Vol. 39 (2008) ACTA PHYSICA POLONICA B No 11 [610094]

Vol. 39 (2008) ACTA PHYSICA POLONICA B No 11
UPSCALING IN DYNAMICAL HEAT TRANSFER
PROBLEMS IN BIOLOGICAL TISSUES
Claudia Timofte
Department of Mathematics, Faculty of Physics, University of Bucharest
P.O. Box MG-11, Bucharest-Magurele, Romania
(Received February 5, 2008; revised version received May 19, 20 08)
The asymptotic behaviour of the solution of a nonlinear dyna mical
boundary-value problem describing the bio-heat transfer i n microvascular
tissues is analysed. Our domain Ωis an ε-periodic structure, consisting
of two parts: a solid tissue part Ωεand small regions of blood Ω\Ωε
of a certain temperature, εrepresenting a small parameter related to the
characteristic size of the blood regions. In such a domain, w e shall consider
a heat equation, with nonlinear sink and source terms and wit h a dynam-
ical condition imposed on the boundaries of the blood zones. The limit
equation, as ε→0, is a new heat equation, with extra-terms coming from
the influence of the non-homogeneous dynamical boundary con dition.
PACS numbers: 02.60.Lj, 47.63.Jd, 44.35.+c
1. Introduction and setting of the problem
The aim of this paper is to study the asymptotic behaviour of t he solu-
tion of a dynamical nonlinear boundary-value problem model ling thermoreg-
ulation phenomena in the human microvascular system. Such d ynamical
boundary-value problems, although not too widely consider ed in the litera-
ture, are very natural in many other mathematical models, su ch as partially
saturated flows in porous media, heat transfer in a solid in co ntact with
a moving fluid, diffusion phenomena in porous media (see [2,4, 16] and [17]
and the references therein).
The bio-heat transport in living tissues is a complex proces s involving
multiple mechanisms, such as conduction, convection, radi ation, metabolism,
etc.Bio-heat transfer models have significant applications in ma ny clinical
and environmental sciences. In particular, the heat transf er mechanism in
biological tissues is important for therapeutic practices , such as cancer hy-
perthermia, burn injury, brain hypothermia resuscitation , disease diagnos-
tics, cryosurgery, etc.
(2811)

2812 C. Timofte
LetΩbe an open bounded set in Rn(n≥2). For the model we intent
to analyse, i.e.the problem of bio-heat transfer in microvascular tissues,
we can consider that Ωis anε-periodic structure, consisting of two parts:
a solid tissue part Ωεof temperature uεand small regions of blood Ω\Ωε
of a certain temperature. εrepresents a small parameter related to the
characteristic size of the blood regions.
The nonlinear problem studied in this paper concerns the non -stationary
heat transfer in the solid tissue part, in contact with the bl ood regions. We
shall assume that we have some external thermal sources finside Ωεand
some nonlinear sink term describing heat loss (cell-destru ction energy, gene-
rated, possible, by special chemical reactions), given by a nonlinear func-
tionβ. Also, due to the fact that this complicated microstructure is dynam-
ically evolving, we shall impose a dynamical nonlinear boun dary condition
on the boundaries of the blood zones.
If we denote by (0,T)the time interval of interest, we shall analyse the
asymptotic behaviour, as ε→0, of the solution of the following problem:
ρcp∂uε
∂t−D0∆uε+β(uε) =f(t,x), inΩε×(0,T),(1)
D0∂uε
∂ν+αε∂uε
∂t=εa(uε
b−uε),onSε×(0,T), (2)
uε(0,x) =u0(x), inΩε, (3)
uε(0,x) =v0(x), onSε, (4)
uε= 0, on∂Ω×(0,T).(5)
Here, νis the exterior unit normal to Ωε,f∈L2(0,T;L2(Ω)),u0∈H1
0(Ω),
v0∈L2(Sε),a >0,D0>0,cp>0,ρ >0,α >0,uε
b∈H1(Ω)andSεis
the boundary of the blood regions. Notice that on Sεwe assume that the
temperature v0(x)is equal to the trace of u0(x). Also, we shall assume that
the nonlinear function βis given (see Section 2).
The existence and uniqueness of a weak solution of problem (1 )–(5) can
be settled by using the theory of nonlinear monotone problem s (see Sec-
tion 2).
Let us notice that the scaling in front of the blood perfusion term in
the boundary condition (2) is the unique one that enables us t o preserve at
the macroscale such an effect and to get the classical Helmhol tz term in the
macroscopic bio-heat equation. If we scale the dynamic term in (2) by ε2as
in [29], we lose at the macroscale the extra term capturing th e dynamical
character of this complicated microstructure.

Upscaling in Dynamical Heat Transfer Problems in Biologica l Tissues 2813
Also, let us mention that we can get similar results if the ter m−D0∆uε
in (1) is replaced by a general strong elliptic operator −div(Aε∇uε), where Aε
isY-periodic and satisfies strong ellipticity conditions. The positive param-
eterεwill also define a length scale measuring how densely the inho mo-
geneities are distributed in Ω.
In fact, this means to assume that we are dealing with heterog eneous
tissues. From a mathematical point of view, we shall conside r the case of
a general medium, having discontinuous properties, repres ented by a coercive
periodic matrix with rapidly oscillating coefficients. Let A∈L∞
#(Ω)n×nbe
a symmetric matrix whose entries are Y-periodic, bounded and measurable
real functions. We use the symbol #to denote periodicity properties. Let
us assume that for some 0< α < β ,
α|ξ|2≤A(y)ξ ξ≤β|ξ|2∀ξ, y∈Rn.
We shall denote by Aε(x)the value of A(y)at the point y=x/ε,i.e.
Aε(x) =A/parenleftBigx
ε/parenrightBig
.
Since the period of the structure is small compared to the dim ension
ofΩ, or in other words, since the non-homogeneities are small co mpared
to the global dimension of the structure, an asymptotic anal ysis becomes
necessary. Two scales are important for a suitable descript ion of the given
structure: one which is comparable with the dimension of the period, called
themicroscopic scale and denoted by y=x/ε, and another one which is of
the same order of magnitude as the global dimension of our sys tem, called
themacroscopic scale and denoted by x.
The main goal of the homogenisation method is to pass from the mi-
croscopic scale to the macroscopic one; more precisely, usi ng the homogeni-
sation method, we try to describe the macroscopic propertie s of our non-
homogeneous system in terms of the properties of its microsc opic structure.
Intuitively, the non-homogeneous real system, having a ver y complicated
microstructure, is replaced by a fictitious homogeneous one , whose global
characteristics represent a good approximation of the init ial system. Hence,
the homogenisation method provides a general framework for obtaining these
macroscale properties, eliminating the difficulties relate d to the explicit de-
termination of a solution of the problem at the microscale an d offering a less
detailed description, but one which is applicable to much mo re complex
systems.
Also, from the point of view of numerical computation, the ho mogenised
equation, defined on a fixed domain Ωand describing the effective behaviour
of our system, will have constant coefficients, called effecti ve or homogenised

2814 C. Timofte
coefficients (see Section 2), and, hence, it will be easier to b e solved numer-
ically than the original equation, which was an equation wit h rapidly oscil-
lating coefficients defined on a perforated domain and satisfy ing nonlinear
conditions on the boundaries. The dependence on the real mic rostructure is
given through the homogenised coefficients.
Hence, we shall be interested in getting the asymptotic beha viour, when
ε→0, of the solution of problem (1)–(5). Using a classical homog enisa-
tion method, i.e.Tartar’s method of oscillating test functions, coupled wit h
monotonicity methods and results from the theory of semilin ear problems,
we can prove that the solution of problem (1)–(5), properly e xtended to
the whole of Ω, converges to the unique solution of a new nonlinear prob-
lem, defined all over the domain Ω, given by a new operator and containing
extra zero order terms, capturing the effect of the blood perf usion and of
the influence of the non-homogeneous dynamical condition im posed on the
boundaries of the blood regions (see Section 2).
We can treat in the same manner the case in which we consider va riable
metabolic heat generation, given by suitable nonlinear fun ctions f(t,x,u).
Other nonlinear problems modelling various physical pheno mena arising
in radiophysics, filtration theory, rheology, elasticity t heory, in the theory of
composites, polycrystals and smart materials and in other d omains of me-
chanics, physics and technology can benefit from a similar eff ective medium
approach (see, for instance, the monograph [20] and [7]).
The results of this paper constitute a generalisation of som e of the re-
sults obtained in [15], by considering non-stationary proc esses, dynamical
conditions on the boundaries of the blood regions and a nonli near sink term
acting inside the solid tissue, modelling cell-destructio n energy, which can
be of huge importance, for instance, in destroying malignan t cells by hyper-
thermia (see [24]).
In 1948, based on experimental observation, Pennes (see [22 ]) proposed
a simple linear mathematical model for describing the therm al interaction
between human tissues and perfused blood, taking also into a ccount the
effects of the metabolism. Later on, alternative models for d escribing the
heat exchange between tissues and blood have been developed (see [8,15,24]
and the references therein).
From a mathematical point of view, problems closed to this on e have been
considered by Cioranescu and Donato [9], Cioranescu, Donat o and Ene [11],
Conca and Donato [14], Conca, Díaz and Timofte [13], Timofte [26–28], Ene
and Polisevski [18], Timofte [25], Bourgeat and Pankratov [5 ], Pankratov,
Piatnitskii and Rybalko [21] for the deterministic case, Wa ng and Duan [29]
for the stochastic one.

Upscaling in Dynamical Heat Transfer Problems in Biologica l Tissues 2815
The plan of the paper is as follows: in Section 2 we introduce s ome useful
notations and assumptions and we give the main convergence r esult of this
paper. For obtaining it, we need some preliminary results, w hich are given
in Section 3. The last section is devoted to the proof of the co nvergence
result.
2. Notation and assumptions
LetΩbe a bounded connected open subset of Rn(n≥2), with ∂Ωof
classC2and let [0,T]be the time interval of interest. Let Y= [0,l1[×…[0,ln[
be the representative cell in RnandFan open subset of Ywith boundary
∂Fof class C2, such that F⊂Y.
We shall denote by F(ε,k)the translated image of εFby the vector εkl,
k∈Zn,kl= (k1l1,….,k nln):
F(ε,k) =ε(kl+F).
Also, we shall denote by Fεthe set of all the holes contained in Ω. So
Fε=/uniondisplay
k∈Zn{F(ε,k)|F(ε,k)⊂Ω}.
LetΩε=Ω\Fε. Hence, Ωεis a periodically perforated domain with holes
of the same size as the period. Let us remark that the holes do n ot intersect
the boundary ∂Ω.
We shall use the following notations:
Y∗=Y
F, (6)
θ=|Y∗|
|Y|. (7)
Also, we shall denote by χεthe characteristic function of the domain Ωεand
throughout the paper, by Cwe shall denote a generic fixed strictly positive
constant, whose value can change from line to line.
As already mentioned, we are interested in studying the asym ptotic be-
haviour, as ε→0, of the solution of the parabolic problem (1)–(5).
We shall consider that the function βin (1) is continuous, monotonously
non-decreasing and such that β(0) = 0 . Moreover, we shall assume that
there exist C≥0and an exponent qsuch that
|β(v)| ≤C(1 +|v|q), (8)
with0≤q < n/ (n−2)ifn≥3and0≤q <+∞ifn= 2.

2816 C. Timofte
For the blood temperature uε
bwe shall assume that uε
b∈H1(Ω)and
/ba∇dbluε
b/ba∇dblH1(Ω)≤C.
Remark 1. The results of this paper will be obtained for the case
n≥3. All of them are still valid, under our assumptions, in the ca se in
which n= 2. Of course, for this case, n/(n−2)has to be replaced by +∞.
Let us notice that due to the compactness injection theorems in Sobolev
spaces, it would be enough, with the same reasoning as in the p aper, to
assume that βsatisfies, for n≥3, the growth condition (8) for some 0≤
q <(n+ 2)/(n−2). Forn= 2,(n+ 2)/(n−2)have to be replaced by +∞.
The existence and uniqueness of a weak solution of (1)–(5) ca n be settled
by using the classical theory of semilinear monotone proble ms (see [1,3,5,6,
25,29]). As a result, we know that there exists a unique weak s olution
uε∈C/parenleftbig
[0,T];H1
∂Ω(Ωε)/parenrightbig/intersectiondisplay
L2(0,T;Y1(Ωε)),
with∂uε
∂t∈L2(0,T;L2(Ωε))
and
∂γ(uε)
∂t∈L2/parenleftbig
0,T;L2(Sε)/parenrightbig
.
Here, H1
∂Ω(Ωε)is the space of elements of H1(Ωε)which vanish (in the sense
of traces) on ∂Ω,
Y1(Ωε) =/braceleftbigg
v∈H1
∂Ω(Ωε)| −∆v∈L2(Ωε),R∂v
∂n∈L2
∂Ω(∂Ωε)/bracerightbigg
andγ:H1(Ωε)→L2(Sε)is the trace operator with respect to Sε, which
is continuous. Moreover, for a function ϕdefined on ∂Ωε,Rϕdenotes its
restriction to Sε.
The main convergence result of this paper is given by the foll owing the-
orem:
Theorem 1. One can construct an extension Pεuεof the solution uεof
the problem (1)–(5) such that Pεuε⇀ uweakly in L2(0,T;H1
0(Ω)), where
uis the unique solution of the following nonlinear problem:


ρcp(1 +δ)∂u
∂t−n/summationdisplay
i,j=1qij∂2u
∂xi∂xj+β(u)
+a|∂F|
|Y∗|(u−ub) =f , x ∈Ω , t ∈(0,T),
u= 0, x ∈∂Ω , t ∈(0,T),
u(0,x) =u0(x), x ∈Ω .(9)

Upscaling in Dynamical Heat Transfer Problems in Biologica l Tissues 2817
Here,
δ=α
ρcp|∂F|
|Y∗|
andQ= ((qij))is the homogenised matrix, whose entries are defined by:
qij=D0
δij+1
|Y∗|/integraldisplay
Y∗∂χj
∂yidy
, (10)
in terms of the functions χi, i= 1,…,n, solutions of the cell problems


−∆χi= 0inY∗,
∂(χi+yi)
∂ν= 0on∂F,
χiY−periodic.
Thus, in the limit, when ε→0, we get a constant coefficient heat equa-
tion, with a Dirichlet boundary condition and with a constan t (due to the
periodicity) extra-term in front of the time derivative, co ming from the well-
balanced contribution of the dynamical part of our boundary condition on
the surface of the blood regions. Also, the effect of the blood perfusion, de-
scribed by the linear Newton’s cooling law, is captured in th e limit equation.
Remark 2. There exists a unique solution of the macromodel prob-
lem (9).
Remark 3. As mentioned in the Introduction, in the general case of
an heterogeneous medium, it is not difficult to see that the lim it problem is
the following one:


ρcp(1 +δ)∂u
∂t−div(A0∇u) +β(u)
+a|∂F|
|Y∗|(u−ub) =f , x ∈Ω , t ∈(0,T),
u= 0, x ∈∂Ω , t ∈(0,T),
u(0,x) =u0(x), x ∈Ω .(11)
Here, A0= (a0
ij)is the classical homogenized matrix, whose entries are
defined as follows:
a0
ij=1
|Y∗|/integraldisplay
Y∗/parenleftbigg
aij(y) +aik(y)∂χj
∂yk/parenrightbigg
dy ,

2818 C. Timofte
in terms of the functions χj, j= 1,…,n, solutions of the cell problems


−divyA(y)(Dyχj+ej) = 0 inY∗,
A(y)(Dχj+ej)·ν= 0on∂F ,
χj∈H1
#Y(Y⋆),/integraldisplay
Y⋆χj= 0,
where ei,1≤i≤n, are the elements of the canonical basis in IRn. The
constant matrix A0is symmetric and positive-definite.
In this limit problem, the periodic heterogeneous structur e of our medium
is reflected by the presence of the homogenized matrix A0.
3. Preliminary results
As already mentioned, there exists a unique solution for the nonlinear
problem (1)–(5),
uε∈C([0,T];H1
∂Ω(Ωε))/intersectiondisplay
L2(0,T;Y1(Ωε)),
with∂uε
∂t∈L2/parenleftbig
0,T;L2(Ωε)/parenrightbig
and
∂γ(uε)
∂t∈L2/parenleftbig
0,T;L2(Sε)/parenrightbig
.
In order to extend it to the whole of Ω, let us recall the following result
(see [12]):
Lemma 1. There exists a linear continuous extension operator Pε∈
L(L2(Ωε);L2(Ω))∩ L(Vε;H1
0(Ω))and a positive constant C, independent
ofε, such that, for any v∈Vε,
/ba∇dblPεv/ba∇dblL2(Ω)≤C/ba∇dblv/ba∇dblL2(Ωε)
and
/ba∇dbl∇Pεv/ba∇dblL2(Ω)≤C/ba∇dbl∇v/ba∇dblL2(Ωε).
Here,
Vε={v∈H1(Ωε)|v= 0on∂Ω}.

Upscaling in Dynamical Heat Transfer Problems in Biologica l Tissues 2819
For getting the effective behaviour of our solution uε, we have to pass
to the limit in the variational formulation of problem (1)–( 5) (see (15)). To
this end, let us introduce, for any h∈Ls′(∂F),1≤s′≤ ∞, the linear form
µε
honW1,s
0(Ω)defined by
/a\}b∇acketle{tµε
h,ϕ/a\}b∇acket∇i}ht=ε/integraldisplay
Sεh/parenleftBigx
ε/parenrightBig
ϕdσ ∀ϕ∈W1,s
0(Ω),
with1/s+ 1/s′= 1. It is proven in [9] that
µε
h→µhstrongly in/parenleftBig
W1,s
0(Ω)/parenrightBig′
, (12)
where
/a\}b∇acketle{tµh,ϕ/a\}b∇acket∇i}ht=µh/integraldisplay
Ωϕdx,
with
µh=1
|Y|/integraldisplay
∂Fh(y)dσ .
Ifh∈L∞(∂F)or even if his constant, we have (see [11])
µε
h→µhstrongly in W−1,∞(Ω). (13)
We denote by µεthe above introduced measure in the case in which h= 1.
Also, for obtaining the limit behaviour of our homogenisati on problem,
let us recall another result from [13].
LetHbe a continuously differentiable function, monotonously no n-de-
creasing and such that H(v) = 0 if and only if v= 0. We shall suppose that
there exists a positive constant Cand an exponent q, with 0≤q < n/ (n−2),
such that |H| ≤C(1 +|v|q). If we denote by q= (2n)/(q(n−2) +n), one
can prove (see [13]) that for any zε⇀ zweakly in H1
0(Ω), we get
H(zε)⇀ H(z)weakly in W1,q
0(Ω). (14)
4. Proof of the main result
Let us consider the variational formulation of problem (1)– (5):
ρcpT/integraldisplay
0/integraldisplay
Ωε˙uεϕdxdt +D0T/integraldisplay
0/integraldisplay
Ωε∇uε· ∇ϕdxdt +T/integraldisplay
0/integraldisplay
Ωεβϕdxdt
+αεT/integraldisplay
0/integraldisplay
Sε˙uεϕdxdt +aεT/integraldisplay
0/integraldisplay
Sε(uε−uε
b)ϕdxdt =T/integraldisplay
0/integraldisplay
Ωεfϕdxdt , (15)

2820 C. Timofte
for any ϕ∈C∞
0([0,T]×Ωε). Here, we have denoted by ˙the partial derivative
with respect to the time.
By classical existence and uniqueness results, we know that t here exists
a unique weak solution of (15). Taking it as a test function in (15) and using
our assumptions on the data and Cauchy–Schwartz, Poincaré’ s and Young’s
inequalities, we can obtain suitable energy estimates, ind ependent of ε, for
our solution (see [4,5,13,23,25,29]).
Denoting by Pεuεthe extension of uεgiven by Lemma 1, one can see
thatPεuεis bounded in L2(0,T;H1
0(Ω))and(∂Pεuε)/∂tis bounded in
L2(0,T;L2(Ω))(see, for details, [13, 25, 29]). So, by passing to a sub-
sequence, we have Pεuε⇀ u weakly in L2(0,T;H1
0(Ω))and strongly in
L2(0,T;L2(Ω))and(∂Pεuε)/∂t ⇀ ∂u/∂t weakly in L2(0,T;L2(Ω)).
It is well-known by now how to pass to the limit, with ε→0, in the linear
terms of (15) defined on Ωε(see, for instance [13], [25] and [29]). Also, recall
thatθis the weak- ⋆limit in L∞(Ω)ofχε. Thus, we get:
T/integraldisplay
0/integraldisplay
Ωε˙uεϕdxdt →T/integraldisplay
0/integraldisplay
Ω˙uθϕdxdt , (16)
D0T/integraldisplay
0/integraldisplay
Ωε∇uε· ∇ϕdxdt →T/integraldisplay
0/integraldisplay
ΩθQ∇u· ∇ϕdxdt , (17)
T/integraldisplay
0/integraldisplay
Ωεfϕdxdt →T/integraldisplay
0/integraldisplay
Ωθfϕdxdt . (18)
Let us see now how we can pass to the limit in the nonlinear term s in (15).
For the third term in the left-hand side of (15), let us notice that, exactly
like in [13] (see (14)), one can prove that for any zε⇀ zweakly in H1
0(Ω),
we see that β(zε)→β(z)strongly in Lq(Ω), where q= (2n)/(q(n−2) +n).
Therefore, we have
T/integraldisplay
0/integraldisplay
Ωεβ(uε)ϕdxdt →T/integraldisplay
0/integraldisplay
Ωβ(u)θϕdxdt . (19)
For the other nonlinear term in (15), using the convergence ( 13) written
forh= 1, we obtain that
ε/integraldisplay
Sεuεϕdx=/a\}b∇acketle{tµε,Pεuεϕ/a\}b∇acket∇i}ht →|∂F|
|Y|/integraldisplay
Ωuϕdx.

Upscaling in Dynamical Heat Transfer Problems in Biologica l Tissues 2821
Since uε
b∈H1(Ω)and/ba∇dbluε
b/ba∇dblH1(Ω)≤C, then, up to a subsequence, we get
uε
b⇀ u bweakly in H1(Ω).
Hence, integrating in time and using Lebesgue’s convergenc e theorem, it is
not difficult to see that
aεT/integraldisplay
0/integraldisplay
Sε(uε
b−uε)ϕdxdt →a|∂F|
|Y|T/integraldisplay
0/integraldisplay
Ω(ub−u)ϕdxdt . (20)
Also, we have
αεT/integraldisplay
0/integraldisplay
Sε˙uεϕdxdt →α|∂F|
|Y|T/integraldisplay
0/integraldisplay
Ω˙uϕdxdt . (21)
Putting together (16)–(21), we can pass to the limit in all th e terms in
(15) and we obtain exactly the variational formulation of th e limit problem
(9). As uis uniquely determined, the whole sequence Pεuεconverges to u
and this completes the proof of Theorem 1.
This work was supported by the CNCSIS Grant Ideas 992, under c on-
tract 31/2007. The author is grateful to the anonymous refer ee for his/her
valuable comments and suggestions, which improved the cont ent and the
presentation of this paper.
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