Electrical Conduction in Biological Tissues

Homogenization Techniques and Asymptotic Decay for Linear and Nonlinear

Problems

M. Amar

, D. Andreucci

, P. Bisegna

and R. Gianni

Dipartimento di Scienze di Base e Applicate per l’Ingegneria - Sapienza Universit`a di Roma

Via A. Scarpa 16, 00161 Roma, Italy

Dipartimento di Ingegneria Civile - Universit`a di Roma “Tor Vergata”

Via del Politecnico 1, 00133 Roma, Italy

Keywords:

Homogenization, Asymptotic Expansions, Two-scale Techniques, Txponential Decay, Electrical Impedance

Tomography.

Abstract:

We collect some results concerning electrical conduction problems in biological tissues. These problems are

set in a ﬁnely mixed periodic medium and the unknown electric potentials solve standard elliptic equations set

in different conductive regions (the intracellular and extracellular spaces), separated by an interface (the cell

membrane), which exhibits both a capacitive and a conductive behavior. As the spatial period of the medium

goes to zero, the problems approach a homogenization limit. The macroscopic models are obtained by using

the technique of asymptotic expansions, in the case where the conductive behavior of the cell membrane is

linear, and by means of two-scale convergence, in the case where, due to its biochemical structure, the cell

membrane performs a strongly nonlinear conductive behavior. The asymptotic behavior of the macroscopic

potential for large times is investigated, too.

1 INTRODUCTION

This is a review article concerning the results obtained

by the authors in several papers dealing with some

aspects of electrical conduction in biological tissues.

It is well knownthat electric potentials can be used

in diagnostic devices to investigate the properties of

biological tissues. Besides the well-known diagnos-

tic techniques such as magnetic resonance, X-rays

and so on, it plays an important role a more recent,

cheap and noninvasive technique known as electric

impedance tomography (EIT). Such a technique is es-

sentially based on the possibility of determining the

physiological properties of a living body by means of

the knowledge of its electrical behavior.

This leads to an inverse problem for an elliptic

equation, usually the Laplacian, which is the equation

satisﬁed by the electrical potential, when the body is

assumed to display only a resistive behavior. How-

ever, it has been observed that, applying high fre-

quency potentials to the body, a capacitive behavior

appears, due to the electric polarization at the inter-

face of the cell membranes produced by the lipidic

composition of the membranes themselves, which act

as capacitors. This phenomenon (known in physics

as Maxwell-Wagner effect) is studied modeling the

biological tissue as a composite medium with a peri-

odic microscopic structure of characteristic length ε,

where two ﬁnely mixed conductive phases (the intra-

and the extra-cellular phase) are separated by a di-

electric interface (the cellular membrane). From the

mathematical point of view, the electrical current ﬂow

through the tissue is described by means of a system

of decoupled elliptic equations in the two conductive

phases (obtained from the Maxwell equations, under

the quasi-static assumption; i.e., we assume that the

magnetic effects are negligible). The solutions of this

system are coupled because of the interface condi-

tions at the membrane, whose physical behavior is de-

scribed by means of a dynamical boundary condition

(which takes into account both the conductive and the

capacitive behavior of the cell membrane), together

with the ﬂux-continuity assumption. Because of the

complex geometry of the domain, these models are

not easily handled, for example from the numerical

point of view. This justiﬁes the need of the homoge-

nization approach, with the aim of producing macro-

scopic models for the whole medium as ε → 0, since

696

Amar M., Andreucci D., Bisegna P. and Gianni R..

Electrical Conduction in Biological Tissues - Homogenization Techniques and Asymptotic Decay for Linear and Nonlinear Problems.

DOI: 10.5220/0004634006960703

In Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (BIOMED-2013), pages

696-703

ISBN: 978-989-8565-69-3

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

the typical scale ε of the microstructure is very small

with respect to the tissue macroscopic scale analyzed

in the experiments.

We present in the following two different cases:

in the ﬁrst one the conductive behavior of the cell

membrane is assumed to be linear and the approach

used in order to obtain the macroscopic equation is

the asymptotic expansion introduced in (Bensoussan

et al., 1978); in the second one, we assume a strongly

nonlinear conductive behavior of the cell membrane,

which actually appears in some physical situation and

which is due to the presence of ionic channels, i.e.

to the biochemical structure of the cell membrane it-

self. The technique used in this last case in order to

obtain the effective potential of the tissue is the two-

scale convergence technique introduced in (Nguet-

seng, 1989) and in (Allaire, 1992).

In the ﬁrst case, the macroscopic equation ob-

tained with this approach is an elliptic equation with

memory, as it could be expected in any electrical cir-

cuit in which a capacitor is present. In the second

case, we obtain a strictly coupled system of equa-

tions for the macroscopic and microscopic potentials,

as usual when the two-scale convergence technique is

applied.

2 SETTING OF THE PROBLEM

Let Ω be an open connected bounded subset of R

Let us introduce a periodic open subset E of R

, so

that E + z = E for all z ∈ Z

. For all ε > 0 deﬁne

Ω

int

= Ω ∩ εE, Ω

out

= Ω \

εE. We assume that Ω, E

have regular boundary, say of class C

∞

for the sake

of simplicity. Moreover, we set Ω = Ω

int

∪ Ω

out

∪ Γ

where Γ

= ∂Ω

int

∩ Ω = ∂Ω

out

∩ Ω. We also employ

the notation Y = (0,1)

, and E

int

= E ∩ Y, E

out

Y \

E, Γ = ∂E ∩ Y. As a simplifying assumption,

we stipulate that E

int

is a connected smooth subset of

Y such that dist(

int

,∂Y) > 0. Some generalizations

may be possible, but we do not dwell on this point

here. Finally, we assume that dist(Γ

,∂Ω) > γε for

some constant γ > 0 independent of ε, by dropping

the inclusions contained in the cells ε(Y + z), z ∈ Z

which intersect ∂Ω (see Figure 1). Finally, let T > 0

be a given time.

We are interested in the homogenization limit as ε ց

0 of the problem for u

(x,t) (here the operators div

and ∇ act only with respect to the space variable x)

−div(σ

int

∇u

) = 0, in Ω

int

; (1)

−div(σ

out

∇u

) = 0, in Ω

out

; (2)

int

∇u

(int)

· ν = σ

out

∇u

(out)

· ν, on Γ

; (3)

Figure 1: An examples of admissible periodic structures in

. Left: Y is the dashed square, and E ∩Y is the shaded

region. Right: the domain Ω.

∂

∂t

] + f



]



= σ

out

∇u

(out)

· ν, on Γ

; (4)

](x,0) = S

(x), on Γ

; (5)

(x,t) = 0, on ∂Ω. (6)

The notation in (1)–(4), (6), means that the indi-

cated equations are in force in the relevant spatial do-

main for 0 < t < T.

Here σ

int

, σ

out

and α are positive constants, and

ν is the normal unit vector to Γ

pointing into Ω

out

Since u

is not in general continuous across Γ

have set

(int)

:= trace of u

ε|Ω

int

on Γ

;

(out)

:= trace of u

ε|Ω

out

on Γ

Indeed we refer conventionally to Ω

int

as to the inte-

rior domain, and to Ω

out

as to the outer domain. We

also denote

] := u

(out)

− u

(int)

Similar conventions are employed for other quanti-

ties; for example (3) can be rewritten as

[σ∇u

· ν] = 0, on Γ

where

σ = σ

int

in Ω

int

, σ = σ

out

in Ω

out

The function f and the initial data S

will be dis-

cussed below.

Under the assumptions above, we prove existence

and uniqueness of a weak solution to (1)–(6), in the

class

ε|Ω

∈ L

(0,T;H

(Ω

)), i = 1,2, (7)

and u

ε|∂Ω

= 0 in the sense of traces (Amar et al.,

2005).

In the following, we will show that, if γ

−1

ε ≤

(x) ≤ γε, where S

is the initial jump prescribed

in (5), for a ﬁxed constant γ > 1, then u

becomes

stable as ε → 0 (i.e., it converges to a nonvanishing

bounded function). Therefore, let us stipulate that

∈ H

1/2

(Γ

) and

(x) = εS





+ εR

(x), (8)

ElectricalConductioninBiologicalTissues-HomogenizationTechniquesandAsymptoticDecayforLinearandNonlinear

Problems

697

where S

: Ω × ∂E → R , and

∞

(Ω×∂E)

< ∞, kR

∞

(Ω)

→ 0, as ε → 0;

(x,y) is continuous in x, uniformly over y ∈ ∂E,

and periodic in y, for each x ∈ Ω.

3 THE LINEAR CASE

In this section we assume that



−1



t , t ∈ R ,

with β ≥ 0. Firstly, we remark that, up to a change

of unknown function, we can assume β = 0; indeed,

setting v

(x,t) = u

(x,t) · exp





, it follows that v

satisﬁes

−div(σ

int

∇v

) = 0, in Ω

int

;

−div(σ

out

∇v

) = 0, in Ω

out

;

int

∇v

(int)

· ν = σ

out

∇v

(out)

· ν, on Γ

;

∂

∂t

] = σ

out

∇v

(out)

· ν, on Γ

;

](x,0) = S

(x), on Γ

;

(x,t) = 0, on ∂Ω.

Hence, from now on, we assume β = 0 in (4).

3.1 Homogenization

The weak formulation of Problem (1)–(6) is

Ω

σ∇u

· ∇ψdxdt

−

]

∂

∂t

[ψ]dσdt

−

](0)[ψ](0)dσ = 0, (9)

for each ψ ∈ L

(Ω× (0,T)) such that ψ is in the class

(7), [ψ] ∈ H

(0,T;L

(Γ

)), and ψ vanishes on ∂Ω ×

(0,T), as well as at t = T.

Moreover, multiplying (1), (2) by u

, integrating

by parts and using (3)–(6), for all 0 < t < T, we obtain

the energy estimate

Ω

σ|∇u

dxdτ+

2ε

]

(x,t)dσ

2ε

(x)dσ ≤ C < +∞, (10)

where C does not depend on ε and the last inequality

is due to (8), taking into account that |Γ

N−1

∼ 1/ε.

Inequality (10) together with a suitable Poincar´e

type lemma assures that, up to a subsequence, u

→ u

weakly in L



Ω × (0,T)



. It remains to identify the

limit function u, and this will be done in the following

theorem.

Theorem 3.1. Under the assumptions listed in Sec-

tion 2, as ε → 0, we have that u

→ u, weakly

in L

(Ω × (0, T)), and strongly in L

loc

(0,T;L

(Ω)),

where the limit u ∈ L

(0,T;H

(Ω)) solves in Ω

−div

∇

u+ A

∇

B(t − τ)∇

u(x,τ)dτ

with u = 0 on ∂Ω. Here F is a source depending

on the initial condition S

in (8) and the two matri-

ces A

,B are symmetric and A := σ

I + A

is positive

deﬁnite.

The proof of this theorem can be found in (Amar

et al., 2003) and (Amar et al., 2004b) where F ,A

are explicitly deﬁned.

Remark 3.2. In this regard, different models are ob-

tained corresponding to different scaling with respect

to ε (where ε denotes the length of the periodicity cell)

of the relevant physical quantity α, entering in the dy-

namical interface condition given by

∂

∂t

] = σ∇u

out

· ν, on Γ

, (11)

with k ∈ Z. As we state in the previous theorem, the

case k = 1 leads to an elliptic equation with memory,

while the case k = −1 leads to a degenerate parabolic

system, the well known bidomain model for the car-

diac syncithial tissue (Krassowska and Neu, 1993),

(Pennacchio et al., 2005). In turn, the case k = 0 leads

to a standard elliptic equation (Lipton, 1998),(Amar

et al., 2006).

In (Amar et al., 2006) we analyze in details the

whole family k ∈ Z, proving that, for k ≥ 2, the cor-

responding homogenized model reduces to a standard

diffraction problem, while for k ≤ −2, in the limit we

obtain two independent standard Neumann problems.

We would like to observe that only the cases cor-

responding to k = 1 and k = −1 in (11), preserve

memory,in the limit, of the membraneproperties (i.e.,

of the constant α). This is not true for all the other

choices of k.

It is not yet clear which one of these two mod-

els is more appropriate to describe the physical situ-

ation. Indeed, it seems that both of them are valid in

their respective frequency ranges. However, the one

presented here (i.e., model (1)–(6)) seems to be more

suitable to describe the response of a biological tissue

when high frequencies of alternating currents (of the

order of Megahertz) are applied, since in this case the

relevance of the capacitive properties of the dielectric

membrane increases. In the case of frequencies of

SIMULTECH2013-3rdInternationalConferenceonSimulationandModelingMethodologies,Technologiesand

Applications

698

the order of hundreds of Megahertz an improved ver-

sion of this model has been developed in (Amar et al.,

2009b) and (Amar et al., 2010). On the contrary, the

case k = −1 has been applied to low frequencies in

the context of activation of cardiac muscle.

The applicability of this model to real physical sit-

uations is connected to the study of an inverse prob-

lem, which for the elliptic equation is tipically related

to the study of the Neumann-Dirichlet map. This

problem has been widely studied. On the contrary

(apart from some geometrically simple cases), the in-

verse problem for the homogenized equation in The-

orem 3.1 is still open; in this case, the usual Dirichlet-

Neumann map should be replaced with a map in

which we assign the Dirichlet boundary condition to-

gether with the condition:

∂u

∂n

∂u

∂x

(t−τ)

∂u

∂x

(x,τ)n

dτ = h(x,t),

where n is the outward normal to ∂Ω and h is a given

function.

3.2 Concentration of the Physical

Problem

We point out that in the physical setting, the cell mem-

brane has a nonzero thickness, even if it is very small

with respect to the characteristic length of the cell.

Hence, we denote by η the ratio between these two

quantities and remark that η << 1. Moreover, we

write Ω as Ω = Ω

ε,η

∪ Γ

ε,η

∪ ∂Γ

ε,η

, where Ω

ε,η

and

ε,η

are two disjoint open subsets of Ω, Γ

ε,η

is the

tubular neighborhood of Γ

with thickness εη, and

∂Γ

ε,η

is its boundary. In addition, we assume also that

Ω

ε,η

= Ω

ε,η

int

∪ Ω

ε,η

out

and ∂Γ

ε,η

= (∂Ω

ε,η

int

∪ ∂Ω

ε,η

out

) ∩ Ω.

Again, Ω

ε,η

out

, Ω

ε,η

int

correspond to the conductive re-

gions, and Γ

ε,η

to the dielectric shell. We assume

that, for η → 0 and ε > 0 ﬁxed, |Γ

ε,η

| ∼ εη|Γ

N−1

Ω

ε,η

→ Ω

out

∪ Ω

int

and ∂Γ

ε,η

→ Γ

. We employ also

the notation Y = E

∪ Γ

∪ ∂Γ

, where E

and Γ

are two disjoint open subsets of Y, Γ

is the tubu-

lar neighborhood of Γ with thickness η, and ∂Γ

its boundary. Moreover, E

= E

int

∪ E

out

(see Fig-

ure 2). For η → 0, E

→ E

int

∪ E

out

, |Γ

| ∼ η|Γ|

N−1

and ∂Γ

→ Γ.

The classical governing equation is derived from

the Maxwell system in the quasi-static approximation,

which gives

−div(A

∇u

) = 0, in Ω

ε,η

; (12)

−div(B

∇u

) = 0, in Γ

ε,η

; (13)

∇u

· ν

= B

∇u

· ν

, on ∂Γ

ε,η

; (14)

out

int

∂Γ

out

int

Figure 2: The periodic cell Y.

Left

: before concentration;

is the shaded region, and E

= E

int

∪ E

out

is the white

region.

Right

: after concentration; Γ

shrinks to Γ as η →

∇u

(x,0) = S

(x), in Γ

ε,η

; (15)

(x,t) = 0, on ∂Ω. (16)

We assume that the conductivity A

> 0 is such

that A

= σ

int

in Ω

ε,η

int

, A

= σ

out

in Ω

ε,η

out

; the perme-

ability B

> 0 is such that B

= αη; and S

= ∇

for some

∈ H

(Γ

ε,η

) with |S

| ∼ 1/η.

Remark 3.3. We are interested in preserving, in the

limit η → 0, the conduction across the membrane Γ

instead of the tangential conduction on Γ

. To this

purpose, we need to preserve the ﬂux B

∇u

εt

· ν and

the jump [u

εt

] across the dielectric shells to be con-

centrated. This is the reason for which we rescale

= αη, instead of scaling B

= α/η in Γ

ε,η

, as more

usual in concentrated-capacity literature.

We are next interested in passing to the limit for

η → 0

, keeping ε > 0 ﬁxed. In (Amar et al., 2006)

we proved the following result.

Theorem 3.4. Under the previous assumptions, when

η → 0

, it follows that the concentration of Problem

(12)–(16) is given by (1)–(6) (with f ≡ 0). More pre-

cisely, as η → 0

it follows that u

→ u

, weakly in

loc

(Ω × (0,T)), where u

ε|Ω

int

∈ L

loc

(0,T;H

(Ω

int

)),

ε|Ω

out

∈ L

loc

(0,T;H

(Ω

out

)) and u

is the unique so-

lution of (1)–(6) (with f ≡ 0). Moreover, as η →

, ∇u

→ ∇u

, weakly in L

loc

(Ω

int

× (0, T)) and in

loc

(Ω

out

× (0,T)).

3.3 Well-posedness Results

The ﬁrst result of this section is connected with the

existence and uniqueness of the solution of the mi-

croscopic problem; hence ε is assumed to be ﬁxed and

equal to 1.

Theorem 3.5. Let Ω be an open connected bounded

subset of R

such that Ω = Ω

∪ Ω

∪ Γ, where

Ω

and Ω

are two disjoint open subset of Ω, Γ =

∂Ω

∩ Ω = ∂Ω

∩ Ω is a compact regular set, and

Γ ∩ ∂Ω =

0. Assume also that Ω, Ω

and Ω

have

Lipschitz boundaries. Let α > 0 and β ≥ 0. Let

ElectricalConductioninBiologicalTissues-HomogenizationTechniquesandAsymptoticDecayforLinearandNonlinear

Problems

699

f ∈ L

(Ω × (0,T)), q,h ∈ L

(0,T;L

(Γ)), and S ∈

1/2

(Γ). Therefore, problem

−σ∆v = f(t), in Ω

, Ω

; (17)

[σ∇v· ν] = q(t) , on Γ; (18)

∂

∂t

[v] = σ

out

∇v

(out)

· ν+ h(t), on Γ; (19)

[v](x,0) = S, on Γ; (20)

v(x,t) = 0, on ∂Ω; (21)

admits a unique solution v ∈ L

(0,T;H

(Ω)) with

[v] ∈ C(0,T;L

(Γ)), where H

(Ω) = {u = (u

) |

:= u

|Ω

int

, u

:= u

|Ω

out

with u

∈ H

(Ω)}.

The technique employed to prove this theorem re-

lies on a result of existence and uniqueness for ab-

stract parabolic equations, to which Problem (17)–

(21) can be reduced by means of a suitable identiﬁ-

cation of the function spaces there involved (Zeidler,

1990, Chapter 23). For the details see (Amar et al.,

2005).

Remark 3.6. Note that the same result as in Theorem

3.5 holds if we assume that Ω = Y = (0, 1)

, g(·,t)

is Y-periodic for a.e. t ∈ (0,T), f and q satisfy the

compatibility condition

f(y,t)dy =

q(y,t)dy for a.e. t ∈ (0,T),

and we replace (21) with the requirement that v(·,t) is

Y-periodic.

For the homogenized problem an existence and

uniqueness theorem, both for weak and classical so-

lutions, is available.

Theorem 3.7. Let A ∈ L

∞

(Ω;R

) be a symmet-

ric matrix such that λ|ξ|

≤ A(x)ξ · ξ ≤ Λ|ξ|

, for

suitable 0 < λ < Λ < +∞, for almost every x ∈ Ω

and every ξ ∈ R

; let B ∈ L

(0,T;L

∞

(Ω;R

)),

and let g ∈ L

(0,T;H

(Ω)). Assume that f : Ω ×

(0,T) → R is a Carath

eodory function such that

f ∈ L

(0,T;H

−1

(Ω)) and g ∈ L

(0,T;H

(Ω)).

Then, there exists a unique function u ∈

(0,T;H

(Ω)) satisfying in the sense of distribu-

tions

−div



A(x)∇

B(x,t − τ)∇

u(x,τ) dτ



= f(x,t)

in Ω × (0,T) with u = g on ∂Ω × (0,T).

Theorem 3.8. Let m ≥ 0 be any ﬁxed integer and let

also 0 < γ < 1. Let A ∈ C

1+γ

(

Ω;R

) satisfy the as-

sumption of Theorem 3.7 and

B ∈ C

([0,T];C

1+γ

(

Ω;R

))

be such that

′

∈ L

(0,T;W

1,∞

(Ω;R

)).

Assume that f ∈ C

([0,T];C

m+γ

(

Ω)), and

that ∇

f(x,t) and f

(x,t) exist and are

bounded. Let g ∈ C

([0,T];C

m+2+γ

(

Ω)), with

∈ L

∞

(0,T;C

m+2+γ

(Ω)).

Then the solution u given in Theorem 3.8 be-

longs to C

([0,T];C

1+γ

(

Ω)) ∩ L

∞

(0,T;C

m+2+γ

(Ω))

and solves the problem in the classical sense.

Both the proofs can be obtained, for example, with

a standard delay argument or a ﬁxed point theorem,

together with an a-priori estimate in the correspond-

ing function spaces. The a-priori estimates are ob-

tained as in standard elliptic equations, using also the

Gronwall’s Theorem to deal with the memory term

(Amar et al., 2004a).

3.4 Stability

In this section we will give a brief description of the

asymptotic behavior of u

(x,t) and u(x,t) for large

times. The interest in studying the asymptotics of this

model is due to the fact that the diagnostic measure-

ments are in general performed at times signiﬁcantly

longer than the typical relaxation time of the system.

In the case where a homogeneous Dirichlet

boundary condition is satisﬁed, the following results

were proven in (Amar et al., 2009a).

Theorem 3.9. Let Ω

int

,Ω

out

,Γ

, σ

int

,σ

out

,α be as be-

fore. Assume that the initial datum S

satisﬁes (8). Let

be the solution of (1)–(6) (with f ≡ 0). Then

(·,t)k

(Ω)

≤ C(ε + e

−λt

) a.e. in (1,+∞), (22)

where C and λ are independent of ε. Moreover, if S

has null mean average over each connected compo-

nent of Γ

, it follows that

(·,t)k

(Ω)

≤ Ce

−λt

a.e. in (1,+∞). (23)

This theorem easily yields the following exponen-

tial time-decay estimate for u under homogeneous

Dirichlet boundary data.

Corollary 3.10. Under the assumptions of Theorem

3.9, if u

→ u weakly in L

(Ω× (0,

T)) for every T >

0, then

ku(·,t)k

(Ω)

≤ Ce

−λt

a.e. in (1,+∞). (24)

Next we are interested in the case of a nonhomo-

geneous but time-periodic Dirichlet boundary data for

and u. Then we assume

(x,t) = Ψ(x)Φ(t) and u(x,t) = Ψ(x)Φ(t),

(25)

on ∂Ω× (0, +∞), where

Φ(t) ∈ H

(R ), Ψ(x) ∈ H

), ∆Ψ = 0

(26)

SIMULTECH2013-3rdInternationalConferenceonSimulationandModelingMethodologies,Technologiesand

Applications

700

in Ω. Here and in the following a subscript # denotes

a space of T-periodic functions, for some ﬁxed T > 0.

In order to deal with this case, for every ε > 0

we introduce an auxiliary function u

which solves a

time-periodic version of the microscopic differential

scheme introduced in Section 2

−div(σ∇u

) = 0, in (Ω

int

∪ Ω

out

) × R ; (27)

[σ∇u

· ν] = 0, on Γ

× R ; (28)

∂

∂t

] = σ∇u

#,out

· ν, on Γ

× R ; (29)

(x,t) = Ψ(x)Φ(t), on ∂Ω× R ; (30)

(x,·) is T periodic, ∀x ∈ Ω; (31)

(·,t)] − S

(·) has null average over each

connected component of Γ

. (32)

Indeed, this problem is derived from (1)–(6) (with

f ≡ 0), replacing equation (5) with (31). Equation

(32) has been added in order to guarantee the unique-

ness of the solution, and is suggested by the observa-

tion that [u

(·,t)] − S

(·) has null average over each

connected component of Γ

, as a consequence of (1)–

(4), (5).

In (Amar et al., 2009a, Theorem 7) it has been

proved that as ε → 0, the function u

(x,t) approaches

a time-periodic function u

∈ H

(R ;H

(Ω)) solving

−div



A∇u

+∞

B(τ)∇u

(x,t−τ)dτ



= 0, (33)

in Ω × R , with u

= Ψ(x)Φ(t) on ∂Ω × R . Here A

and B the same matrices deﬁned in Theorem 3.1.

Moreover, the following result holds.

Theorem 3.11. Let Ω

int

,Ω

out

,Γ

, σ

int

,σ

out

,α be as

before. Assume that the initial datum S

satisﬁes (8)

and the boundary datum satisﬁes (26). Let {u

} and

} be the sequences of the solutions of (1)–(5) (with

f ≡ 0), (25) and (27)–(32), respectively. Then

(·,t) − u

(·,t)k

(Ω)

≤ Ce

−λt

a.e. in (1,+∞),

where C and λ are positive constants, independent of

ε.

This theorem easily yields the following exponen-

tial time-decay estimate for u− u

Corollary 3.12. Under the assumption of Theorem

3.11, if u

→ u and u

→ u

weakly in L

(Ω× (0,

T)),

for every

T > 0, then the following estimate holds:

ku(·,t) − u

(·,t)k

(Ω)

≤ Ce

−λt

a.e. in (1,+∞),

where C and λ are positive constants, independent of

ε.

Finally, expressing the function Φ by means of its

Fourier series; i.e.,

Φ(t) =

+∞

∑

k=−∞

iω

(34)

where ω

= 2kπ/T is the k-th circular frequency, and

representing the solution u

(x,t) as follows:

(x,t) =

+∞

∑

k=−∞

εk

(x)e

iω

, (35)

we obtain that the complex-valued functions v

εk

(x) ∈

(Ω) are such that v

εk

Ω

∈ H

(Ω

), i = 1,2, and for

k 6= 0 satisfy the problem

−div(σ∇v

εk

) = 0, in Ω

int

∪ Ω

out

; (36)

[σ∇v

εk

· ν] = 0, on Γ

; (37)

iω

εk

] = (σ∇v

εk

· ν)

out

, on Γ

; (38)

εk

= c

Ψ, on ∂Ω, (39)

whereas for k = 0 they satisfy the problem

−div(σ∇v

ε0

) = 0, in Ω

int

∪ Ω

out

; (40)

[σ∇v

ε0

· ν] = 0, on Γ

; (41)

(σ∇v

ε0

· ν)

out

= 0, on Γ

; (42)

ε0

= c

Ψ, on ∂Ω; (43)

ε0

] − S

(·) has null average

over each connected component of Γ

. (44)

Note that any solution v

εk

of Problem (36)–(39) is

such that [v

εk

] has null average over each connected

component of Γ

Finally, in (Amar et al., 2009a) the following ho-

mogenization result is proven:

Theorem 3.13. Let Ω

int

,Ω

out

,Γ

, σ

int

,σ

out

,α be as

before. Assume that the boundary datum satisﬁes

(26). Then, for k ∈ Z \ {0} [respectively, k = 0,

under the further assumptions (8), the solution v

εk

of Problem (36)–(39) [respectively, Problem (40)–

(44)] strongly converges in L

(Ω) to a function v

∈

(Ω) which is the unique solution of the problem

−div(A

∇v

) = 0, in Ω; (45)

= c

Ψ, on ∂Ω; (46)

where

= A+

+∞

B(t) e

−iω

dt , (47)

with A and B the same matrices deﬁned in Theorem

3.1.

ElectricalConductioninBiologicalTissues-HomogenizationTechniquesandAsymptoticDecayforLinearandNonlinear

Problems

701

Remark 3.14. Experimental measurements in clini-

cal applications are currently performed by assigning

time-harmonic boundary data and assuming that the

resulting electric potential is time-harmonic, too. This

assumption, which is often referred to as the limiting

amplitude principle, leads to the commonly accepted

mathematical model based on the complex elliptic

Problem (45)–(46) for the electric potential (Borcea,

2003), (Dehghani and Soni, 2005). In (Amar et al.,

2009a), in view of the preceding theorem, this phe-

nomenological equations have been mathematically

justiﬁed and, moreover, in (47) a quasi-explicit rela-

tion between the circular frequency ω

and the coefﬁ-

cient A

has been found.

4 THE NONLINEAR CASE

In this section the function f appearing in equation

(4) is assumed to be continuous and strictly mono-

tone increasing; moreover, we require that f(0) = 0

and | f(s)| ≤ Λ|s| ∀s ∈ R , where Λ > 0 is a suitable

constant. For later use, let us set

(Y) := {(u

(1)

(2)

) | u

(1)

:= u

int

(2)

:= u

out

with u

(1)

∈ H

), u

(2)

∈ H

), and uY −periodic},

and recall the deﬁnition of two-scale convergence.

Deﬁnition 4.1. Given a sequence {u

} ∈



0,T;L

(Ω)



and a function u ∈ L



0,T;L

(Ω ×



, we say that u

two-scale converges to u

in L



0,T;L

(Ω × Y)



for ε → 0 (and we write

2−sc

→ u) if

lim

ε→0

Ω

(x,t)ϕ





dxdt =

Ω×Y

u(x,y,t)ϕ(x,y,t)dxdydt

for any test function ϕ ∈ L



Y;C (

Ω× [0,T])



Following (Allaire et al., 1995) (see also (Hum-

mel, 2000)), we recall also the notion of two-

scale convergence for sequences of functions deﬁned

on periodic surfaces, suitably adapted to the time-

dependent case.

Deﬁnition 4.2. Given a sequence {v

} ∈



0,T;L

(Γ

)



and a function v ∈ L



Ω ×

(0,T);L

(Γ)



, we say that v

two-scale converges to

v in L



Ω × (0,T);L

(Γ)



for ε → 0 (and we write

2−sc

→ v) if

lim

ε→0

(x,t)ψ





dσdt =

Ω

v(x,y,t)ψ(x,y,t)dxdσ(y)dt

for any test function ψ ∈ C



Ω× [0, T];C

(Y)



A weak formulation and an energy estimate anal-

ogous to the ones in (9) and (10) can be written down

also in this case, so that we can assert again that, up

to a subsequence, u

→ u weakly in L



Ω × (0,T)



where u is identiﬁed in the next theorem (see (Amar

et al., 2013a)).

Theorem 4.3. Let the assumptions listed in Sec-

tion 2 be satisﬁed and let f be as stated above.

Assume, in addition, that S

/ε two-scale converges

in L



Ω;L

(Γ)



to a function S

which satisﬁes

(x,·) = S

|Γ

(x,·) for some S ∈ C



Ω;C

(Y)



, and

lim

ε→0





(x)dσ =

Ω

(x,y)dxdσ(y).

Then there exists u ∈ L



0,T;H

(Ω)



and there exists

∈ L



Ω × (0,T);X

(Y)



such that, as ε → 0, we

have

→ u strongly in L

loc



0,T;L

(Ω)



Ω\Γ

∇u

2−sc

→ ∇u+ ∇

in L



0,T;L

(Ω×Y)



−1

]

2−sc

→ [u

] in L



Ω× (0, T);L

(Γ)



Moreover, the pair (u,u

) solves

−div



∇u+

σ∇



= 0, in Ω; (48)

−div

(σ∇u+σ∇

)=0, in Ω×(E

int

∪ E

out

); (49)

[σ(∇u+ ∇

) · ν] = 0, on Ω × Γ; (50)

∂

∂t

] + f ([u

]) = σ(∇u+ ∇

) · ν, on Ω× Γ;

(51)

](x,y,0) = S

(x,y), on Ω× Γ; (52)

u(x,t) = 0, on ∂Ω. (53)

As in Subsection 3.4, also in this nonlinear case

we are interested in studying the asymptotic behavior

of the macroscopic potential u for large times. In the

case where a homogeneous Dirichlet boundary con-

dition is satisﬁed, the following result is proven in

(Amar et al., 2013b, in preparation), which is anal-

ogous to the one stated in Corollary 3.10.

Theorem 4.4. Let u,u

be the solution of the homog-

enized Problem (48)–(53). Then,

ku(·,t)k

(Ω)

≤ Ce

−λt

a.e. in (1,+∞)

SIMULTECH2013-3rdInternationalConferenceonSimulationandModelingMethodologies,Technologiesand

Applications

702

5 CONCLUSIONS AND FUTURE

PERSPECTIVES

As already pointed out in Remark 3.14 our research

gives, at least in the linear case, a mathematical jus-

tiﬁcation of the phenomenological model (45)–(46)

commonly accepted in clinical applications, when

time-harmonic boundary data are assigned (Borcea,

2003), (Dehghani and Soni, 2005). At the same time,

in (47) a quasi-explicit relation between the circular

frequency ω

and the coefﬁcient A

has been found.

Moreover, we provide also a model for the case of

general periodic boundary data (see (33)). These re-

sults could be useful in clinical applications for the

reduction of the noise problems still affecting the di-

agnostic image reconstruction.

Our future research will be mainly aimed at ob-

taining similar results also in the nonlinear case

where, at present, the asymptotic behavior of the elec-

tric potential, when time-harmonic or periodic bound-

ary data are assigned, has not completely been ex-

ploited.

REFERENCES

Allaire, G. (1992). Homogenization and two-scale conver-

gence. SIAM J. Math. Anal., 23:1482–1518.

Allaire, G., Damlamian, A., and Hornung, U. (1995). Two-

scale convergence on periodic surfaces and applica-

tions. Proceedings of the International Conference on

Mathematical Modelling of Flow through Porous Me-

dia, pages 15–25.