THE COMPLETE ELLIPSOIDAL SHELL-MODEL IN EEG IMAGING

EEG IMAGING

S. N. GIAPALAKI AND F. KARIOTOU

Received 5 December 2004; Accepted 16 December 2004

This work provides the solution of the direct Electroencephalography (EEG) problem for the complete ellipsoidal shell-model of the human head. The model involves four confocal ellipsoids that represent the successive interfaces between the brain tissue, the cerebrospinal fluid, the skull, and the skin characterized by different conductivities. The electric excitation of the brain is due to an equivalent electric dipole, which is located within the inner ellipsoid. The proposed model is considered to be physically complete, since the effect of the substance surrounding the brain is taken into account. The direct EEG problem consists in finding the electric potential inside each conductive space, as well as at the nonconductive exterior space. The solution of this multitransmission prob- lem is given analytically in terms of elliptic integrals and ellipsoidal harmonics, in such way that makes clear the effect that each shell has on the next one and outside of the head. It is remarkable that the dependence on the observation point is not affected by the presence of the conductive shells. Reduction to simpler ellipsoidal models and to the corresponding spherical models is included.

Copyright © 2006 S. N. Giapalaki and F. Kariotou. This is an open access article distrib- uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The method of Electroencephalography (EEG) is the most widely used, noninvasive method for studying the human brain in vivo. The data of an Electroencephalogram are obtained by measuring the electric potentials in the exterior of the head. The in- verse EEG problem consists in determining the location of the electrochemical source inside the brain that produces the externally measured electric potential field. The results obtained from the solution of the forward EEG problem, namely the electric potential field that a given source produces, are of major importance for the inverse problem. The

Hindawi Publishing Corporation Abstract and Applied Analysis

Volume 2006, Article ID 57429, Pages1–18 DOI10.1155/AAA/2006/57429

well-poseness of the mathematical problem demands certain assumptions concerning the physical model approximating the electrochemical source as well as the geometri- cal model used for the brain-head approximation. The most popular model used for the source is that of an equivalent electric dipole current of a given moment.

As far as the geometrical model of the conductor is concerned and for the analytical treatment of the problem, the dominant model for the brain-head system is the one of a homogeneous spherical [3,6,2,13], or a homogeneous spheroidal [2,16] conductor.

The improvement of these models, so that the 3D anisotropy of the system is taken into account, has led to the concept of more realistic volume conductors [12]. Furthermore, the case of a homogeneous ellipsoidal conductor, which fits best to the geometrical char- acteristics of the brain [14], was treated in [8]. Furthermore, the brain is protected by shells consisted of the cerebrospinal fluid, the bone and the skin that are all characterized by different electrical conductivities. As it is expected, this inhomogeneity constitutes an important parameter of the problem that should be taken into account considering lay- ered volume conductors [2]. The case of one confocal ellipsoidal shell, characterised by different conductivity, surrounding the homogeneous ellipsoidal conductor, represents the brain was treated in [7].

In this work, in order to study the effect of inhomogeneity in the measured electric po- tential for the case of ellipsoidal geometry, we assume the physically complete ellipsoidal shell-model. Specifically we consider three confocal ellipsoidal shells, each one character- ized by a different conductivity, which surround the homogeneous ellipsoidal conductor representing the brain. It is observed that the conductivity values, as well as the geomet- rical parameters of the four ellipsoidal boundaries, appear in every term of the multipole expansion of the electric potential, justifying this way the improvement offered by this model.

The postulation of the transmission problem that the electric potential field has to satisfy near the dipole source, in the spaces between the ellipsoidal boundaries and in the exterior space, is presented in Section 2. In Section 3we deal with the solution of this problem using eigenfunction expansions in ellipsoidal coordinates. The solution is expressed in terms of elliptic integrals and ellipsoidal harmonics, while relative expres- sions in Cartesian and in tensorial form are also included. InSection 4the corresponding homogeneous and one shell inhomogeneous ellipsoidal results are recovered through re- duction process and also the electric potential for the two confocal ellipsoidal shell model is provided. The corresponding manipulations needed for the reduction to the spherical- shell model are given inSection 5.

2. Statement of the problem

Following anatomic structure we model the head as an ellipsoid, occupied by the brain, which is surrounded by three confocal ellipsoidal shells, which are filled, starting from the inside, with the cerebrospinal fluid, the skull and the skin. From the physical point of view the above compartments of this realistic model of the head are distinguished by their different values of electric conductivity.

LetSs,Sb,Sf, andScdenote the triaxial ellipsoidal surfaces, which in rectangular coor- dinates are specified by

x²₁ s²₁ +x²₂

s²₂ +x²₃

s²₃ ⁼1, 0< s3< s2< s1<+∞, (2.1) x²₁

b²1

+x²₂ b²2

+x²₃

b²3 =1, 0< b3< b2< b1<+∞, (2.2) x²₁

f₁²+ x²₂ f₂²+ x²₃

f₃^{2 =}1, 0< f3< f2< f1<+∞, (2.3) x²₁

c²₁ +x²₂ c²₂ +x²₃

c₃^{2 =}1, 0< c3< c2< c1<+∞, (2.4) respectively, whereci< fi< bi< si,i=1, 2, 3, are their semiaxes. The ellipsoids (2.1), (2.2), (2.3), (2.4) are confocal and correspond to the ellipsoidal systemρ,μ,ν[5] with semifocal distancesh1,h2,h3, where

h²₁=s²₂−s²₃=b²₂−b²₃= f₂²−f₃²=c²₂−c²₃, h²₂=s²₁−s²₃=b²₁−b²₃= f₁²−f₃²=c²₁−c²₃, h²₃=s²₁−s²₂=b²₁−b²₂= f₁²−f₂²=c²₁−c²₂.

(2.5)

The ellipsoidal coordinatesρ,μ,νare connected to the Cartesian onesx1,x2,x3 by the relations [5]

x1= ρμν h2h3

,

x2=

ρ²−h²3

μ²−h²3

h²3−ν² h1h3

,

x3=

ρ²−h²2

h²2−μ²h²2−ν²

h1h2 ,

(2.6)

and vary in the intervals [h2, +∞), [h3,h2], and [−h3,h3], respectively.

In terms of the variableρ, the surfacesS_s,S_b,S_f, andS_ccorrespond toρ=s1,ρ=b1, ρ= f1, andρ=c1 and represent the boundaries of the skin, the scull (bone), the fluid and the cerebrum, respectively. The interior toScspaceVc corresponds to the interval ρ∈[h2,c1) and is characterized by the conductivityσc. The ellipsoidal shell betweenSc

andS_f, denoted byV_f, corresponds to the intervalρ∈(c1,f1) and is characterized by the conductivityσf. The ellipsoidal shell betweenSf andSb, denoted byVb, corresponds to the intervalρ∈(f1,b1) and is characterized by the conductivityσb. Finally, the ellipsoidal shell bounded byS_b and S_s, is denoted by V_s, corresponds to the intervalρ∈(b1,s1) and is characterized by the conductivityσs. The exterior toSsnonconductive spaceV is described byρ∈(s1, +∞).

At the point r0∈Vcthere exists a primary current dipole source with moment Q. This is specified by the current density function

J^P(r)=Qδr−r0

, (2.7)

whereδstands for the Dirac measure at the point r0.

The primary current J^P induces an electric field E in the interior conductive space, which in turn generates an induction current with density J^V:

J^V(r)=σ_cEc(r)X_Vc(r) +σ_fEf(r)X_Vf(r) +σ_bEb(r)X_Vb(r) +σ_sEs(r)X_Vs(r), (2.8) whereXA(r) denotes the characteristic function of the setA.

Hence, the total current at every point r of the conductor is given by

J(r)=J^P(r) + J^V(r). (2.9)

The current J generates an electromagnetic field, which propagates in the interior as well as in the exterior of the conductive space.

Because of the values of the dielectric constant and the electric conductivity of the brain tissue, the quasistatic approximation of Maxwell’s equations is considered [4,9,13, 15]. Therefore the electric field E and the magnetic induction field B satisfy the following equations [9]:

∇ ×E=0, (2.10)

∇ ×B=μ0J, (2.11)

∇ ·E=0, (2.12)

∇ ·B=0, (2.13)

whereμ0denotes the magnetic permeability in the whole space.

Since E is irrotational, it can be represented by an electric potentialu, via the differen- tial representation

E(r)= −∇u(r). (2.14)

The electric potentialuis the field recorded in any electroencephalogram. In particular, we denote the electric potential in the interior spaceVcbyuc, in the ellipsoidal shellVf

byu_f, in the ellipsoidal shellV_bbyu_b, in the ellipsoidal shellV_sbyu_sand in the exterior spaceV byu. Combining (2.9), (2.14), and (2.11), we obtain the Poisson equation

Δuc(r)= 1

σ_c^{∇ ·}J^P(r), r∈V_c, (2.15) which the interior potentialucmust satisfy inVc.

In the source-free spacesVf,Vb,Vs, andV the potentialsuf,ub,us, andusolve the Laplace equation

Δuf(r)=0, r∈V_f, (2.16)

Δub(r)=0, r∈Vb, (2.17)

Δus(r)=0, r∈Vs, (2.18)

Δu(r)=0, r∈V. (2.19) On the surface S_cthe following transmission conditions hold

uf(r)=uc(r), r∈Sc, (2.20)

σ_f∂_nu_f(r)=σ_c∂_nu_c(r), r∈S_c, (2.21) where the∂_nindicates the outward normal differentiation. Conditions (2.20)-(2.21) state the continuity of the potential function as well as the continuity of the normal component of the electric field onSc.

On the surfaceS_f we demand that

ub(r)=uf(r), r∈Sf, (2.22)

σb∂nub(r)=σf∂nuf(r), r∈Sf (2.23) and similarly onS_b,

ub(r)=us(r), r∈Sb, (2.24)

σ_b∂_nu_b(r)=σ_s∂_nu_s(r), r∈S_b. (2.25) SinceVis characterized by zero conductivity, on the surfaceSsthe continuity conditions read

u_s(r)=u(r), r∈S_s, (2.26)

∂nus(r)=0, r∈Ss. (2.27)

In addition the asymptotic behavior at infinity u(r)=O

1 r

, r−→ ∞, (2.28)

has to be imposed in order to insure uniqueness.

3. The interior and exterior electric potential

The basic notation for the spectral decomposition of the Laplace operator in ellipsoidal coordinates can be found in [1,5,7,8], where all interiorE^mn(ρ,μ,ν) and exteriorF^mn(ρ,μ, ν) ellipsoidal harmonics that are used in this work, as well as useful relations connecting them, can be found. We recall the definition

F^mn(ρ,μ,ν)=(2n+ 1)I_n^m(ρ)E^mn(ρ,μ,ν)=(2n+ 1)I_n^m(ρ)E^m_n(ρ)E^m_n(μ)E^m_n(ν) (3.1)

which connects the ellipsoidal exterior harmonicsF^m_n(r) to the interior ellipsoidal har- monicsE^mn(r) via the elliptic integrals

I_n^m(ρ)= _∞

ρ

dt

E^m_n(t) ²t²−h²₂t²−h²₃, (3.2) whereE^m_n(x) are the Lam´e functions of the first kind.

The solution of (2.19), is an exterior harmonic function which assumes the exterior ellipsoidal expansion

u(ρ,μ,ν)= ∞ n=0

2n+1

m=1

f_n^mF^m_n(ρ,μ,ν), ρ > s1, (3.3) and satisfies automatically the asymptotic condition (2.28).

Inside the ellipsoidal shellsVs,Vb,Vf the electric potentialsus,ub,uf solve (2.18), (2.17), (2.16), respectively, and therefore they assume the following ellipsoidal expansions

us(r)= ∞ n=0

2n+1

m=1

g_n^mE^m_n(ρ,μ,ν) +h^m_nF^m_n(ρ,μ,ν) , b1< ρ < s1,

ub(r)= ∞ n=0

2n+1

m=1

k_n^mE^m_n(ρ,μ,ν) +m^m_nF^m_n(ρ,μ,ν) , f1< ρ < b1,

uf(r)= ∞ n=0

2n+1

m=1

p^m_nE^m_n(ρ,μ,ν) +q^m_nF^m_n(ρ,μ,ν) c1< ρ < f1.

(3.4)

Finally, in the interior spaceVc, which includes the primary source J^p, the interior elec- tric potentialub solves (2.15), and it is given as a superposition of an interior harmonic functionΦ(r) and the particular solution of Poisson’s equation

V(r)= − 1

4πσ_cQ· ∇r 1

r−r0= 1

4πσ_cQ· ∇r0

r−1r0. (3.5) Using the ellipsoidal expansion for the interior harmonic functionΦ(r),

Φ(r)= ∞ n=0

2n+1

m=1

t^m_nE^m_n(ρ,μ,ν), (3.6) we can write the interior electric potential as

u_c(r)= 1

4πσ_cQ· ∇r0

r−1r0+ ∞ n=0

2n+1

m=1

t^m_nE^m_n(ρ,μ,ν), ρ < c1. (3.7) The ellipsoidal expansion of the fundamental solution of the Laplace operator forρ > ρ0

is given in [11] by

r−1r0= ∞ n=0

2n+1

m=1

4π 2n+ 1

1 γ^m_n^E

mn

ρ0,μ0,ν0

F^mn(ρ,μ,ν), (3.8)

whereγ^m_n are the normalization constants of the surface ellipsoidal harmonics. Applying properly the gradient operator on (3.8), we obtain the following form foruc:

uc(r)=t₀¹+ ∞ n=1

2n+1

m=1

t_n^m+ 1

σcγ_n^m

Q· ∇r0E^m_n

ρ0,μ0,ν0 I_n^m(ρ)

E^m_n(ρ,μ,ν). (3.9)

In (3.9) we have further expressed the exterior ellipsoidal harmonics in terms of the cor- responding interior ones, by means of the elliptic integralI_n^m. Expansion (3.9) holds for ρ > ρ0, therefore it holds true on all boundariesSc,Sf,Sb,Ss. In (3.3), (3.4) and (3.9) we have expressed all the potentials in terms of ellipsoidal harmonics and therefore the ap- plication of the transmission conditions (2.20)–(2.27) is straightforward. Furthermore, the homogeneity of (2.21), (2.23), (2.25), and (2.27) in the operator∂nallows for the re- placement of the normal derivative∂_nwith theρ-derivative∂_ρ, since the corresponding metric coefficient cancels out.

Introducing (3.3), (3.4), and (3.9) in the boundary conditions (2.20)–(2.27) and using the orthogonality property of the surface ellipsoidal harmonics, the constants f_n^m,g_n^m,h^m_n, k_n^m,m^m_n,p^m_n,q_n^m,t_n^mare determined as the solutions of a 8×8 linear algebraic system. Long but straightforward calculations, which are not shown here, lead to the expressions:

g0¹=k0¹=p¹0=t0¹=f0¹I0¹

s1

,

h¹₀=m¹₀=q¹₀=0 (3.10)

while forn=1, 2,. . .,m=1, 2,. . ., 2n+ 1, particular expressions for the eight sequences of constants f_n^m,g_n^m,h^m_n,k_n^m,l^m_n,p^m_n,q^m_n,t_n^m, which contain the Lam´e functions of the first and the second kind evaluated at specific points are obtained. Introducing the notation

I_n^m(x,y)=I_n^m(x)−I_n^m(y)= _y

x

dt E_n^m(t) ²

t²−h²2

t²−h²3

, (3.11)

S^m_n =E^m_ns1

E_n^ms1

s2s3, (3.12)

B^m_n =E^m_nb1 E^m_nb1

b2b3, (3.13)

F_n^m=E^m_nf1

E^m_nf1

f2f3, (3.14)

C^m_n =E^m_nc1

E^m_nc1

c2c3, (3.15)

where the prime denotes differentiation with respect to the variable and using the cor- responding values of the coefficients in (3.3), (3.4), and (3.9) we obtain the following expressions for the potential fieldsu,us,ub,uf, anducwhich hold true in the indicated regions. In particular for the exterior space we obtain

u(r)=g0¹

I₀¹(ρ) I0¹

s1+ ∞ n=1

2n+1

m=1

I_n^m(ρ) I_n^ms1

1 S^m_n

1 G^m_3,n

Q· ∇E^mn

r0

γ^m_n ^E

mn(ρ,μ,ν), ρ > s1, (3.16)

whereg₀¹is an arbitrary constant. For the skin region we obtain u_s(r)=us1

+ ∞ n=1

2n+1

m=1

I_n^mρ,s1

1 G^m_3,n

Q· ∇E^m_n r0

γ^m_n ^E

mn(ρ,μ,ν), b1< ρ < s1, (3.17) for the scull region

u_b(r)=u_sb1

+ ∞ n=1

2n+1

m=1

I_n^mρ,b1

1 σ_b

G^m_1,n G^m_3,n

Q· ∇E^m_n r0

γ^m_n ^E

mn(ρ,μ,ν), f1< ρ < b1, (3.18) for the fluid region

uf(r)=ub

f1

+ ∞ n=1

2n+1

m=1

I_n^mρ,f1

1 σ_f

G^m_2,n G^m_3,n

Q· ∇E^mn

r0

γ_n^{m E}

mn(ρ,μ,ν), c1< ρ < f1, (3.19) and finally for the region occupied by the cerebrum we obtain

uc(r)=uf

c1

+ ∞ n=1

2n+1

m=1

I_n^mρ,c1

1 σc

Q· ∇E^m_n r0

γ_n^{m E}

mn(ρ,μ,ν), ρ < c1. (3.20) The constantsG^m_1,n,G^m_2,nandG^m_3,nare given by

G^m_1,n=σ_b+σ_b−σ_sI_n^mb1,s1

+ 1 S^m_n ⁻

1 B_n^m

B^m_n, (3.21)

G^m_2,n=σf+σf−σb

I_n^mf1,s1

+ 1 S^m_n ⁻

1 F_n^m

F_n^m+σb−σs

I_n^mb1,s1

+ 1 S^m_n ⁻

1 B_n^m

B_n^m

+

σf−σb

σb−σs

σb I_n^mf1,b1

I_n^mb1,s1

+ 1 S^m_n ⁻

1 B^m_n

B_n^mF_n^m,

(3.22) G^m_3,n=σ_c+σ_c−σ_fI_n^mc1,s1

+ 1 S^m_n ⁻

1 C^m_n

C_n^m

+σ_f−σ_bI_n^mf1,s1

+ 1 S^m_n ⁻

1 F_n^m

F_n^m+σ_b−σ_sI_n^mb1,s1

+ 1 S^m_n ⁻

1 B_n^m

B^m_n

+

σc−σf

σf−σb

σ_f I_n^mc1,f1

I_n^mf1,s1

+ 1 S^m_n ⁻

1 F_n^m

F_n^mC^m_n

+

σc−σf σb−σs

σb I_n^mc1,b1

I_n^mb1,s1 + 1

S^m_n ⁻ 1 B^m_n

B_n^mC^m_n

+

σf−σb σb−σs

σb I_n^mf1,b1

I_n^mb1,s1

+ 1 S^m_n ⁻

1 B^m_n

B_n^mF_n^m

+

σc−σf

σf−σb σb−σs σfσb

×I_n^mc1,f1

I_n^mb1,s1

+ 1 S^m_n ⁻

1 B^m_n

I_n^mf1,b1

− 1 F_n^m

B_n^mF_n^mC^m_n.

(3.23)

In trying to interpret (3.16) to (3.20) we observe the following. Expression (3.16) pro- vides the electric potential at any point outside the conductor. Then the potential within the outmost shell is expressed as the exterior potentialuevaluated on the surfaceS_sof the skin, plus an expansion evaluated at the observation point r, which represents the con- tribution that comes from the shellVs. In a similar fashion, the potentials (3.17)–(3.19) within the following succesive shells, as well as the potential (3.20) inside the cerebrum region, are expressed as the potential of the exterior shell evaluated at their common boundary plus a contribution from the particular shell, always in the form of the appro- priate eigenfunction expansion.

Furthermore, the form of each one of these expansions remains the same. They only differ by the constant ratios involving the conductivity profiles and by the fact that the corresponding elliptic integrals are evaluated on different surfaces. The above ratios spec- ify the effect of the surrounding shells normalized by the effect of all shells considered in the model. Each ratio is multiplied by a conductivity factor which is what the equivalent homogeneous conductor would impose to the exterior electric potential.

It is worth noticing though that the part of the solution which is depended on the location of the observation point remains unaltered by the presence of the shells.

In the sequel we are going to work further on the expression (3.16), since the exterior potential is what it is registered on an electroencephalogram. Therefore, elaborating fur- ther on (3.16) by using the interior Lam´e functions and the interior ellipsoidal harmonics in terms of the more tractable Cartesian coordinates and by calculating the action of the gradient onE^m_n and onE^m_n, we obtain the following analytic form ofuexpressed in Carte- sian coordinates and elliptic integrals

u(ρ,μ,ν)

=g₀¹I₀¹(ρ) I0¹

s1

+ 3 4πs1s2s3

3

m=1

Qmxm

G^m_3,1 I₁^m(ρ) I₁^ms1

− 5

8πs1s2s3

Λs−Λs

3

m=1

Q_mx_0m 1

G¹3,2

I₂¹(ρ) I2¹

s1

E¹2(r) Λs

Λs−s²_m⁻ 1 G²3,2

I₂²(ρ) I2²

s1

E²2(r) Λs

Λs−s²_m

+ 15

4πs1s2s3

3

i,j=1 i=j

Q_ix_0jx_ix_j G⁶3,2^−i−j

s²_i+s²_j I₂^i+j(ρ) I2^i+j

s1

+Oel3

.

(3.24) The notationO(el3) in (3.24) denotes ellipsoidal terms of degree greater or equal to three.

The constants

Λs

Λs

=s²₁−1 3

h²₂+h²₃±

h⁴₁+h²₂h²₃

(3.25) satisfy the equation

3

m=1

1

Λs−s²_m ⁼0 (3.26)

and generate the constant dyadics Λs= 1

G¹_3,2 3

m=1

xm⊗xm

Λs−s²_m , Λs= 1

G²_3,2 3

m=1

xm⊗xm

Λ_s−s²_m.

(3.27)

Furthermore, in terms of the dyadic fields A(ρ) = 3

4πs1s2s3

3

m=1

1 G^m_3,1

I₁^m(ρ) I₁^ms1

x_m⊗x_m, B(r) = − 5

8πs1s2s3

Λs−Λ_s I2¹(ρ)

I₂¹s1

Λs

ΛsE¹2(r)− I2²(ρ) I₂²s1

Λs

ΛsE²2(r)

(3.28)

and the tetradic field

Γ(ρ)= 15 4πs1s2s3

3

i,j=1 i=j

1 G⁶3,2^−i−j

I₂^i+j(ρ) I2^i+j

s1

xi⊗xj⊗xi⊗xj

s²_i+s²_j (3.29)

we rewrite the electric field as u(r)=g₀¹I₀¹(ρ)

I₀¹s1+ Q·A·r + Q⊗r0:B(r) + Q ⊗r0:Γ(ρ) : r ⊗r +Oel3

, (3.30)

where the double contraction is defined by

a⊗b : c⊗d=(a·c)(b·d). (3.31) The use of the polyadic notation in expressing the exterior electric potential offers the ad- vantage of a unified and compact form in which the source enters in a distinctive and clear way. In fact, the polyadic fieldsA(ρ), B(r), Γ(ρ) include all the geometric and physical characteristics of the conductor while the moment and position of the source is obtained from them via simple and double contraction.

4. Physical degeneracies

Our purpose here is to recover from results (3.16), (3.17), (3.18), (3.19), and (3.20) for the electric potential fields in the four compartment ellipsoidal model, the corresponding results for the one shell model [7]. In the notation of the present work the corresponding results read as

u1(r)=g0¹

I₀¹(ρ) I0¹

s1+ ∞ n=1

2n+1

m=1

I_n^m(ρ) I_n^ms1

1 S^m_n

1 G^m_1,n

Q· ∇E^mn

r0

γ_n^{m E}

mn(ρ,μ,ν) (4.1)

forρ > s1,

u1,s(r)=u1

s1

+ ∞ n=1

2n+1

m=1

I_n^mρ,s1

1 G^m_1,n

Q· ∇E^mn

r0

γ^m_n ^E

mn(ρ,μ,ν) (4.2) forc1< ρ < s1and

u1,c(r)=u1,s c1

+ ∞ n=1

2n+1

m=1

I_n^mρ,c1

1 σc

Q· ∇E^m_n r0

γ^m_n ^E

mn(ρ,μ,ν) (4.3) forρ0< ρ < c1, wheres1,c1appear in (2.1) and (2.4), respectively.

In (4.1), g₀¹ is an arbitrary constant and the rest of the notation in (4.1), (4.2), and (4.3) remain identical with the present work. In order to reduce the three shells-ellipsoidal model to the one shell-ellipsoidal model we need to unify appropriately the spacesVs,Vb, V_f andV_c. This is obtained by the following three options. One corresponds to taking the limits

σ_f −→σ_b−→σ_s (4.4)

while the conductivity of the core remainsσ_c. The second choice corresponds to

σb−→σs, σf −→σc (4.5)

and the third one is obtained by

σ_b−→σ_f −→σ_c (4.6)

while we preserve the conductivity of the outer boundary to beσ_s.

Whichever of these three settings we choose, the results for the one shell-ellipsoidal model are recovered. Indicatively we select the first alternative, which geometrically cor- responds to

fi−→bi−→si, i=1, 2, 3. (4.7) and it is denoted by 3sh→1sh. As a consequence, from (3.21) we obtain

3shlim_→1sh

G^m_1,n=σs (4.8)

while from (3.22) we obtain

3shlim→1sh

G^m_2,n=σ_s (4.9)

and finally, in view of (3.23),

3shlim→1sh

G^m_3,n=σ_c+σ_c−σ_sI_n^mc1,s1

+ 1 S^m_n ⁻

1 C_n^m

C_n^m (4.10)

which is the conductivity term for the one shell model.