`+1

.

.

.bDL−1,L

bNL−1,L

bNL,L+1

(D``,`+1)ij = K(pi, ξj) pi ∈ P

D`,`+1, ξj ∈ Ξ`

(D`+1`,`+1

)ij = −K(pi, ξj) pi ∈ PD`,`+1, ξj ∈ Ξ`+1

(N``,`+1)ij = σ`n(pi) · ∇K(pi, ξj) pi ∈ P

N`,`+1, ξj ∈ Ξ`

(N`+1`,`+1

)ij = −σ`+1n(pi) · ∇K(pi, ξj) pi ∈ PN`,`+1, ξj ∈ Ξ`+1

(bD`,`+1)i = α`+1φp,`+1(pi)− α`φp,`(pi) pi ∈ P

D`,`+1

(bN`,`+1)i = α`+1σ`+1n(pi) · ∇φp,`+1(pi)− α`σ`n(pi) · ∇φp,`(pi) pi ∈ P

N`,`+1


Numerical Results Homogeneous Sphere

Simulation data

Homogeneous sphereAnalytical solution in [Yao (2000)]Problem data:

Sphere radius: 0.1 m

Conductivity: 0.2 S/m

Source position: (0, 0, 0.06) [m]

Source moment: (1, 0, 0) [Am]

Simulations with different ratios RMFS between the no. source points andthe number of collocation points



Convergence

103

10-12

10-10

10-8

10-6

10-4

10-2

100

Current Dipole in Homogeneus Sphere - Convergence test

Total Collocation Points or Elements

Rel

ativ

e E

rro

r o

n S

urf

ace

Po

ten

tial

Symmetric BEMMFS, R

MFS =0.8

MFS, RMFS

=0.4

MFS, RMFS

=0.2



Cost vs. Relative Error

10-12

10-10

10-8

10-6

10-4

10-2

100

10-2

10-1

100

101

102

103

Current Dipole in Homogeneus Sphere - Cost per Accuracy

Relative Error on Surface Potential

CP

U T

ime

[s]

Symmetric BEMMFS, R

MFS =0.8

MFS, RMFS

=0.4

MFS, RMFS

=0.2


Numerical Results Three layered Sphere

Simulation data

Three layered sphereSemi-analytical solution in [Zhang (1995)]Problem data:

Sphere radii: R1 = 0.087 m, R2 = 0.092 m, R3 = 0.1 m

Conductivities: σ1 = 0.33 S/m, σ2 = 0.0125 S/m, σ3 = 0.33 S/m

Source position: (0, 0, 0.052) [m]

Source moment: (1, 0, 0) [Am]



Convergence

103

10-5

10-4

10-3

10-2

10-1

100

Current Dipole in Three Layered Sphere - Convergence test

Total Collocation Points or Elements

Rel

ativ

e E

rro

r o

n S

urf

ace

Po

ten

tial

Symmetric BEMMFS, R

MFS =0.8

MFS, RMFS

=0.4

MFS, RMFS

=0.2



Cost vs. Relative Error

10-5

10-4

10-3

10-2

10-1

100

101

102

Current Dipole in Three Layered Sphere - Cost per Accuracy

Relative Error on Surface Potential

CP

U T

ime

[s]

Symmetric BEMMFS, R

MFS =0.8

MFS, RMFS

=0.4

MFS, RMFS

=0.2


Conclusions

Conclusions

1 The MFS via MPS has been proposed to address the EEG/MEG forwardproblem

2 This permits to get rid of complex and time consuming meshing algo-rithms, mesh related artifacts and troublesome numerical integration

3 The implementation of the presented method is straightforward: unlikeBEM solvers, the code is very flexible

4 Simulations results for simplified head geometries showed:

very good agreement with (semi)analytic solutionsclear superiority with respect to BEM from a cost vs. accuracystandpoint


References

References I

Adde, G., Clerc, M., Faugeras, O., Keriven, R., Kybic, J., and Papadopoulo,T. (2003). Symmetric bem formulation for the m/eeg forward problem.In Taylor, C. and Noble, J., editors, Information Processing in MedicalImaging, volume 2732 of Lecture Notes in Computer Science, pages 524–535. Springer Berlin Heidelberg.

Cheng, R. (1987). Delta-trigonometric and spline methods using the single-layer potential representation. PhD thesis, University of Maryland.

Katsurada, M. (1994). Charge simulation method using exterior mappingfunctions. Japan journal of industrial and applied mathematics, 11(1):47–61.

Katsurada, M. and Okamoto, H. (1996). The collocation points of thefundamental solution method for the potential problem. Computers &Mathematics with Applications, 31(1):123–137.


References

References II

Kupradze, V. (1967). On the approximate solution of problems in mathe-matical physics. Russian Mathematical Surveys, 22:58–108.

Kupradze, V. and Aleksidze, M. (1964a). A method for the approximatesolution of limiting problems in mathematical physics. U.S.S.R. Compu-tational Mathematics and Mathematical Physics, 4:199–205.

Kupradze, V. and Aleksidze, M. (1964b). The method of functional equa-tions for the approximate solution of certain boundary value problems.U.S.S.R. Computational Mathematics and Mathematical Physics, 4:82–126.

Oostenveld, R., Fries, P., Maris, E., and Schoffelen, J. (2011). Fieldtrip:open source software for advanced analysis of meg, eeg, and invasiveelectrophysiological data. Computational intelligence and neuroscience,2011:1.

Sarvas, J. (1987). Basic mathematical and electromagnetic concepts of thebiomagnetic inverse problem. Physics in Medicine and Biology, 32(1):11.


References

References III

Tadel, F., Baillet, S., Mosher, J., Pantazis, D., and Leahy, R. (2011). Brain-storm: a user-friendly application for meg/eeg analysis. Computationalintelligence and neuroscience, 2011:8.

Yao, D. (2000). Electric potential produced by a dipole in a homoge-neous conducting sphere. IEEE Transactions on Biomedical Engineering,47(7):964–966.

Zhang, Z. (1995). A fast method to compute surface potentials generatedby dipoles within multilayer anisotropic spheres. Physics in Medicine andBiology, 40(3):335.


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