MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2006; 29:767–787Published online 8 December 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mma.709MOS subject classi�cation: 35B 27; 35C 20; 78A 70

On a hierarchy of models for electrical conductionin biological tissues

M. Amar1;∗;†, D. Andreucci1, P. Bisegna2 and R. Gianni1

1Dipartimento di Metodi e Modelli Matematici; Universit�a di Roma ‘La Sapienza’; Via A. Scarpa 16;00161 Roma; Italy

2Dipartimento di Ingegneria Civile; Universit�a di Roma ‘Tor Vergata’; Via del Politecnico 1;00133 Roma; Italy

Communicated by P. Colli

SUMMARY

In this paper we derive a hierarchy of models for electrical conduction in a biological tissue, whichis represented by a periodic array of period ” of conducting phases surrounded by dielectric shells ofthickness ”� included in a conductive matrix. Such a hierarchy will be obtained from the Maxwellequations by means of a concentration process �→ 0, followed by a homogenization limit with respectto ”. These models are then compared with regard to their physical meaning and mathematical issues.Copyright ? 2005 John Wiley & Sons, Ltd.

KEY WORDS: homogenization; asymptotic expansion; dynamical condition; electrical conduction inbiological tissues

1. INTRODUCTION

In this paper we deal with models of electrical conduction in composite media and, speci�cally,biological tissues. The classical governing equation is

−div(A∇u+B∇ut)=0 (1)

which is derived from the Maxwell equation in the quasi-stationary approximation (see, e.g.Reference [1]). Here, u is the electrical potential. We look at a material which is a mixture ofa conductive phase of conductivity A and a dielectric phase of permeability B; we extend theconductivity A (respectively, the permeability B) by zero in the dielectric phase (respectively,in the conductive phase). More precisely, we look at a conductive phase containing a �nely

∗Correspondence to: M. Amar, Dipartimento di Metodi e Modelli Matematici, Universit�a di Roma ‘La Sapienza’,Via A. Scarpa 16, 00161 Roma, Italy.

†E-mail: amar@dmmm.uniroma1.it

Copyright ? 2005 John Wiley & Sons, Ltd. Received 8 November 2005

768 M. AMAR ET AL.

mixed periodic array of conductive inclusions coated by dielectric shells. The typical structureof the periodic cell we have in mind is given in Figure 1. This leads to the assumption

A=A(x=”0); B=B(x=”0)

where ”0 is the period of the physical structure.On the other hand, a second small parameter appears in the models we look at, that is the

width of the dielectric shell, which we denote by �0”0, with �0 � 1.For example, in the case of a biological tissue, the conductive phases are given, respectively,

by the extracellular �uid and the cell cytosol, while the dielectrical phase is given by the cellmembrane. In this case, the diameter of the cell is of the order of tens of micrometers, whilethe width of the membrane is of the order of ten nanometers, so that �0∼10−3.A concentration of capacity is performed to replace the thin dielectric shell with a two-

dimensional surface, in order to simplify the model, and, possibly, get a better understandingof the e�ect of the geometrical properties of the microscopic structure. Then a homogenizationlimit is taken, replacing ”0 with a parameter 0¡”¡”0 which is sent to 0.Material properties are usually rescaled in concentration processes. In particular, we will

assume that in the dielectric B= ��, where � is the ratio between the physical permeabilityof the membrane and the constant �0 (this scaling is motivated in Remark 2.1). On the otherhand, we note that A and B are physical properties of the material and, in principle, shouldnot change in the homogenization limit: a kind of stability which is standard in homogenizationtheory. However, for reasons we explain below, in the following we allow for dependenceof B on ” (see (8) and the ensuing discussion).Letting �→ 0 yields

−div(A∇u”) = 0 in �”T (2)

A∇uint” · �=A∇uout” · � on �”T (3)

�”

@@t[u”] =A∇uout” · � on �”

T (4)

[u”](x; 0) = S”(x) on �” (5)

where �” denotes the union of the two disjoint conductive phases, �” is the separating inter-face, T is a positive time, �”

T =�” × (0; T ), �”

T =�” × (0; T ), and uint” , u

out” are the potential in

the internal and the external conductive phase, respectively. Let � be the normal unit vectorto �” pointing into the external conductive phase. We also denote

[u”]= uout” − uint” (6)

S” denotes suitable bounded initial data which will be more precisely described in Section 4.If the dielectric phase has also a conductive behaviour, which can be modelled simply

by letting A= �� in the dielectric phase (� is a non-negative constant), condition (4) isreplaced with

�”

@@t[u”] +

�”[u”]=A∇uout” · � (7)

However for the sake of simplicity we consider the case �=0 in the sections below. Thewell posedness of this problem for each ”¿ 0 has been investigated in References [2,3].

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Clearly, (2)–(5) should be complemented with boundary conditions for u” on @�. Since allour arguments are independent of such boundary conditions, we omit them in the following(see Reference [4] for the Dirichlet problem).This model, which has been investigated in References [4,5], is a special case (k=1) of a

hierarchy of models where Equation (4) is replaced by

�”k

@@t[u”]=A∇uout” · � on �” (8)

with k ∈Z. For example, the case k= − 1 is used in Reference [6] in order to obtain, in thehomogenization limit, the well known bidomain model for the cardiac syncithial tissue, wherehowever, in the left-hand side of (8) an extra term depending on [u”] appears, modelling thenon-linear conductive behaviour of the membrane (see Reference [7]); see also Reference [8]for the delicate mathematical investigation of this non-linear behaviour. The case k=0 (moreprecisely, its stationary version where the term on the left-hand side of (8) is replaced by�[u”]) is considered in Reference [9] in connection with a heat conduction problem in presenceof thermal barrier resistance. The problem with k=1 has been applied to the case of currentsin the radiofrequency range (see Reference [5]), where a conductive term linearly dependingon [u”] is usually added in the left-hand side of (8) as in (7) (see Reference [10]).We show that this family of concentrated problems can be derived from Equation (1) by

means of the uni�ed approach sketched above, when we rescale in ” the physical constant B,i.e. we take B= ��=”k−1.We observe that the homogenized limit equation for the cases k6 − 2, k=0, k¿ 2, do

not preserve any trace of the permeability B of the dielectric phase, accounting for the capac-itive behaviour of the dielectric shell. Conversely, the limit equations for the cases k= − 1and k=1 keep trace of the permeability B. However, these two models di�er substantiallyfrom a mathematical point of view, since the former is described by a system of degenerateparabolic equations, whereas the latter consists of an elliptic equation with memory. Thesemodels have been employed to study electrical conduction in biological tissues in di�erentfrequency ranges [4–6].Our analysis presents a uni�ed derivation of di�erent schemes, thus allowing a comparison

among their underlying physical assumptions, and could be useful to assess the viability ofeach scheme as a model for a speci�c experimental setting.The paper is organized as follows: the problem to be concentrated and homogenized,

together with the relevant geometrical assumptions, is stated in Section 2. The concentra-tion limit �→ 0 is performed in Section 3, in a rigorous mathematical way, yielding thefamily of concentrated problems (20)–(23), depending on k ∈Z. These problems are thenformally homogenized (”→ 0) in Section 4, with the aim of drawing a comparison amongthem. The homogenization limit are rigorously performed for the case k=1 in Reference [4]and for the case k= − 1 in Reference [8]. Finally, in Section 5, the so obtained models arecompared with regard to their physical meaning and mathematical issues.

2. POSITION OF THE PROBLEM

The typical periodic geometrical setting is displayed in Figure 1. Here we give a detailedformal de�nition of it.

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770 M. AMAR ET AL.

Figure 1. The periodic cell Y . Left: before concentration; �� is the shaded region, andE� = E�

int ∪E�out is the white region. Right: after concentration; �

� shrinks to � as �→ 0.

Let � be an open connected bounded subset of RN , and let �=�”int ∪�”

out ∪�”, where �”int

and �”out are two disjoint open subsets of �, and �

”= @�”int ∩�= @�”

out ∩�. The region �”out

[respectively, �”int] corresponds to the outer conductive phase [respectively, to the conductive

inclusions], while �” is the dielectric interface. Let � denote the normal unit vector to �”

pointing into �”out and [u”] be de�ned as in (6).

For �¿ 0, let us write � as �=�”;� ∪�”;� ∪ @�”;�, where �”;� and �”;� are two disjoint opensubsets of �, �”;� is the tubular neighbourhood of �” with thickness ”�, and @�”;� is the partof the boundary of �”;� which intersects �. Moreover, we assume also that �”;�=�”;�

int ∪�”;�out

and @�”;�=(@�”;�int ∪ @�”;�

out)∩�. Again, �”;�out, �

”;�int correspond to the conductive regions, and

�”;� to the dielectric shell. We assume that, for �→ 0 and ”¿ 0 �xed, |�”;�|∼”�|�”|N−1,�”;� →�”

out ∪�”int and @�”;� →�”.

More speci�cally, let us introduce a periodic open subset E of RN , so that E+ z=E for allz ∈ZN . For all ”¿ 0 de�ne �”

int =�∩ ”E, �”out =� \ ”E. We assume that �, E have regular

boundary, say of class C∞ for the sake of simplicity. We also employ the notation Y =(0; 1)N ,and Eint =E ∩Y , Eout =Y \ �E, �= @E ∩ �Y . As a simplifying assumption, we stipulate that|�∩ @Y |N−1 = 0.For every �¿ 0, let Y =E� ∪�� ∪ @��, where E� and �� are two disjoint open subsets of Y ,

�� is the tubular neighbourhood of � with thickness �, and @�� is the part of the boundary of�� which intersects Y . Moreover, E�=E�

int ∪E�out (see Figure 1). For �→ 0, E� →Eint ∪Eout,

|��|∼�|�|N−1 and @�� →�.Let T ¿ 0 be a given time. For any spatial domain G, we denote by GT =G × (0; T ) the

corresponding space–time cylindrical domain over the time interval (0; T ).We start from the problem considered in Section 1, i.e.

−div(A�∇u�”) = 0 in �”;�

T (9)

−div(B�∇u�”t) = 0 in �”;�

T (10)

A�∇u�” · �� = B�∇u�

”t · �� on @�”;�T (11)

∇u�”(x; 0) =S�

”(x) in �”;� (12)

where u�”(t) ∈ H 1(�) at each time level t, S

�” =∇S

�”, for some S

�” ∈H 1(�”;�) and |S�

”|∼1=�.Here A� (A in the previous section) satis�es A�(x)=�int in �”;�

int , A�(x)=�out in

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HIERARCHY OF MODELS FOR ELECTRICAL CONDUCTION 771

�”;�out and A�(x)=0 in �”;�; B� (B in the previous section) satis�es B�(x)= �� in �”;� and

B�(x)=0 in �”;� (see Remark 2.1 and Section 5.1); �� is the unit normal vector to @�”;�

pointing into �”;� and �int, �out, � are positive constants. We also de�ne � : �→R as

�=�int in �”int ; �=�out in �”

out

We note that (9)–(11) can be compactly written as

−div(A�∇u�” + B�u�

”t)=0 in �T

coinciding with (1).

Remark 2.1We are interested in preserving, in the limit �→ 0, the conduction across the membrane �”

instead of the tangential conduction on �”. To this purpose, we need to preserve the �uxB�∇u�

”t · � and the jump [u�”t] across the dielectric shells to be concentrated. Hence, we rescale

B�= ��, instead of scaling B�= �=� in �”;�, as more usual in concentrated-capacity literature(see, e.g. References [11,12]; we comment further on some di�erent scalings in Remark 3.2).

3. DERIVATION OF THE CONCENTRATED PROBLEMS

For the sake of de�niteness, we assume here zero Dirichlet boundary data on the boundaryof �. However, the argument is essentially local and could be reproduced once uniformL2-estimates for u�

” and ∇u�” are available.

The rigorous formulation of problem (9)–(11) is∫ T

0

∫�”;�out ∪�”;�

int

A�(x)∇u�” · ∇’ dx d�=

∫ T

0

∫�”;�

��∇u�” · ∇’� dx d� (13)

where

u�” ∈L2(0; T ;H 1

o (�))

for all ’∈C2o (�× (0; T )).The rigorous formulation of the concentrated problem (2)–(4) is moreover∫ T

0

∫�”out ∪�”

int

�(x)∇u” · ∇� dx d�=∫ T

0

∫�”

�”[u”][�]� d� d� (14)

where

u”|�”out

∈L2(0; T ;H 1(�”out)); u”|�”

int∈L2(0; T ;H 1(�”

int))

for all � such that

�|�”int

∈C2o ((�”int ∪�”)× (0; T )); �|�”

out∈C2o ((�

”out ∪�”)× (0; T ))

By an integration by parts, one can derive from (13) the energy inequality∫ T

t0

∫�”;�out ∪�”;�

int

|∇u�”|2 dx d�+ sup

0¡t¡T

∫�”;�(t)

�|∇u�”|2 dx6 � (15)

where � does not depend on �.

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772 M. AMAR ET AL.

As a consequence, the L2-norm of u�” |�”

outcan be bounded by means of the usual Poincar�e

inequality and a standard extension technique. Then, the L2-norm of the trace of u�” |�”

inton

@�”;� ∩�”int can be bounded in terms of the L2-norm of the trace of u�

” |�”outand the bound

in (15).Hence, as �→ 0, we may assume, extracting a subsequence if needed

u�” → u” weakly in L2loc(�× (0; T ))

where

u”|�”int

∈L2loc(0; T ;H1(�”

int)); u”|�”out

∈L2loc(0; T ;H1(�”

out))

Moreover, we may also assume

∇u�” → ∇u” weakly in L2loc(�

”int × (0; T )); L2loc(�

”out × (0; T ))

3.1. Di�eomorphisms

The following fact is well known.

Theorem 3.1There exists an �0¿ 0 such that for �¡�0, the application

: �” ×[−”�2;”�2

]→RN ; (; r)= + r�()

is a di�eomorphism onto its image, which equals (by de�nition) �”;�.

Locally, we can represent �” as a graph

xN =F(x′); x′=(x1; : : : ; xN−1) (16)

perhaps after relabelling the co-ordinates. In the following we always assume this represen-tation, i.e. we choose a testing function whose support is contained in the region where (16)is valid. The general case can then be recovered by means of a standard partition of unityargument.Let � denote also the normal as a function of the local co-ordinates; then

�(x′)= (−∇x′F(x′); 1)g(x′)−1=2; g(x′)=1 + |∇x′F(x′)|2

Let us introduce the notation

−1 : �”;� →�” ×[−”�2;”�2

]; −1(y)= (0(y); F(0(y)); �(y))

with

0 : �”;� →RN−1; � : �”;� →[−”�2;”�2

]We know that, for y∈�”;�, if we let �(y) be the closest point of �” to y, then y−�(y) mustbe normal to �”. Since is a di�eomorphism, (0(y); F(0(y))) is the only point with thisproperty. It follows that in �”;�

�(y)= (signed) distance from �”

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HIERARCHY OF MODELS FOR ELECTRICAL CONDUCTION 773

Hence

∇�(y)= �(0(y)) for y=(0(y); F(0(y))∈�”; i.e. for �(y)=0

while in general, by virtue of the assumed regularity of �”

∇�(y)= �(0(y)) + R�(y); y∈�”;� (17)

where we denote by R� any quantity such that

|R�|6 ��; 0¡�¡�0

3.2. Piecewise smooth testing function

Let

’1 ∈C2o ((�”int ∪�”)× (0; T )); ’2 ∈C2o ((�

”out ∪�”)× (0; T ))

Then de�ne for 0¡�¡�0, y∈�”;�, the functions ’3� and J from

’3�(y) = [’2( (0(y); ”�=2); t)− ’1( (0(y);−”�=2); t)]�(y) + ”�=2

”�

+’1( (0(y);−”�=2); t)=:J�(y) + ”�=2

”�+ ’1( (0(y);−”�=2); t)

where for the sake of notational simplicity we use to denote the di�eomorphism written inlocal co-ordinates.The function

’�(y; t)=

⎧⎪⎪⎨⎪⎪⎩

’1(y; t); y∈�”;�int

’3�(y; t); y∈�”;�

’2(y; t); y∈�”;�out

is an admissible testing function in (13).Owing to our smoothness assumptions, we have

∇’3�(y) = (∇J)�(y) + ”�=2

”�+J(”�)−1∇�(y) +∇’1( (0(y);−”�=2); t)

=C�(y) +J(”�)−1∇�(y)=C�(y) +J(”�)−1�(0(y)) (18)

where we have used (17) and denoted by C� any quantity that stays bounded for all 0¡�¡�0.

3.3. The limit �→ 0

Write (13) for ’�, i.e.∫ T

0

∫�”;�out ∪�”;�

int

A�(x)∇u�” · ∇’� dx d�=

∫ T

0

∫�”;�

��∇u�” · ∇’3�� dx d� (19)

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774 M. AMAR ET AL.

The left-hand side of (19) approaches as �→ 0

∫ T

0

∫�”out ∪�”

int

�(x)∇u · ∇’ dx d�

where

’(y; t)=

{’1(y; t); y∈�”

int

’2(y; t); y∈�”out

Indeed, this follows from the local convergence of ∇u�” and the uniform boundedness given

by (15).The right-hand side of (19) equals, by virtue of (18)

∫ T

0

∫�”;�

��∇u�” · {C�(y; �) +J�(”�)−1�(0(y))} dy d�

= �∫ T

0

∫�”;�

�∇u�” ·C�(y; �) dy d�+

�”

∫ T

0

∫�”;�

J�∇u�” · �(0(y)) dy d�= I1 + I2

As �→ 0, owing to (15)

|I1|6 ��1=2(�

∫ T

0

∫�”;�

|∇u�”|2 dy d�

)1=2 (∫ T

0

∫�”;�

|C�(y; �)|2 dy d�)1=2

6 ��1=2 → 0

Let us next change co-ordinates in I2, according to y �→(0(y); �(y))= (x′; r). The Jacobianmatrix of this change of co-ordinates is given by

J (x′; r)=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(e′1;

@F@x1

)+ r

@�(x′)@x1

· · ·(e′N−1;

@F@xN−1

)+ r

@�(x′)@xN−1

�(x′)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

where e′k is the standard basis in RN−1. Then the Jacobian determinant is (recall the de�nitionof g)

|det J (x′; r)|= |det J (x′; 0)|+ R�(x′; r)=√

g(x′) + R�(x′; r)

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HIERARCHY OF MODELS FOR ELECTRICAL CONDUCTION 775

Hence

I2 =�”

∫ T

0

∫RN−1

J�

√g(x′)

∫ ”�=2

−”�=2∇u�

”( (x′; r); �) · �(x′) dr dx′ d�

+�”

∫ T

0

∫RN−1

J�

∫ ”�=2

−”�=2R�(x′; r)∇u�

”( (x′; r); �) · �(x′) dr dx′ d�= I21 + I22

Invoking again (15), we get from an application of H�older’s inequality

|I22|6C�∫ T

0

∫�”;�

|∇u�”| dx d�6C�1=2 → 0 as �→ 0

Finally, by the bound on the traces mentioned above, it follows

I21 =�”

∫ T

0

∫RN−1

[’2�( (x′; ”�=2); �)− ’1�( (x′;−”�=2); �)]

×√1 + |∇x′F(x′)|2[u�

”( (x′; ”�=2); �)− u�

”( (x′;−”�=2); �)] dx′ d�

→ �”

∫ T

0

∫RN−1

[’2�( (x′; 0); �)− ’1�( (x′; 0); �)]

×[uout” ((x′; F(x′)); �)− uint” ((x

′; F(x′)); �)]√1 + |∇x′F(x′)|2 dx′ d�

=�”

∫ T

0

∫�”[’]�[u”] d� d�

Collecting the limiting relations above, we see that indeed (13) leads to (14) as �→ 0.

Remark 3.2It follows immediately from the technique displayed above that, if we de�ne B�= ��p, p¿ 1,in our setting, i.e. if we replace in (13) �� with ��p, the limiting model is the one obtainedby formally setting the right-hand side of (14) equal to zero. In other words, the membranecontribution disappears in the limit �→ 0.

4. HOMOGENIZATION OF THE CONCENTRATED PROBLEMS

We look at the homogenization limit (”→ 0) of the problem for u”(x; t) stated in Section 1.We give here a complete formulation for convenience (the operators div and ∇ act only withrespect to the space variable x)

−div(�∇u”) = 0 in �”int ;�

”out (20)

[�∇u” · �] = 0 on �” (21)

�”k

@@t[u”] = �out∇uout” · � on �” (22)

[u”](x; 0) = S”(x) on �” (23)

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776 M. AMAR ET AL.

We are interested in understanding how the limiting behaviour of the problem above when”→ 0 depends on the parameter k ∈ Z.First, we have to determine the admissible order of S” with respect to ”. Multiply (20)

by u” and integrate by parts. When, for the sake of simplicity, we look at the case of Dirichletboundary conditions u”=0 on @�, we arrive, for all 0¡t¡T , to the energy estimate∫ t

0

∫��|∇u”|2 dx d�+ �

2”k

∫�”[u”]2(x; t) d�=

�2”k

∫�”

S2” (x) d� (24)

Since |�”|N−1∼1=”, we assume thatS”=O(”(k+1)=2) (25)

so that the right-hand side of (24) is stable as ”→ 0. In fact (24), coupled with suitablePoincare’s inequalities, is a main tool in the rigorous proof of convergence of u” to its limit[4,13].

4.1. The two-scale approach

We summarize here, to establish the notation, some well known asymptotic expansions neededin the two-scale method (see, e.g. References [14,15]), when applied to stationary, or evolu-tive, problems involving second-order partial di�erential equations. Introduce the microscopicvariables y∈Y , y= x=”, assuming

u”= u”(x; y; t)= u0(x; y; t) + ”u1(x; y; t) + ”2u2(x; y; t) + · · · (26)

Note that u0, u1, u2 are periodic in y, and u1, u2 are assumed to have zero integral averageover Y . Recalling that

div=1”divy + divx; ∇= 1

”∇y +∇x (27)

we compute

u”=1”2

A0u0 +1”(A0u1 + A1u0) + (A0u2 + A1u1 + A2u0) + · · · (28)

Here

A0 =y; A1 = divy∇x + divx∇y; A2 =x (29)

Let us recall explicitly that

∇u”=1”∇yu0 + (∇xu0 +∇yu1) + ”(∇yu2 +∇xu1) + · · · (30)

We also stipulate

S”= S”(x; y)= S0(x; y) + ”S1(x; y) + ”2S2(x; y) + · · · (31)

We consider here only expansions in integral powers of ”, though (25) would allow, inprinciple, for an expansion in powers of

√”.

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HIERARCHY OF MODELS FOR ELECTRICAL CONDUCTION 777

4.2. The two extreme cases k= +∞, k= − ∞The following two cases, though they are essentially stationary problems, provide a frame ofmind we �nd helpful to understand the overall picture of our class of models. Moreover, theyare strictly connected, from the technical point of view, to the homogenization process wedevelop.

4.2.1. The Di�raction problem. We look here at the case of perfectly conducting interfaces,i.e. to (20)–(23) with S” ≡ 0 and k= +∞. More explicitly, we look at the problem obtainedwhen (20), (21) are complemented with

[u”]= 0 on �”

which is assumed to hold for every t ∈ [0; T ], replacing (22) and (23). Then, time plays onlythe role of a parameter. On substituting in this formulation the expansion (26), and applying(27)–(30), one readily obtains by matching corresponding powers of ”, that u0 solves [u0]= 0on �, and

P0[u0] :

{ −�yu0 = 0 in Eint ; Eout

[�∇yu0 · �] = 0 on �

Hence u0 is independent of y, i.e. u0 = u0(x; t).Moreover u1 satis�es [u1] = 0 on �, and

P1[u1] :

{ −�yu1 = 0 in Eint ; Eout

[�∇yu1 · �] =−[�∇xu0 · �] on �

Finally, u2 solves [u2]= 0 on �, and

P2[u2] :

⎧⎪⎨⎪⎩

−�yu2 = �xu0 + 2�@2u1

@xj@yjin Eint ; Eout

[�∇yu2 · �] =−[�∇xu1 · �] on �

Following a classical approach, one introduces the factorization

u1(x; y; t)= − D(y) · ∇xu0(x; t) (32)

for a vector function D : Y →RN , satisfying

−�y Dh =0 in Eint ; Eout (33)

[�(∇y Dh − eh) · �] = 0 on � (34)

[ Dh ] = 0 on � (35)

The components Dh are also required to be periodic functions in Y , with zero integral averageon Y . The limiting equation for u0 is �nally obtained as a solvability condition for P2[u2],and amounts to

−div((�0I + AD)∇xu0)=0 in �T (36)

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778 M. AMAR ET AL.

where

(AD)=∫�[�]� ⊗ D d�; �0 =�int|Eint|+ �out|Eout| (37)

To avoid repetition, we refer the reader to the classical texts (e.g. Reference [14]) for thedetails of the proof of (36), or to Section 4.4 below, where similar calculations are carriedout in a somehow more complex setting.

4.2.2. The Neumann problem. This is the case corresponding to k=−∞, when the interfacesare perfectly insulating. Here (20), (21) are complemented with

�out∇uout” · �=0 on �”

so that, actually, we are solving two independent Neumann problems in �”int and in �

”out.

Of course, this is meaningful if each one of the two phases is connected (see Remark 5.3).Again, time acts only as a parameter: the problem is in fact essentially stationary, and theinitial condition (23) can no longer be assigned.Reasoning as above, we �nd for u0{

P0[u0]

�out∇yuout0 · �=0 on �

In this case [u0] =0 on �, but u0 is independent of y in each phase, i.e.

u0(x; y; t)=

{uint0 (x; t) in Eint

uout0 (x; t) in Eout(38)

The term u1 solves {P1[u1]

�out∇yuout1 · �+ �out∇xuout0 · �=0 on �

Analogously, u2 is given by{P2[u2]

�out∇yuout2 · �+ �out∇xuout1 · �=0 on �

We introduce again the factorization for u1, which is given by

u1(x; y; t)=

{− N(y) · ∇xuint0 (x; t) + uint1 (x; t) y∈Eint

− N(y) · ∇xuout0 (x; t) + uout1 (x; t) y∈Eout(39)

where N is given by

−�y Nh =0 in Eint ; Eout (40)

[�(∇y Nh − eh) · �] = 0 on � (41)

�out(∇y Nh;out − eh) · �=0 on � (42)

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The components Nh are also required to be periodic functions in Y , with zero integral averageon Y . This implies that only one of the two constants up to which Nint and Nout are determinedby (40)–(42) is �xed, but this does not a�ect (43) and (44). Note that, in general, [ Nh ] =0on �.In the limit, we obtain (as compatibility conditions for the problem solved by u2) two

partial di�erential equations for the two components of u0, i.e.

−div((�int|Eint|I + ANint )∇xuint0 ) = 0 in �T (43)

−div((�out|Eout|I + ANout)∇xuout0 ) = 0 in �T (44)

where

ANint = −

∫��int� ⊗ Nint d�; AN

out =∫��out� ⊗ Nout d� (45)

See, e.g. Reference [16] for a treatment of the homogenization problem with Neumann bound-ary conditions.

4.3. The case k¿ 2: essentially the same as k= +∞According to (25), we stipulate S0 ≡ 0, S1 ≡ 0 in (31). If k ¿ 2, we should further assumethat Sh ≡ 0 for every h up to the integer part of (k + 1)=2. The problem for u0 is given by⎧⎨

⎩P0[u0]

�@[u0]@t

=0; [u0]|t=0 =0 on �

It follows that [u0]= 0 for all t. Then, P0[u0] implies as in Section 4.2 that u0 is independentof y, i.e. u0 = u0(x; t).Next, one checks that u1 solves⎧⎨

⎩P1[u1]

�@[u1]@t

=�out∇yu0 · �; [u1]|t=0 =0 on �

Since ∇yu0 ≡ 0, we obtain [u1]= 0 for all times. It is a simple matter to verify that u1 maybe represented as

u1(x; y; t)= − D(y) · ∇xu0(x; t) (46)

The problem for u2 amounts to⎧⎨⎩

P2[u2]

�@[u2]@t

=�out∇xu0 · �+ �out∇yuout1 · �; [u2]|t=0 = S2 on �

The compatibility condition for this problem is obtained integrating the partial di�erentialequation by parts as in Section 4.4. On using also expansion (46), we �nally get in the limit

−div((�0I + AD)∇xu0)=0 (47)

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780 M. AMAR ET AL.

where the matrix AD was de�ned in (37). Note that (47) is the same limiting equation weobtained in the Di�raction problem k=+∞. Moreover, the limiting problem does not dependon the initial data Si, i¿ 2, for t ¿ 0: indeed, the dependence of u0 on t is merely parametrical.In other words, the homogenization limit in the case k¿ 2 is the same as in the case k=∞.

4.4. The case k=1: a limiting equation with memory

This case has been treated in Reference [4]. According to Equation (25), we assume S0 ≡ 0in (31).As a consequence, and reasoning as in Section 4.2, we �nd for u0⎧⎨

⎩P0[u0]

�out∇yuout0 · �= �@[u0]@t

; [u0]|t=0 =0 on �

It follows (see Reference [4]) that [u0]= 0 for all times, and

u0 = u0(x; t)

Next we �nd for u1⎧⎨⎩

P1[u1]

�out∇yuout1 · �+ �out∇xuout0 · �= �@[u1]@t

; [u1]|t=0 = S1(x; y) on �

We want to represent u1 in a suitable way, in the spirit of (32), though the representationformula must be more complicated here. Let s : �→R be a jump function, and consider theproblem

−�yv=0 in Eint ; Eout (48)

[�∇yv · �] = 0 on � (49)

�@@t[v] = �out∇yvout · � on � (50)

[v](y; 0) = s(y) on � (51)

where v is a periodic function in Y , such that∫Y v=0. De�ne the transform T by

T(s)(y; t)= v(y; t); y∈Y; t ¿ 0

and extend the de�nition of T to vector (jump) functions, by letting it act componentwiseon its argument.Introduce also the function 1 : Y →RN de�ned by

� 1h=T(�out(eh − ∇y Dh ) · �) (52)

Straightforward calculations show that u1 may be written in the form

u1(x; y; t) =− D(y) · ∇xu0(x; t) +T(S1(x; · ))(y; t)

−∫ t

0∇xu0(x; �) · 1(y; t − �) d� (53)

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HIERARCHY OF MODELS FOR ELECTRICAL CONDUCTION 781

The term u2 in the expansion of u” satis�es⎧⎨⎩

P2[u2]

�out∇yuout2 · �+ �out∇xuout1 · �= �@[u2]@t

; [u2]|t=0 = S2(x; y) on �

Let us �nd the solvability conditions for this problem. Integrating by parts the partial dif-ferential equations solved by u2, both in Eint and in Eout, and adding the two contributions,we get [∫

Eint

+∫Eout

] {�xu0(x; t) + 2�

@2u1@xj@yj

}dy

=∫�{�out∇yuout2 · � − �int∇yuint2 · �} d�=

∫�[�∇yu2 · �] d�= −

∫�[�∇xu1 · �] d�

Thus for �0 de�ned as in (37), we get

�0xu0 = 2∫�[�∇xu1 · �] d� −

∫�[�∇xu1 · �] d�=

∫�[�∇xu1 · �] d� (54)

We use next the expansion (53); namely, we recall that, in it, only the terms T(: : :) and 1

have a non-zero jump across �. Thus we infer from the equality above

�0xu0 =∫�[�∇xu1 · �] d�= −

∫�[�] Dh (y)�j d�u0xhxj(x; t)

+@@xj

∫�[�T(S1(x; · ))](y; t)�j d� −

∫ t

0u0xhxj(x; �)

∫�[� 1h](y; t − �)�j d� d�

We �nally write the partial di�erential equation for u0 in �× (0; T ) as

−div((�0I + AD)∇xu0 +

∫ t

0A1(t − �)∇xu0(x; �) d� − F

)= 0 (55)

The constant matrix AD is the same as in the limiting equation of the Di�raction problem(case k= +∞; see (37)). The matrix A1 is de�ned by

A1(t)=∫�[�]�⊗ 1(y; t) d� (56)

The matrices AD and A1 are symmetric, and �0I + AD is positive de�nite [4]. The vector Fis de�ned by

F=∫�[�T(S1(x; · ))](y; t)� d� (57)

4.5. The case k=0: a threshold case

For a stationary analog of this case, see Reference [9]. According to Equation (25), we letS0 ≡ 0 in (31).

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782 M. AMAR ET AL.

The problem for u0 is given by{P0[u0]

�out∇yuout0 · �=0; [u0]|t=0 =0 on �

It follows that u0(x; · ; t) solves two independent homogeneous Neumann problems in Eintand Eout, as in Section 4.2.2, so that u0(x; · ; t)|Eint and u0(x; · ; t)|Eout are independent of y, and(38) holds.The problem for u1 is given by⎧⎨

⎩P1[u1]

�out∇yuout1 · �+ �out∇xuout0 · �= �@[u0]@t

; [u1]|t=0 = S1(x; y) on �(58)

On integrating by parts the partial di�erential equation in P1[u1], and using also (38), wehave

|�|(�@@t[u0]

)=

∫��out∇yuout1 · � d�+

∫��out∇xuout0 · � d�=0 (59)

Hence, recalling [u0]|t=0 =0, we have [u0]= 0 on � for all t. As a consequence, in (38)the two components are actually equal, and u0 = u0(x; t). Then, the problem for u1 may bewritten as {

P1[u1]

�out∇yuout1 · �+ �out∇xuout0 · �=0; [u1]|t=0 = S1(x; y) on �(60)

Apart from the initial data S1, this is the same problem satis�ed by u1 in the Neumannproblem (k= −∞). Then we may represent u1 as in (39), where uint0 = uout0 =: u0 and N wasde�ned in (40)–(42). The initial data for [u1] cannot, in general, be satis�ed, suggesting theonset of an initial layer as ”→ 0. One condition on the two functions uint1 and uout1 followsfrom the requirement that the integral average of u1 is zero. A second condition will be foundin (63) below.The problem for u2 is given by⎧⎨

⎩P2[u2]

�out∇yuout2 · �+ �out∇xuout1 · �= �@[u1]@t

; [u2]|t=0 = S2(x; y) on �

Since at this stage the terms containing u0 and u1 in the problem for u2 are regarded asknown, this reduces to two independent Neumann problems in Eint and in Eout, respectively,so that two independent compatibility conditions will be enforced in the following.First, integrating by parts the partial di�erential equation for u2 both in Eint and in Eout,

adding the two contributions, and using (39), we obtain

−div((�0I + AN)∇xu0)=0 (61)

The matrix AN is de�ned as

AN=∫�[��⊗ N] d� (62)

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HIERARCHY OF MODELS FOR ELECTRICAL CONDUCTION 783

that is, as matrix AD, with D formally substituted with N. Note instead the de�nition (45)of the limiting di�usion matrices in the Neumann problem. Thus, we may see the case k=0as a threshold case in our sequence of problems: the insulation provided by the interface issu�cient to modify the limiting di�usion matrix, with respect to the Di�raction case, but itis not strong enough to force the existence of two di�erent limit phases, as in the Neumannproblem.Second, on integrating the evolution equation for u2 over �, and using again the elliptic

di�erential equation for u2, we obtain the second compatibility condition∫�

(−�[ N] · @

@t∇xu0

)d�+ |�|

(�@@t[u1]

)

= �out|Eout|xu0 −∫��out∇xuout1 · � d�

= �out|Eout|xu0 +∫��out∇x( Nout · ∇xu0) · � d� −

∫��out∇xu

out1 · � d�

= div((�out|Eout|I + ANout)∇xu0) (63)

where the matrix ANout was de�ned in (45). Equation (63) governs the evolution of [u1] on

�. The choice of the initial data for [u1] is connected to the appearance of an initial layer.We observe that, di�erently from the case k= − 1, Equation (63) is not used to derive thehomogenized problem for u0.

4.6. The case k= − 1: a limit degenerate parabolic systemThe rigorous derivation of this limit can be found in Reference [8]. Recalling (25), we remarkthat no one of the terms S0; S1; : : : ; needs to vanish.The problem for u0 is{

P0[u0]

�out∇yuout0 · � = 0; [u0]|t=0 = S0 on �(64)

As a consequence, u0 is independent of y, separately in Eint and in Eout, so that (38) is inforce. It follows that the initial data [u0]|t=0 = S0 must be independent of y, or an initial layermust occur.The term u1 solves the same problem (60) as in case k=0. Moreover, the representa-

tion (39) for u1 is still valid. The argument exploited in Section 4.5 to infer that [u0]= 0 forall times cannot be repeated in the present case, since it relies on the formulation (58) whichis no longer valid. Indeed, u” does approach in the case k6 − 1 two di�erent componentsuint0 and uout0 , as ”→ 0 (clearly, in the two di�erent compartments �”

int and �”out, respectively).

A further di�erence with the case k = 0 appears in the problem for u2, which is nowgiven by ⎧⎨

⎩P2[u2]

�out∇yuout2 · �+ �out∇xuout1 · �= �@[u0]@t

; [u2]|t=0 = S2(x; y) on �(65)

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784 M. AMAR ET AL.

However, as in the case k=0, this scheme reduces to two independent Neumann problemsfor u2, in Eint and in Eout, so that we have to enforce two independent compatibility conditions.Calculations similar to the ones outlined above, starting from (65), yield

−div((�int|Eint|I + ANint )∇xuint0 + (�out|Eout|I + AN

out)∇xuout0 )=0 (66)

The matrices ANint and AN

out were de�ned in (45). This equation should be compared with thelimiting equations (43)–(44) obtained for the Neumann problem k= − ∞: in that case, uint0and uout0 solve two independent equations.As in case k=0, we get the second compatibility condition by integrating the evolution

equation for u2 over �, and using again the other equations of (65). We �nd

|�|� @@t(uout0 − uint0 ) = �out|Eout|xuout0 −

∫��out∇xuout1 · � d�

= �out|Eout|xuout0 +∫��out∇x( Nout · ∇xuout0 ) · � d� −

∫��out∇xu

out1 · � d�

=div((�out|Eout|I + ANout)∇xuout0 ) (67)

Equations (66) and (67) constitute a system of degenerate parabolic equations for theunknowns uint0 (x; t) and uout0 (x; t). They are complemented with the initial condition

uout0 (x; 0)− uint0 (x; 0)= S0(x) on � (68)

which follows from the second of (64).

4.7. The case k6 − 2: essentially the same as k= − ∞The expansion of S” is the same as in the case k= − 1.The problems for u0 and for u1 are the same as in the case k= −1. Then, the representation

(38) for u0, and (39) for u1 are still in force.The problem for u2 in this case reduces to{

P2[u2]

�out∇yuout2 · �+ �out∇xuout1 · �=0; [u2]|t=0 =S2 on �(69)

i.e. to the same scheme obtained in the Neumann problem k= − ∞. Therefore we obtainin the limit the two independent di�erential equations (43) and (44). In other words, thehomogenization limit in the case k6 − 2 is the same as in the case k= − ∞.

5. DISCUSSION

5.1. ”-Scaling of the dielectric constant

As remarked in Section 1, we rescale in ” the dielectric constant B� as follows:

B�=��”k−1

; k ∈Z (70)

in order to derive the concentrated equations (20)–(23) from Equation (1).

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HIERARCHY OF MODELS FOR ELECTRICAL CONDUCTION 785

We point out that di�erent choices of k keep di�erent physical quantities constant in thehomogenization limit ”→ 0. For instance, according to (70), the following physical quantitiesare preserved:

• the permeability of the dielectric phase, in the case k=1;• the capacity of the dielectric shell per unit of area, in the case k=0;• the total capacity of the intracellular phase with respect to the extracellular phase, orequivalently the total capacity per unit of volume, in the case k= − 1.

5.2. Comparison among the models

Following the analysis in the previous sections, we obtained �ve models, which can be clas-si�ed according to di�erent criteria. As a �rst criterium, we could divide the models corre-sponding to k6−1 from the ones corresponding to k¿ 0. The former are known as bidomainmodels since they involve two unknown macroscopic functions uint0 and uout0 ; while the latterinvolve only one macroscopic function u0 and henceforth they are known as monodomainmodels. A similar classi�cation can be found in Reference [17] in the case of static interfaceconditions.A second criterium takes into account the mathematical structure of the involved cell func-

tions and divides the models corresponding to k6 0 from the ones corresponding to k¿ 1.The former involve the cell function D, while in the latter we �nd N. Accordingly, wenote that in the �rst class the model k=1 is the most general, since it formally gives modelk¿ 2 for �→ +∞; while in the second class the most general model is k = −1, since itformally gives models k = 0 and k6 −2 on letting �→ +∞ or �→ 0, respectively. Becauseof the di�erent nature of D and N no continuous passage from one class to the other seemspossible.The last criterium, which is the most physical one, corresponds to single out the models

which preserve in the limit the membrane properties (i.e. the constant �). They are just two:the model k = 1 and the model k= − 1, which, remarkably, were the most general of theirown class in the previous classi�cation. These models are also the only ones able to take intoaccount the in�uence of the initial data.

5.3. Geometry of the problem

In the cases k¿ 1, the matrix �0I+AD, obtained by means of the cell function D and involvedinto the homogenized equations (36) and (55) is positive de�nite for any geometry (seeReference [4]). On the contrary, for k6 0, the matrices involved into the limiting problemsdepend on the cell function N and may degenerate depending on the geometry of the media.Indeed, if either one of the phases �”

int or �”out is not connected, it can be easily veri�ed that

the matrix �int|Eint|I + ANint or �out|Eout|I + AN

out, respectively, may degenerate or even vanish.For example, in the geometrical setting displayed in Figure 1, we have �int|Eint|I+AN

int = 0. Ina layered material, the two limiting di�usion matrices would degenerate along the directionorthogonal to the layers.However, in the case k= − 1, the two matrices cited above appear simultaneously in the

homogenized equation (66). Hence the limiting problem is meaningful if at least one ofthe two matrices above does not vanish. Analogously, in the case k=0, the homogenized

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786 M. AMAR ET AL.

equation (61) is not trivial when at least one of the two matrices is not zero, since their sumgives the matrix de�ning the elliptic operator in (61).On the other hand, in the cases k6 −2, both matrices in (43) and (44) may not degenerate

in order to obtain a meaningful limit problem. This is the case, for instance, when both phasesare connected.

5.4. Some simpli�ed cases

Many papers in the literature, devoted to the study of reconstruction of the interior of thehuman body from exterior electrical measurements, rely on the study of the Dirichlet–Neumannmap for an elliptic equation. This fact corresponds, in the Maxwell equation (1), to set B=0and to have A di�erent from zero in the membrane. The choice B=0 amounts to neglect thee�ect of the displacement currents. Nevertheless, experimental data show that, for su�cientlyhigh frequency range, displacement currents play a relevant role; i.e. memory e�ects appear(see Reference [10]). This justi�es the study of models (55) (see References [4,5]) and (66)(see References [2,6,8]).However, taking into account, for instance, problem (55), it is worthwhile noticing that, for

some simple geometries, it reduces to the study of an equation for the Laplacian. This occurs,for example, when the periodicity cell is invariant with respect to rotations of =2, since therelevant matrix AD and A1(t) are scalar multiples of the identity, which we denote by aD anda1(t), respectively. In this case, assigning the current �ux h on the boundary corresponds tothe equation

−(�0 + aD)@u0@n

−∫ t

0a1(t − �)

@u0@n(x; �) d�= h(x; t) (71)

which can be solved with respect to @u0=@n, thus obtaining a standard Neumann boundarycondition. Accordingly, the study of the inverse problem reduces to the standard study ofthe Dirichlet–Neumann map, with a relevant physical di�erence; i.e. the value of the normalderivative is not given by h(x; t) itself, but it can be obtained by means of a clever use ofthe Laplace transformation and its inverse applied to h and a1. This implies that the value ofthe normal derivative at time t is in�uenced by the �ux values at previous time and this factin itself produces the appearance of physically relevant memory e�ects.On the other hand, for general geometry, when we cannot assume B=0 (as in the case

of high frequency range) the study of the inverse problem for Equation (55) is still an openproblem.

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