Electrical Conduction in Biological Tissues
Homogenization Techniques and Asymptotic Decay for Linear and Nonlinear
Problems
M. Amar
1
, D. Andreucci
1
, P. Bisegna
2
and R. Gianni
1
1
Dipartimento di Scienze di Base e Applicate per l’Ingegneria - Sapienza Universit`a di Roma
Via A. Scarpa 16, 00161 Roma, Italy
2
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
Copyright
c
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 be an open connected bounded subset of R
N
.
Let us introduce a periodic open subset E of R
N
, so
that E + z = E for all z Z
N
. For all ε > 0 define
ε
int
= εE,
ε
out
= \
εE. We assume that , E
have regular boundary, say of class C
for the sake
of simplicity. Moreover, we set =
ε
int
ε
out
Γ
ε
,
where Γ
ε
= ∂Ω
ε
int
= ∂Ω
ε
out
. We also employ
the notation Y = (0,1)
N
, 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(
E
int
,Y) > 0. Some generalizations
may be possible, but we do not dwell on this point
here. Finally, we assume that dist(Γ
ε
,∂Ω) > γε for
some constant γ > 0 independent of ε, by dropping
the inclusions contained in the cells ε(Y + z), z Z
N
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
ε
(x,t) (here the operators div
and act only with respect to the space variable x)
div(σ
int
u
ε
) = 0, in
ε
int
; (1)
div(σ
out
u
ε
) = 0, in
ε
out
; (2)
σ
int
u
(int)
ε
· ν = σ
out
u
(out)
ε
· ν, on Γ
ε
; (3)
i
Figure 1: An examples of admissible periodic structures in
R
2
. Left: Y is the dashed square, and E Y is the shaded
region. Right: the domain .
α
ε
t
[u
ε
] + f
[u
ε
]
ε
= σ
out
u
(out)
ε
· ν, on Γ
ε
; (4)
[u
ε
](x,0) = S
ε
(x), on Γ
ε
; (5)
u
ε
(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
ε
pointing into
ε
out
.
Since u
ε
is not in general continuous across Γ
ε
we
have set
u
(int)
ε
:= trace of u
ε|
ε
int
on Γ
ε
;
u
(out)
ε
:= trace of u
ε|
ε
out
on Γ
ε
.
Indeed we refer conventionally to
ε
int
as to the inte-
rior domain, and to
ε
out
as to the outer domain. We
also denote
[u
ε
] := u
(out)
ε
u
(int)
ε
.
Similar conventions are employed for other quanti-
ties; for example (3) can be rewritten as
[σ∇u
ε
· ν] = 0, on Γ
ε
,
where
σ = σ
int
in
ε
int
, σ = σ
out
in
ε
out
.
The function f and the initial data S
ε
will be dis-
cussed below.
Under the assumptions above, we prove existence
and uniqueness of a weak solution to (1)–(6), in the
class
u
ε|
ε
i
L
2
(0,T;H
1
(
ε
i
)), 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
S
ε
(x) γε, where S
ε
is the initial jump prescribed
in (5), for a fixed constant γ > 1, then u
ε
becomes
stable as ε 0 (i.e., it converges to a nonvanishing
bounded function). Therefore, let us stipulate that
S
ε
H
1/2
(Γ
ε
) and
S
ε
(x) = εS
1
x,
x
ε
+ εR
ε
(x), (8)
ElectricalConductioninBiologicalTissues-HomogenizationTechniquesandAsymptoticDecayforLinearandNonlinear
Problems
697
where S
1
: × E R , and
kS
1
k
L
(×E)
< , kR
ε
k
L
()
0, as ε 0;
S
1
(x,y) is continuous in x, uniformly over y E,
and periodic in y, for each x .
3 THE LINEAR CASE
In this section we assume that
f
ε
1
t
=
β
ε
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
ε
(x,t) = u
ε
(x,t) · exp
β
α
t
, it follows that v
ε
satisfies
div(σ
int
v
ε
) = 0, in
ε
int
;
div(σ
out
v
ε
) = 0, in
ε
out
;
σ
int
v
(int)
ε
· ν = σ
out
v
(out)
ε
· ν, on Γ
ε
;
α
ε
t
[v
ε
] = σ
out
v
(out)
ε
· ν, on Γ
ε
;
[v
ε
](x,0) = S
ε
(x), on Γ
ε
;
v
ε
(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
Z
T
0
Z
σ∇u
ε
· ∇ψdxdt
α
ε
Z
T
0
Z
Γ
ε
[u
ε
]
t
[ψ]dσdt
α
ε
Z
Γ
ε
[u
ε
](0)[ψ](0)dσ = 0, (9)
for each ψ L
2
(× (0,T)) such that ψ is in the class
(7), [ψ] H
1
(0,T;L
2
(Γ
ε
)), and ψ vanishes on ∂Ω ×
(0,T), as well as at t = T.
Moreover, multiplying (1), (2) by u
ε
, integrating
by parts and using (3)–(6), for all 0 < t < T, we obtain
the energy estimate
Z
t
0
Z
σ|u
ε
|
2
dxdτ+
α
2ε
Z
Γ
ε
[u
ε
]
2
(x,t)dσ
=
α
2ε
Z
Γ
ε
S
2
ε
(x)dσ C < +, (10)
where C does not depend on ε and the last inequality
is due to (8), taking into account that |Γ
ε
|
N1
1/ε.
Inequality (10) together with a suitable Poincar´e
type lemma assures that, up to a subsequence, u
ε
u
weakly in L
2
× (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
ε
u, weakly
in L
2
( × (0, T)), and strongly in L
1
loc
(0,T;L
1
()),
where the limit u L
2
(0,T;H
1
0
()) solves in
div
σ
0
x
u+ A
0
x
u+
Z
t
0
B(t τ)
x
u(x,τ)dτ
!
=F
with u = 0 on ∂Ω. Here F is a source depending
on the initial condition S
1
in (8) and the two matri-
ces A
0
,B are symmetric and A := σ
0
I + A
0
is positive
definite.
The proof of this theorem can be found in (Amar
et al., 2003) and (Amar et al., 2004b) where F ,A
0
,B
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
α
ε
k
t
[u
ε
] = σ∇u
out
ε
· ν, on Γ
ε
, (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:
σ
0
u
n
+A
0
ij
u
x
i
n
j
+
Z
t
0
B
ij
(tτ)
u
x
i
(x,τ)n
j
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 as =
ε,η
Γ
ε,η
∂Γ
ε,η
, where
ε,η
and
Γ
ε,η
are two disjoint open subsets of , Γ
ε,η
is the
tubular neighborhood of Γ
ε
with thickness εη, and
∂Γ
ε,η
is its boundary. In addition, we assume also that
ε,η
=
ε,η
int
ε,η
out
and ∂Γ
ε,η
= (∂Ω
ε,η
int
∂Ω
ε,η
out
) .
Again,
ε,η
out
,
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
ε
|
N1
,
ε,η
ε
out
ε
int
and ∂Γ
ε,η
Γ
ε
. We employ also
the notation Y = E
η
Γ
η
∂Γ
η
, where E
η
and Γ
η
are two disjoint open subsets of Y, Γ
η
is the tubu-
lar neighborhood of Γ with thickness η, and ∂Γ
η
is
its boundary. Moreover, E
η
= E
η
int
E
η
out
(see Fig-
ure 2). For η 0, E
η
E
int
E
out
, |Γ
η
| η|Γ|
N1
and ∂Γ
η
Γ.
The classical governing equation is derived from
the Maxwell system in the quasi-static approximation,
which gives
div(A
η
u
η
ε
) = 0, in
ε,η
; (12)
div(B
η
u
η
ε
t
) = 0, in Γ
ε,η
; (13)
A
η
u
η
ε
· ν
η
= B
η
u
η
ε
t
· ν
η
, on ∂Γ
ε,η
; (14)
E
η
out
E
η
int
Γ
η
Γ
η
ν
η
ν
η
ν
E
out
E
int
i
Figure 2: The periodic cell Y.
Left
: before concentration;
Γ
η
is the shaded region, and E
η
= E
η
int
E
η
out
is the white
region.
Right
: after concentration; Γ
η
shrinks to Γ as η
0.
u
η
ε
(x,0) = S
η
ε
(x), in Γ
ε,η
; (15)
u
η
ε
(x,t) = 0, on ∂Ω. (16)
We assume that the conductivity A
η
> 0 is such
that A
η
= σ
int
in
ε,η
int
, A
η
= σ
out
in
ε,η
out
; the perme-
ability B
η
> 0 is such that B
η
= αη; and S
η
ε
=
e
S
η
ε
,
for some
e
S
η
ε
H
1
(Γ
ε,η
) with |S
η
ε
| 1/η.
Remark 3.3. We are interested in preserving, in the
limit η 0, the conduction across the membrane Γ
ε
instead of the tangential conduction on Γ
ε
. To this
purpose, we need to preserve the flux B
η
u
η
εt
· ν and
the jump [u
η
εt
] across the dielectric shells to be con-
centrated. This is the reason for which we rescale
B
η
= αη, instead of scaling B
η
= 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
η
ε
u
ε
, weakly in
L
2
loc
( × (0,T)), where u
ε|
ε
int
L
2
loc
(0,T;H
1
(
ε
int
)),
u
ε|
ε
out
L
2
loc
(0,T;H
1
(
ε
out
)) and u
ε
is the unique so-
lution of (1)(6) (with f 0). Moreover, as η
0
+
, u
η
ε
u
ε
, weakly in L
2
loc
(
ε
int
× (0, T)) and in
L
2
loc
(
ε
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 be an open connected bounded
subset of R
N
such that =
1
2
Γ, where
1
and
2
are two disjoint open subset of , Γ =
∂Ω
1
= ∂Ω
2
is a compact regular set, and
Γ ∂Ω =
/
0. Assume also that ,
1
and
2
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
2
( × (0,T)), q,h L
2
(0,T;L
2
(Γ)), and S
H
1/2
(Γ). Therefore, problem
σv = f(t), in
1
,
2
; (17)
[σ∇v· ν] = q(t) , on Γ; (18)
α
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
2
(0,T;H
1
o
()) with
[v] C(0,T;L
2
(Γ)), where H
1
o
() = {u = (u
1
,u
2
) |
u
1
:= u
|
int
, u
2
:= u
|
out
with u
1
,u
2
H
1
o
()}.
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 = Y = (0, 1)
N
, g(·,t)
is Y-periodic for a.e. t (0,T), f and q satisfy the
compatibility condition
Z
Y
f(y,t)dy =
Z
Γ
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
(;R
N
2
) be a symmet-
ric matrix such that λ|ξ|
2
A(x)ξ · ξ Λ|ξ|
2
, for
suitable 0 < λ < Λ < +, for almost every x
and every ξ R
N
; let B L
2
(0,T;L
(;R
N
2
)),
and let g L
2
(0,T;H
1
()). Assume that f : ×
(0,T) R is a Carath
´
eodory function such that
f L
2
(0,T;H
1
()) and g L
2
(0,T;H
1
()).
Then, there exists a unique function u
L
2
(0,T;H
1
()) satisfying in the sense of distribu-
tions
div
A(x)
x
u+
Z
t
0
B(x,t τ)
x
u(x,τ) dτ
= f(x,t)
in × (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
(
;R
N
2
) satisfy the as-
sumption of Theorem 3.7 and
B C
0
([0,T];C
1+γ
(
;R
N
2
))
be such that
B
L
2
(0,T;W
1,
(;R
N
2
)).
Assume that f C
0
([0,T];C
m+γ
(
)), and
that
x
f(x,t) and f
t
(x,t) exist and are
bounded. Let g C
0
([0,T];C
m+2+γ
(
)), with
g
t
L
(0,T;C
m+2+γ
()).
Then the solution u given in Theorem 3.8 be-
longs to C
0
([0,T];C
1+γ
(
)) L
(0,T;C
m+2+γ
())
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
ε
(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
ε
int
,
ε
out
,Γ
ε
, σ
int
,σ
out
,α be as be-
fore. Assume that the initial datum S
ε
satisfies (8). Let
u
ε
be the solution of (1)(6) (with f 0). Then
ku
ε
(·,t)k
L
2
()
C(ε + e
λt
) a.e. in (1,+), (22)
where C and λ are independent of ε. Moreover, if S
ε
has null mean average over each connected compo-
nent of Γ
ε
, it follows that
ku
ε
(·,t)k
L
2
()
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
ε
u weakly in L
2
(× (0,
T)) for every T >
0, then
ku(·,t)k
L
2
()
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
u
ε
and u. Then we assume
u
ε
(x,t) = Ψ(x)Φ(t) and u(x,t) = Ψ(x)Φ(t),
(25)
on ∂Ω× (0, +), where
Φ(t) H
1
#
(R ), Ψ(x) H
1
(R
N
), ∆Ψ = 0
(26)
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700
in . 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
#
ε
which solves a
time-periodic version of the microscopic differential
scheme introduced in Section 2
div(σ∇u
#
ε
) = 0, in (
ε
int
ε
out
) × R ; (27)
[σ∇u
#
ε
· ν] = 0, on Γ
ε
× R ; (28)
α
ε
t
[u
#
ε
] = σ∇u
#,out
ε
· ν, on Γ
ε
× R ; (29)
u
#
ε
(x,t) = Ψ(x)Φ(t), on ∂Ω× R ; (30)
u
#
ε
(x,·) is T periodic, x ; (31)
[u
#
ε
(·,t)] S
ε
(·) has null average over each
connected component of Γ
ε
. (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
ε
(·,t)] S
ε
(·) has null average over each
connected component of Γ
ε
, 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
#
ε
(x,t) approaches
a time-periodic function u
#
H
1
#
(R ;H
1
()) solving
div
Au
#
+
Z
+
0
B(τ)u
#
(x,tτ)dτ
= 0, (33)
in × 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
ε
int
,
ε
out
,Γ
ε
, σ
int
,σ
out
,α be as
before. Assume that the initial datum S
ε
satisfies (8)
and the boundary datum satisfies (26). Let {u
ε
} and
{u
#
ε
} be the sequences of the solutions of (1)(5) (with
f 0), (25) and (27)(32), respectively. Then
ku
ε
(·,t) u
#
ε
(·,t)k
L
2
()
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
ε
u and u
#
ε
u
#
weakly in L
2
(× (0,
T)),
for every
T > 0, then the following estimate holds:
ku(·,t) u
#
(·,t)k
L
2
()
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=
c
k
e
iω
k
t
(34)
where ω
k
= 2kπ/T is the k-th circular frequency, and
representing the solution u
#
ε
(x,t) as follows:
u
#
ε
(x,t) =
+
k=
v
εk
(x)e
iω
k
t
, (35)
we obtain that the complex-valued functions v
εk
(x)
L
2
() are such that v
εk
|
ε
i
H
1
(
ε
i
), i = 1,2, and for
k 6= 0 satisfy the problem
div(σ∇v
εk
) = 0, in
ε
int
ε
out
; (36)
[σ∇v
εk
· ν] = 0, on Γ
ε
; (37)
iω
k
α
ε
[v
εk
] = (σ∇v
εk
· ν)
out
, on Γ
ε
; (38)
v
εk
= c
k
Ψ, on ∂Ω, (39)
whereas for k = 0 they satisfy the problem
div(σ∇v
ε0
) = 0, in
ε
int
ε
out
; (40)
[σ∇v
ε0
· ν] = 0, on Γ
ε
; (41)
(σ∇v
ε0
· ν)
out
= 0, on Γ
ε
; (42)
v
ε0
= c
0
Ψ, on ∂Ω; (43)
[v
ε0
] S
ε
(·) has null average
over each connected component of Γ
ε
. (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
ε
.
Finally, in (Amar et al., 2009a) the following ho-
mogenization result is proven:
Theorem 3.13. Let
ε
int
,
ε
out
,Γ
ε
, 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
2
() to a function v
0k
H
1
() which is the unique solution of the problem
div(A
ω
k
v
0k
) = 0, in ; (45)
v
0k
= c
k
Ψ, on ∂Ω; (46)
where
A
ω
k
= A+
Z
+
0
B(t) e
iω
k
t
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
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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
k
and the coeffi-
cient A
ω
k
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
X
1
#
(Y) := {(u
(1)
,u
(2)
) | u
(1)
:= u
|E
int
,u
(2)
:= u
|E
out
,
with u
(1)
H
1
(E
1
), u
(2)
H
1
(E
2
), and uY periodic},
and recall the definition of two-scale convergence.
Definition 4.1. Given a sequence {u
ε
}
L
2
0,T;L
2
()
and a function u L
2
0,T;L
2
( ×
Y)
, we say that u
ε
two-scale converges to u
in L
2
0,T;L
2
(× Y)
for ε 0 (and we write
u
ε
2sc
u) if
lim
ε0
Z
T
0
Z
u
ε
(x,t)ϕ
x,
x
ε
,t
dxdt =
Z
T
0
Z
×Y
u(x,y,t)ϕ(x,y,t)dxdydt
for any test function ϕ L
2
#
Y;C (
× [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
ε
}
L
2
0,T;L
2
(Γ
ε
)
and a function v L
2
×
(0,T);L
2
(Γ)
, we say that v
ε
two-scale converges to
v in L
2
× (0,T);L
2
(Γ)
for ε 0 (and we write
v
ε
2sc
v) if
lim
ε0
ε
Z
T
0
Z
Γ
ε
v
ε
(x,t)ψ
x,
x
ε
,t
dσdt =
Z
T
0
Z
Z
Γ
v(x,y,t)ψ(x,y,t)dxdσ(y)dt
for any test function ψ C
× [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
ε
u weakly in L
2
× (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
/ε two-scale converges
in L
2
;L
2
(Γ)
to a function S
1
which satisfies
S
1
(x,·) = S
|Γ
(x,·) for some S C
;C
1
#
(Y)
, and
lim
ε0
ε
Z
Γ
ε
S
ε
ε
2
(x)dσ =
Z
Z
Γ
S
2
1
(x,y)dxdσ(y).
Then there exists u L
2
0,T;H
1
o
()
and there exists
u
1
L
2
× (0,T);X
1
#
(Y)
such that, as ε 0, we
have
u
ε
u strongly in L
2
loc
0,T;L
2
()
,
1
\Γ
ε
u
ε
2sc
u+
y
u
1
in L
2
0,T;L
2
(×Y)
,
ε
1
[u
ε
]
2sc
[u
1
] in L
2
× (0, T);L
2
(Γ)
.
Moreover, the pair (u,u
1
) solves
div
σ
0
u+
Z
Y
σ∇
y
u
1
dy
= 0, in ; (48)
div
y
(σ∇u+σ∇
y
u
1
)=0, in ×(E
int
E
out
); (49)
[σ(u+
y
u
1
) · ν] = 0, on × Γ; (50)
α
t
[u
1
] + f ([u
1
]) = σ(u+
y
u
1
) · ν, on × Γ;
(51)
[u
1
](x,y,0) = S
1
(x,y), on × 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
1
be the solution of the homog-
enized Problem (48)(53). Then,
ku(·,t)k
L
2
()
Ce
λt
a.e. in (1,+)
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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
k
and the coefficient A
ω
k
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.
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ElectricalConductioninBiologicalTissues-HomogenizationTechniquesandAsymptoticDecayforLinearandNonlinear
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