Spectroscopic imaging of a dilute cell suspension Journal de Mathématiques Pures et Appliquées

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Contents lists available atScienceDirect

Journal de Mathématiques Pures et Appliquées

www.elsevier.com/locate/matpur

Spectroscopic imaging of a dilute cell suspension

^✩

Habib Ammari^a,∗, Josselin Garnier^b, Laure Giovangigli^c, Wenjia Jing^d, Jin-Keun Seo^e

a DepartmentofMathematics,ETHZürich,Rämistrasse101,CH-8092Zürich,Switzerland

bLaboratoiredeProbabilitésetModèlesAléatoires&LaboratoireJacques-LouisLions,Université Paris VII,75205ParisCedex13,France

cDepartmentofMathematics,340RowlandHall,UniversityofCaliforniaatIrvine,Irvine, CA 92697-3875,USA

dDepartmentofMathematics,TheUniversityofChicago,5734S.UniversityAvenue,Chicago,IL60637, USA

eDepartmentofComputationalScienceandEngineering,YonseiUniversity,50Yonsei-Ro, Seodaemun-Gu,Seoul 120-749,RepublicofKorea

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received15July2015

Availableonline19November2015

MSC:

35R30 35B30 65W21

Keywords:

Electricalimpedancespectroscopy Stochastichomogenization Maxwell–Wagner–Frickeformula Debyerelaxationtime

The paper aims at analytically exhibiting for the first time the fundamental mechanismsunderlyingthefactthateffectivebiologicaltissueelectricalproperties andtheirfrequencydependencereflectthetissuecompositionandphysiology.For doingso,ahomogenizationtheoryisderivedtodescribetheeffectiveadmittivityof cellsuspensions.A newformulaisreportedfordilutecasesthatgivesthefrequency- dependenteffectiveadmittivitywithrespecttothemembranepolarization.Different microstructuresareshowntobedistinguishableviaspectroscopicmeasurementsof theoveralladmittivityusingthespectralpropertiesofthemembranepolarization.

The Debyerelaxationtimesassociatedwiththemembranepolarizationtensorare showntobeabletogivethemicroscopicstructureofthemedium.A naturalmeasure oftheadmittivityanisotropyisintroducedanditsdependenceonthefrequencyof appliedcurrentisderived.A Maxwell–Wagner–Frickeformulaisgivenforconcentric circularcells, andtheresults canbe extendedto therandom cases. A randomly deformedperiodicmediumisalsoconsideredandanewformulaisderivedforthe overalladmittivityofadilutesuspensionofrandomlydeformedcells.

r é s u m é

Dans cet article, on introduit une approche spectroscopique aﬁn d’imager les propriétés électriques d’un tissu biologique. On construit un dévéloppement asymptotiquedel’admittivitéeﬀectivedutissuenfonctiondelafractionvolumique des cellules et du tenseur de polarisation de la membrane cellulaire. On étudie les propriétés de ce tenseur et on introduit le concept de temps de relaxation pour des géométries de cellules quelconques. Ce concept permet de distinguer

✩ ThisworkwassupportedbytheERCAdvancedGrantProjectMULTIMOD-267184.

* Correspondingauthor.

E-mailaddresses:habib.ammari@math.ethz.ch(H. Ammari),garnier@math.jussieu.fr(J. Garnier),lgiovang@uci.edu (L. Giovangigli),wjing@math.uchicago.edu(W. Jing),seoj@yonsei.ac.kr(J.-K. Seo).

http://dx.doi.org/10.1016/j.matpur.2015.11.009

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diﬀérentesorganisationscellulairesdutissuàpartirdemesuresspectroscopiquesde l’admittivitéeﬀective.

1. Introduction

The electric behavior of biologicaltissue under theinfluence of an electricfield at frequency ω canbe characterized by its frequency-dependent effective admittivity kef := σef(ω)+iωef(ω), where σef and εef are respectivelyits effectiveconductivity and permittivity. Electricalimpedance spectroscopy assesses the frequency dependence of the effective admittivity by measuring it across a range of frequencies from a few Hz to hundreds of MHz. Effective admittivity of biological tissues and its frequency dependence vary withtissue composition, membranecharacteristics, intra- and extra-cellular fluids and other factors.

Hence, the admittance spectroscopyprovides information about the microscopicstructure of the medium and physiologicalandpathologicalconditionsofthetissue.

The determinationof theeﬀective,or macroscopic, property of asuspensionis anenduringproblem in physics[44].Ithasbeenstudiedbymanydistinguishedscientists,includingMaxwell,Poisson[52],Faraday, Rayleigh [54], Fricke [31], Lorentz, Debye, and Einstein [26]. Many studies have been conducted on ap- proximateanalytic expressionsforoveralladmittivityofacellsuspensionfromtheknowledgeofpointwise conductivity distribution, and these studies were mostly restricted to the simpliﬁed model of a strongly dilutesuspensionofsphericalorellipsoidalcells.

In this paper, we consider aperiodic suspension of identical cellsof arbitrary shape. We apply at the boundaryofthemediumanelectricfieldoffrequencyω.The medium outsidethecellshasanadmittivity of k₀ :=σ₀+iω₀. Each cell is composed of anisotropic homogeneous core of admittivityk₀ and athin membrane of constantthickness δ andadmittivity km:=σm+iωm. The thicknessδ isconsidered to be very small relative to the typical cell size and the membrane is consideredvery resistive, i.e., σ_m σ₀. In this context, the potentialin themedium passes aneffective discontinuity over the cell boundary;the jumpisproportional toitsnormalderivativewithacoefficientoftheeffectivethickness, givenbyδk₀/k_m. The normalderivativeofthepotentialiscontinuous acrossthecellboundaries.

Weusehomogenizationtechniqueswithasymptoticexpansionstoderiveahomogenizedproblemandto defineaneffectiveadmittivityofthemedium.Weprovearigorousconvergenceoftheoriginalproblemtothe homogenizedproblemviatwo-scaleconvergence.Fordilutecellsuspensions,weuselayerpotentialtechniques toexpandtheeffectiveadmittivityintermsofcellvolumefraction.Throughtheeffectivethickness,δ k0/km, the first-orderterminthisexpansioncanbeexpressedintermsofamembranepolarizationtensor,M,that depends on the operating frequency ω. We retrieve the Maxwell–Wagner–Fricke formula for concentric circular-shaped cells. This explicit formula has been generalized in many directions: in three dimension for concentric spherical cells;to include higherpower termsof the volumefraction for concentric circular and spherical cells; and to include various shapes such as concentric, confocal ellipses and ellipsoids; see [14,15,28–30,43,55–57].

The imaginary part of M is positive for δ small enough. Its two eigenvalues are maximal for frequencies 1/τ_i,i= 1,2,oforderofafewMHzwithphysicallyplausibleparametersvalues.Thisdispersion phenomenon well knownbythe biologists isreferred to as theβ-dispersion.The associated characteristic timesτ_icorrespond toDebyerelaxationtimes.Giventhis,weshowthatdifferentmicroscopicorganizations of themediumcanbedistinguishedviaτi,i= 1,2,alone.The relaxationtimesτiarecomputednumericallyfor differentconfigurations:onecircularorellipticcell,twoorthreecellsincloseproximity.The obtainedresults illustratetheviabilityofimagingcellsuspensionsusingthespectralpropertiesofthemembranepolarization.

The Debye relaxationtimesareshowntobeabletogivethemicroscopicstructureof the medium.

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In the second part of this paper, we show that our results can be extended to the random case by considering arandomly deformed periodicmedium. We also derive arigorous homogenizationtheory for cells (and hence interfaces) that are randomly deformed from a periodic structure by random, ergodic, and stationary deformations. Weprove a newformula for theoverall conductivity of adilute suspension of randomly deformedcells. Again, the spectral properties of the membrane polarization canbe used to classify diﬀerent microscopic structures of the mediumthrough their Debye relaxation times. For recent worksoneﬀectivepropertiesofdiluterandommedia,wereferto[7,17].

Ourresultsinthispaperhavepotential applicabilityincancerimaging,foodsciencesandbiotechnology [41,42],andappliedandenvironmentalgeophysics.TheycanbeusedtomodelandimprovetheMarginProbe system for breast cancer [61], which emits an electric field and senses the returning signal from tissue underevaluation. The greater vascularization, differently polarizedcell membranes,and other anatomical differencesoftumorscomparedwithhealthytissuecausethemtoshowdifferentelectromagneticsignatures.

The abilityof theprobe to detectsignalscharacteristic of cancerhelps surgeonsensure theremovalof all unwantedtissuearoundtumormargins.

Another commercial medical system to which our results can be applied is ZedScan [62]. ZedScan is based on electrical impedance spectroscopy for detecting neoplasias in cervical disease [1,20]. Malignant whitebloodcellscanbealsodetectedusinginducedmembranepolarization[53].In foodqualityinspection, spectroscopicconductivityimagingcanbeusedtodetectbacterialcells[12,59].In appliedandenvironmental geophysics,inducedmembrane polarizationcanbe usedto probe uptosubsurface depths ofthousands of meters [58,60].

Thestructureoftherest ofthispaperisas follows.Section2introducestheproblemsettings andstate themain resultsofthis work.Section3is devotedtothe analysisof theproblem.Weproveexistenceand uniquenessresultsandestablishusefulaprioriestimates.In section4weconsideraperiodiccellsuspension andderivespectralpropertiesoftheoverallconductivity.In section5weconsidertheproblemofdetermining theeffective property of asuspensionof cellswhen thevolumefraction goes to zero.Section6is devoted to spectroscopic imagingofadilutesuspension.Wemake useof theasymptoticexpansionof theeffective admittivity in terms of the volume fraction to image a permittivity inclusion. We also discuss selective spectroscopicimagingusingapulsedapproach.Finally,weintroduceanaturalmeasureoftheconductivity anisotropyandderiveitsdependenceonthefrequencyofappliedcurrent.In section7weextendourresults tothecaseofrandomlydeformedperiodicmedia.In section8weprovidenumericalexamplesthatsupport our findings. A few concluding remarks are given in the last section. For simplicity, we only treat the two-dimensionalcase.Ourresultscanbeextendedinto thethree dimensionalsetting [35].

2. Problemsettingsandmainresults

Theaimofthissectionisto introducetheproblem settingsandstatethemainresultsofthispaper.

2.1. Periodic domain

We consider the probe domain Ω to be a bounded open set of R² of class C². The domain contains a periodic array of cells whose size is controlled by ε. Let C be aC^2,η domain being contained inthe unit squareY = [0,1]²,seeFig. 2.1.Here,0< η <1 andC representsareferencecell.Wedividethedomain Ω periodicallyineachdirectioninidenticalsquares(Yε,n)n ofsize ε,where

Yε,n=εn+εY.

Here,n∈N_ε:=

n∈Z²|Y_ε,n∩Ω=∅ .

Weconsider thatacellC_ε,nlives ineachsmall squareY_ε,n. As showninFig. 2.4, allcellsareidentical, uptoatranslationandscalingofsizeε, tothereferencecellC:

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Fig. 2.1.Schematic illustration of a unit periodY.

∀n∈N_ε, C_ε,n=εn+ε C.

Soaretheirboundaries (Γ_ε,n)_n∈N_ε totheboundaryΓ of C:

∀n∈Nε, Γε,n=εn+εΓ.

Let usalso assumethatall thecellsarestrictly containedinΩ,thatisforeveryn∈Nε,the boundary Γε,nofthecellCε,ndoesnotintersecttheboundary∂Ω:

∂Ω∩(

n∈Nε

Γ_ε,n) =∅.

2.2. Electricalmodelofthecell Setforany opensetD ofR²:

L²₀(D) :=

⎧⎨

⎩f ∈L²(D)

∂D

f(x)ds(x) = 0

⎫⎬

⎭

and

H¹(D) :=

f ∈L²(D)|∇f| ∈L²(D)

.

WeconsiderinthissectionthereferencecellCimmersedinadomainD.Weapplyasinusoidalelectrical current g∈L²₀(∂D) withangularfrequencyω attheboundaryofD.

Themediumoutsidethecell,D\C,isahomogeneousisotropicmediumwithadmittivityk0:=σ0+iω0. The cellCiscomposedofanisotropichomogeneouscoreofadmittivityk0andathinmembraneofconstant thickness δwithadmittivitykm:=σm+iωm.Wemakethefollowing assumptions:

σ0>0, σm>0, 0>0, m≥0.

If we apply a sinusoidal current g(x)sin(ωt) on the boundary ∂D in the low frequency range below 10 MHz,the resultingcomplex-valuedtimeharmonicpotentialuˇisgovernedby

⎧⎨

⎩

∇ ·(k0+ (km−k0)χ_Γ^δ)∇u) = 0ˇ inD k₀∂uˇ

∂n

∂D =g,

where Γ^δ :={x∈C : dist(x,Γ)< δ}andχ_Γδ isthecharacteristic functionoftheset Γ^δ.

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The membrane thickness δ is considered to be very small compared to the typical size ρ of the cell, i.e.,δ/ρ1.Accordingtothetransmissioncondition,the normalcomponentofthecurrentdensityk₀∂uˇ canbeapproximatelyregardedascontinuousacross thethinmembraneΓ. ∂n

Weset β := δ km

. Sincethemembraneis veryresistive,i.e.σ_m/σ₀ 1,the potentialuˇinD undergoes ajumpacrossthecellmembraneΓ,which canbe approximatedatﬁrstorder byβk0

∂uˇ

∂n. A rigorousproof ofthisresult,basedonasymptoticexpansionsoflayerpotentials, canbefound in[37].

Moreprecisely,weapproximateuˇbyudeﬁnedasthesolutionofthefollowing equations[37,50,51]:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∇ ·k0∇u= 0 in D\C,

∇ ·k₀∇u= 0 in C, k₀∂u

∂n

+=k₀∂u

∂n

− on Γ,

u|+−u|−− βk₀∂u

∂n = 0 on Γ,

k0

∂u

∂n

∂D=g,

∂D

g(x)ds(x) = 0,

D\C

u(x)dx= 0.

(2.1)

Here n is the outward unit normal vector and u|±(x) denotes lim

t→0⁺u(x±tn(x)) for x on the concerned boundary.Likewise, ∂u

∂n

± := lim

t→0⁺∇u(x±tn(x))·n(x).

Equation(2.1)isthestartingpoint ofouranalysis.

Foranyopen setB inR²,wedenote H_C¹(B) theSobolevspaceH¹(B)/Cwhichcanberepresentedas:

H_C¹(B) =

⎧⎨

⎩u∈H¹(B)|

B

u(x)dx= 0

⎫⎬

⎭. Thefollowingresultholds:

Lemma2.1. Thereexistsaunique solutionu:= (u⁺,u⁻)inH_C¹(D⁺)×H¹(D⁻)to(2.1).

Proof. To provethewell-posednessof(2.1)weintroducethefollowingHilbertspace:V :=H_C¹(D)×H¹(D) equippedwiththefollowing naturalnormforourproblem:

∀u∈VuV =∇u⁺L²(D⁺)+∇u⁻L²(D⁻)+u⁺−u⁻L²(Γ). Wewritethevariationalformulationof(2.1)asfollows:

Findu∈Vsuch that for allv:= (v⁺, v⁻)∈V :

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

D⁺

k0∇u⁺(x)· ∇v⁻(x)dx+

D⁻

k0∇u⁺(x)· ∇v⁻(x)dx + 1

βk0 Γ

(u⁺−u⁻)(v⁺−v⁻)dσ(x) = 1 ko

∂Ω

gv dσ(x).

Since (k0) = σ0 > 0 and ( 1 βk0

) = σmσ0+εmε0

δ|k0| > 0, we can apply Lax–Milgram theory to obtain existenceanduniquenessofasolutiontoproblem (2.1). 2

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Fig. 2.2.Real and imaginary parts of the potentialuoutside and inside the cell.

Fig. 2.3.Gradient vector ﬁelds of the real and imaginary parts ofu.

We conclude this subsection with a few numerical simulations to illustrate the typical proﬁle of the potential u. Weconsider anelliptic domain D inwhich lives anelliptic cell.Wechooseto virtually apply at theboundaryofD anelectricalcurrent g=eⁱ^∗^30r.

Weuseforthediﬀerentparametersthefollowingrealisticvalues:

• thetypicalsizeofeukaryotescells:ρ10–100 μm;

• theratiobetweenthemembranethickness andthesizeofthecell:δ/ρ= 0.7·10⁻³;

• theconductivityofthemediumand thecell:σ₀= 0.5 S m⁻¹;

• themembraneconductivity:σm= 10⁻⁸ S m⁻¹;

• thepermittivityofthemediumand thecell:0= 90×8.85·10⁻¹² F m⁻¹;

• themembranepermittivity:ε_m= 3.5×8.85·10⁻¹² F m⁻¹;

• thefrequency:ω= 10⁶Hz.

Note thattheassumptionsofourmodelδρandσ_mσ₀ areveriﬁed.

Therealand imaginaryparts ofuoutsideandinsidethecellarerepresentedinFig. 2.2.

We can observe that the potential jumps across the cell membrane. We plot the outside and inside gradientvectorﬁelds;seeFig. 2.3.

2.3. Governingequation

Wedenote byΩ⁺_ε themediumoutsidethecellsandΩ⁻_ε themediuminsidethecells:

Ω⁺_ε = Ω∩(

n∈Nε

Yε,n\Cε,n), Ω⁻_ε =

n∈Nε

Cε,n.

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Fig. 2.4.Schematic illustration of the periodic medium Ω.

SetΓ_ε:=

n∈Nε

Γ_ε,n.Bydeﬁnition,the boundaries ∂Ω⁺_ε and∂Ω⁻_ε ofrespectivelyΩ⁺_ε andΩ⁻_ε satisfy:

∂Ω⁺_ε =∂Ω∪Γ_ε, ∂Ω⁻_ε = Γ_ε.

We apply asinusoidal current g(x)sin(ωt) at x ∈∂Ω, where g ∈ L²₀(∂Ω). The inducedtime-harmonic potentialuεinΩ satisﬁes:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∇ ·k0∇u⁺_ε = 0 in Ω⁺_ε,

∇ ·k₀∇u⁻_ε = 0 in Ω⁻_ε, k0

∂u⁺_ε

∂n =k0

∂u⁻_ε

∂n on Γε,

u⁺_ε −u⁻_ε −ε βk0

∂u⁺_ε

∂n = 0 on Γε, k0

∂u⁺_ε

∂n

∂Ω=g,

∂Ω

g(x)ds(x) = 0,

Ω⁺ε

u⁺_ε(x)dx= 0,

(2.2)

whereuε=

u⁺_ε in Ω⁺_ε, u⁻_ε in Ω⁻_ε.

Note thatthe previouslyintroduced constantβ, i.e., the ratiobetween thethickness of the membrane ofC anditsadmittivity,becomesεβ.Becausethecells(C_ε,n)_n∈N_ε areinsquaresofsizeε,the thicknessof theirmembranesisgivenbyεδandconsequently,afactorεappears.

2.4. Main resultsintheperiodiccase

We set Y⁺ := Y \C and Y⁻ := C. For any open set D in R², we denote H_C¹(D) the Sobolev space H¹(D)/Cwhichcanberepresentedas

H_C¹(D) =

⎧⎨

⎩u∈H¹(D)|

D

u(x)dx= 0

⎫⎬

⎭.

Throughoutthis paper,weassumethatdist(Y⁻,∂Y)=O(1).Wewritethesolutionuεas

∀x∈Ω u_ε(x) =u₀(x) +εu₁(x,x

ε) +o(ε) (2.3)

with

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y−→u₁(x, y)Y-periodic andu₁(x, y) =

u⁺₁(x, y) in Ω×Y⁺, u⁻₁(x, y) in Ω×Y⁻. Thefollowing theoremholds:

Theorem 2.1.

(i) The solution uε to (2.2) two-scale converges to u0 and ∇uε(x) two-scale converges to ∇u0(x)+ χ_Y⁺(y)∇yu⁺₁(x,y)+χ_Y−(y)∇yu⁻₁(x,y),whereχ_Y± are thecharacteristicfunctionsof Y^±.

(ii) The functionu₀ in(2.3)isthesolution inH_C¹(Ω) tothefollowinghomogenized problem:

∇ ·K^∗∇u₀(x) = 0 inΩ,

n·K^∗∇u0=g on∂Ω, (2.4)

where K^∗,the eﬀectiveadmittivityof themedium,isgivenby

∀(i, j)∈ {1,2}², K_i,j^∗ =k0

⎛

⎝δ_ij+

Y

(χ_Y⁺∇w⁺_i +χ_Y−∇w⁻_i )·ej

⎞

⎠, (2.5)

andthefunction(w_i)_i=1,2 arethesolutions of thefollowingcellproblems:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∇ ·k0∇(w_i⁺(y) +yi) = 0 inY⁺,

∇ ·k0∇(w_i⁻(y) +yi) = 0 inY⁻, k0

∂

∂n(w⁺_i (y) +yi) =k0

∂

∂n(w⁻_i (y) +yi) onΓ, w_i⁺−w⁻_i −βk0

∂

∂n(w_i⁺(y) +yi) = 0 onΓ, y−→wi(y)Y-periodic.

(2.6)

(iii) Moreover, u1 can be writtenas

∀(x, y)∈Ω×Y, u₁(x, y) = 2

i=1

∂u0

∂x_i(x)w_i(y). (2.7)

Wedeﬁne theintegraloperatorLΓ:C^2,η(Γ)→ C^1,η(Γ),with 0< η <1 by LΓ[ϕ](x) = 1

2π

Γ

∂²ln|x−y|

∂n(x)∂n(y)ϕ(y)ds(y), x∈Γ. (2.8) LΓ isthenormalderivativeofthedouble layerpotentialDΓ.

SinceLΓ ispositive,onecanprovethattheoperatorI+αLΓ:C^2,η(Γ)→ C^1,η(Γ) isaboundedoperator and hasaboundedinverseprovidedthatα >0[23,47].

As the fraction f of the volume occupied by the cells goes to zero, we derive an expansion of the eﬀectiveadmittivityforarbitraryshapedcellsintermsofthevolumefraction.Werefertothesuspension, as periodic dilute.The followingtheorem holds.

Theorem 2.2. The eﬀective admittivity of a periodic dilute suspension admits the following asymptotic expansion:

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K^∗=k₀

I+f M

I−f 2M

−1

+o(f²), (2.9)

whereρ=

|Y⁻|,f =ρ²,

M=

⎛

⎜⎝M_ij =βk₀

ρ⁻¹Γ

n_jψ_i^∗(y)ds(y)

⎞

⎟⎠

(i,j)∈{1,2}²

, (2.10)

andψ^∗_i isdeﬁnedby

ψ_i^∗=−

I+βk0Lρ⁻¹Γ

−1

[ni]. (2.11)

Notethatρ⁻¹Γ istherescaledmembraneandtherefore,M isindependentofρ.

2.5. Descriptionof therandomcells andinterfaces

We describe the domains occupied by the cells. As mentioned earlier, they are formed by randomly deforming aperiodic structure.We transform the aforementioned periodicstructure by arandom diﬀeo- morphismΦ:R²→R².Let

R⁺2 :=

n∈Z²

(n+Y⁺), R⁻2 :=

n∈Z²

(n+Y⁻), Γ:=

n∈Z²

(n+ Γ). (2.12)

Thecells,the environmentandtheinterfacesarehencedeformedtoΦ(R⁻2),Φ(R⁺2) andΦ(Γ).Weemphasize thatthe topologyof these sets arethe sameas before. Finally,the deformed structure is scaledto size ε, where 0 < ε 1, by the dilation operator εI where I is the identity operator. The ﬁnal sets εΦ(R⁻2), εΦ(Γ) andεΦ(R⁺2) thusarerealisticmodelsfortherandom cells,membranesand theenvironmentforthe biologicalproblemathand.

TomodelthecellsinsideanarbitraryboundeddomainΩ asin(2.2),wewouldliketosetΩ⁺_ε := Ω∩εΦ(R⁻2) and Γε:= Ω∩εΦ(Γ). However,atechnicalityis encountered,precisely, the intersectionofεΦ(Γ) with the boundary∂Ω maynotbeempty.In thiscase,somecellsarecutbytheboundaryofthebody,whichisnot physicallyadmissible.Moreover,anarbitrarydiffeomorphismΦ mayallowsomedeformedcellsinεΦ(R⁻2) toget arbitrarilyclosetoeachother. Thisimposes difficultiesfor rigorousmathematical analysis.In order to resolve these issues, we will impose a few conditions on Φ and refine the above construction in the next subsection.

2.6. Stationary ergodicsetting

Let (O,F,P) be some probability space on which Φ(x,γ) : R²× O → R² is deﬁned. Throughout this paper,weassumethatFiscountablygeneratedsothatthespaceL²(O) isseparable.Forarandomvariable X ∈L¹(O,dP),wedenoteitsexpectationby

EX =

O

X(γ)dP(γ).

Throughoutthis paper,weassumethatthegroup(Z²,+) actsonObysomeaction{τ_n:O → O}n∈Z², andthatforalln∈Z²,τ_n isP-preserving,thatis,

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P(A) =P(τ_nA), for allA∈ F.

We assumefurtherthattheaction isergodic, whichmeansthatforany A∈ F,ifτ_nA=A foralln∈Z², then necessarilyP(A)∈ {0,1}.

Following [19],wesaythatarandomprocess F∈L¹_loc(R²,L¹(O)) is(discrete)stationary if

∀n∈Z², F(x+n, γ) =F(x, τ_nγ) for almost everyxandγ. (2.13) Clearly, adeterministic periodicfunctionis aspecial caseof stationary process. However,we precise that theabovenotionofstationarityisdiﬀerentfromtheclassicalone,seeforinstance[49]and[39].Throughout this paper, wepresume stationarityinthesense of(2.13)ifnotstated otherwise.Whatmakes thisnotion useful isthefollowing versionofergodictheorem[25,34].

Proposition 2.1. (i) Let F ∈ L^∞(R²,L¹(O)) be a stationary random process. Equip Z² with the norm

|n|∞= max1≤i≤2|ni| foralln∈Z².Then 1

(2N+ 1)²

|n|∞≤N

F(x, τnγ)−−−−→^L^∞

N→∞ EF(x,·) for a.e.γ∈ O. (2.14) This impliesin particularthat

Fx

ε, γ −−−−−−−→^L^∞_ε→0^weak-^∗ E

⎛

⎝

Y

F(x,·)dx

⎞

⎠ for a.e.γ∈ O. (2.15)

(ii) Forp∈(1,∞),suppose F ∈L^p(O,L^p_loc(R²))isastationaryrandom process,then theaboveconver- gence resultsstillholdif wereplaceL^∞ by L^p_loc.

Weassumethatforeveryγ∈ O, Φ(·,γ) isadiﬀeomorphismfromR² toR² andthatitsatisﬁes

∇Φ(x, γ) is stationary. (2.16)

ess inf

γ∈O,x∈R²det(∇Φ(x, γ)) =κ >0, (2.17)

ess sup

γ∈O,x∈R² |∇Φ(x, γ)|F =κ>0, (2.18)

where | · |F is the Frobenius norm and ess inf and ess sup are the essential inﬁmum and the essential supremum,respectively.WepointoutthatΦ⁻¹ automaticallysatisﬁessimilarconditionswithconstantsκ₁ and κ₁.BytheuniformLipschitzassumptiononΦ andΦ⁻¹ above,wehave

(κ₁)⁻¹|y1−y2| ≤ |Φ(y1, γ)−Φ(y2, γ)| ≤κ|y1−y2|.

So thecellsremain well separatedafter thedeformation. Toavoidthe intersectionof∂Ω and therandom cells εΦ(R⁻2), we erase those intersecting the boundary. More precisely, given a bounded and simply connected open set Ω with smooth boundary and a small numberε 1, we denote by Ω_1/ε the scaled set {x∈R² |εx∈Ω}.LetΩ!1/ε betheshrunkset

Ω!_1/ε:={x∈Ω_1/ε| dist(x, ∂Ω_1/ε)≥dist(Y⁻, ∂Y)}.

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We introduce for n ∈ Z², Y_n and Y_n^± the translated cubes, reference cells and reference environments:

Yn :=n+Y,Y_n^± :=n+Y^±. Let Iε ⊂Z² be the indices of cubes Yn such that Yn ∈ Ω!1/ε. Note that Iε

correspondsto Nεintheperiodiccase.WesetΩ⁻_ε tobe Ω⁻_ε :=

n∈Iε

εΦ(Y_n⁻) (2.19)

andthenΩ⁺_ε = Ω\Ω⁻ε.Wealsodeﬁne thefollowing twonotations:

Eε:=

n∈Iε

εΦ(Yn) and Kε:= Ω\Eε. (2.20)

Clearly, Eε encloses all the cells in εΦ(Y_n⁻),n ∈ Iε and their immediate surroundings εΦ(Y_n⁺); Kε is a cushionlayerneartheboundarythatpreventsthecellsfromtouchingtheboundary.Fromtheconstruction weseethat

x∈Ωinf⁻ε

dist(x, ∂Ω)≥Cε and sup

x∈Kε

dist(x, ∂Ω)≤Cε. (2.21)

Furthermore,wecancheckthat

n,j∈Isupε,n=j inf

x∈εΦ(Yn⁻),y∈εΦ(Y_j⁻)|x−y| ≥Cε. (2.22) Above, the constantsC vary butallof them dependonly ondist(Y⁻,Y), κand κ and areuniform inε.

This shows thatthe cellsinΩ arewell separated, i.e.,with a distance comparableto (if notmuch larger than)thesizeofthecells.

2.7. Main resultsintherandom case

Thefirstimportantresultintherandomcaseconcernsanauxiliaryproblemwhichproduces oscillating testfunctionsthatareusedinthestochastichomogenizationprocedure.In thefollowingtheorem,afunction fêxt in W_loc^1,s(R²) is said to be an extension of f ∈ W_loc^1,s(R⁺2) if fêxt = f on R⁺2 and fêxtW^1,s(K) ≤ C(K,R⁺2)fW^1,s(R⁺2∩K),forany compactsubsetK.

Thefollowingtheorem holds.

Theorem 2.3. Let Φ(·,γ) be a random diffeomorphism from R² to R² defined on the probability space (O,F,P),andassumethat (2.16)–(2.18) hold. Fora.e.γ∈ Oand foranyfixed vectorp∈R²,the system

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∇ ·k0(∇w_p⁺(y) +p) = 0 in Φ(R⁺2, γ),

∇ ·k₀(∇w_p⁻(y) +p) = 0 in Φ(R⁻2, γ), k0

∂w_p⁺

∂n (y)−k0

∂w_p⁻

∂n (y) = 0 on Φ(Γ, γ),

w_p⁺−w⁻_p =βk0

∂w⁺_p

∂n (y) +νy·p

on Φ(Γ, γ),

w_p^±(y, γ) =w"_p^±(Φ⁻¹(y, γ), γ), ∇w"_p^± are stationary,

∃w˜^ext_p ∈H_loc¹ (R²)that extendsw˜_p⁺ s.t. ∇w˜^ext_p =P

∇w˜⁺_p , and E#

Y ∇w˜_p^ext(˜y,·)d˜y

= 0,

(2.23)

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admits a unique (up to an addition of a random variable) weak solution w_p = w_p⁺χ_Φ(R⁺

2) +w⁻_pχ_Φ(_R− 2)

in H_loc¹ (Φ(R⁺2))×H_loc¹ (Φ(R⁻2)) and the operator P above denotes the extension operator of Corollary Appendix A.1.

The precise weak formulation of the system above is postponed to section 7, where the proof of this theorem isgiven;see(7.1).Weremarkthatthenon-uniqueadditiverandomvariableis notimportantand what matters is the fact thatthe gradient ∇wp of the solution is unique. The second main result inthe random caseisthefollowing homogenizationtheorem.

Theorem 2.4. Let Ω be a bounded and connected open subset of R² with regular boundary. Let Φ be a random diﬀeomorphism on(O,F,P)satisfying (2.16)–(2.18).Assumethat thecellsΩ⁻_ε areconstructed as inSection2.6.Thenfora.e.γ∈ O,the solutionuε(·,γ)= (u⁺_ε,u⁻_ε)of(2.2)satisﬁesthefollowingproperties:

(i) We canextend u⁺_ε(·,γ)tou^ext_ε (·,γ)∈H¹(Ω),where u^ext_ε (·,γ)convergesweakly, as ε→0,to adeter- ministic functionu0∈H¹(Ω).

(ii) The function uε(·,γ) convergesstrongly in L²(Ω) to u0 above. Further,let Q be thetrivial extension operatorsetting Qf = 0outside thedomainof f,anddeﬁne

:= det

⎛

⎝E

Y

∇Φ(z,·)dz

⎞

⎠

−1

, θ:=E

Y⁻

det∇Φ(z,·)dz <1, (2.24)

where detdenotesthedeterminant.Then, Qu⁻_ε convergesweakly toθu0 inL²(Ω).

(iii) The functionu₀ istheuniqueweaksolution inH_C¹(Ω) tothehomogenizedequation ∇ ·K^∗∇u₀(x) = 0, x∈Ω,

n(x)·K^∗∇u0(x) =g, x∈∂Ω,

(2.25)

The homogenized admittivitycoeﬃcientK^∗ isgivenby ∀(i,j)∈ {1,2}²,

K_ij^∗ =k0

⎛

⎜⎝δij+E

Φ(Y)

ej·(χΦ(Y⁺)∇w⁺_e_i+χ_Φ(Y−)∇w⁻_e_i)(y,·)dy

⎞

⎟⎠, (2.26)

where{ei}²i=1istheEuclideanbasisofR²andforeachp∈R²,the pairoffunctions(w_p⁺,w⁻_p)istheunique solution totheauxiliary system(2.23).

WementionthefactthatK^∗ isuniformly elliptic,the proofofwhichisstandardandisomitted.In the dilute limit ρ:=

|Y⁻| 1, weobtain the following approximation of the eﬀectivepermittivity for the dilutesuspension:

K_ij^∗ =k0(I+fEMij) +o(f), (2.27) where accountsfortheaveragedchangeofvolumeduetotherandomdiﬀeomorphismandf:=ρ²isthe volumefraction occupiedbythecells;thepolarizationmatrixM isdeﬁnedby

Mij =βk0 ρ⁻¹Φ(Γ)

ψ˜inj ds(˜y), (2.28)

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where

ψ˜i=−(I+βk0n· ∇Dρ⁻¹Φ(Γ))⁻¹[ni],

withDρ⁻¹Φ(Γ)thedoublelayerpotentialassociatedtothedeformedinclusionscaledtotheunitlengthscale.

3. Analysisofthe problem

Foraﬁxed ε,recallthatH_C¹(Ω⁺_ε) denotestheSobolevspaceH¹(Ω⁺_ε)/C,whichcanbe representedas

H_C¹(Ω⁺_ε) =

⎧⎪

⎨

⎪⎩u∈H¹(Ω⁺_ε)|

Ω⁺ε

u(x)dx= 0

⎫⎪

⎬

⎪⎭. (3.1)

Thenaturalfunctional spacefor(2.2)is Wε:=

u=u⁺χ⁺_ε +u⁻χ⁻_ε |u⁺ ∈H_C¹(Ω⁺_ε), u⁻ ∈H¹(Ω⁻_ε)

, (3.2)

whereχ^±_ε arethecharacteristicfunctionsofthesetsΩ^±_ε.Wecanverifythat uWε =

∇u⁺²_L2(Ω⁺ε)+∇u⁻²_L2(Ω⁻ε)+εu⁺−u⁻²L²(Γε)

1

2 (3.3)

deﬁnesanormonW_ε.In fact,as itwill beseeninProposition 3.2,thisnormisequivalenttothestandard normonW_ε whichis

uH¹_C(Ω⁺ε)×H¹(Ω⁻ε)=

∇u⁺²_L²_(Ω⁺_ε₎+∇u⁻²_L2(Ω⁻ε)+u⁻²_L2(Ω⁻ε)

1

2. (3.4)

3.1. Existence anduniquenessof asolution

Problem (2.2) should be understood through its weak formulation as follows: For a ﬁxed ε > 0, ﬁnd uε∈Wεsuchthat

Ω⁺ε

k0∇u⁺_ε(x)· ∇v⁺(x)dx+

Ω⁻ε

k0∇u⁻_ε(x)· ∇v⁻(x)ds(x)

+ 1 εβ

Γε

(u⁺_ε −u⁻_ε)(x)(v⁺−v⁻)(x)ds(x) =

∂Ω

g(x)v⁺(x)ds(x), (3.5)

foranyfunctionv∈Wε.

Deﬁnethesesquilinearforma_ε(·,·) onW_ε×W_ε by aε(u, v) :=

Ω⁺ε

k0∇u⁺· ∇v⁺dx+

Ω⁻ε

k0∇u⁻· ∇v⁻dx+ 1 εβ

Γε

(u⁺−u⁻)(v⁺−v⁻)ds. (3.6)

Associatethefollowinganti-linearform onWεtotheboundarydatag:

(u) :=

∂Ω

gu⁺ds. (3.7)