Insight into the transport mechanism of solute removed in dialysis by a membrane with double functionality

To link to this article : DOI :

10.1016/j.cherd.2017.08.017

URL :

http://dx.doi.org/10.1016/j.cherd.2017.08.017

To cite this version :

Snisarenko, Dmytro and Pavlenko, Denys and Stamatialis, Dimitrios and Aimar,

Pierre and Causserand, Christel and Bacchin, Patrice Insight into the transport

mechanism of solute removed in dialysis by a membrane with double

functionality. (2017) Chemical Engineering Research and Design, vol. 126. pp.

97-108. ISSN 02638762

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Eprints ID : 18602

Insight

into

the

transport

mechanism

of

solute

removed

in

dialysis

by

a

membrane

with

double

functionality

D.

Snisarenko

a

_,

_D.

_Pavlenko

b

_,

_D.

_Stamatialis

b

_,

_P.

_Aimar

a

_,

C.

Causserand

a,∗

,

P.

Bacchin

a

a_{LaboratoiredeGénieChimique,UniversitédeToulouse,CNRS,INPT,UPS,Toulouse,France}

b_{BioartificialOrgans,DepartmentofBiomaterialsScienceandTechnology,MIRAInstitute,FacultyofScienceand}

Technology,UniversityofTwente,7500AEEnschede,TheNetherlands

Keywords:

Hemodialysis Mathematicalmodel Double-layermembrane Bloodtoxinsremoval

a

b

s

t

r

a

c

t

Thepresentstudyaimsatsheddinglightonthetransportmechanismsinvolvedina func-tionalizedmembranedesignedforimprovinghemodialysis.Thismembraneisprepared byembeddingabsorptivemicroparticleswithinitsporousstructure.Tounderstandthe transportmechanismthroughthemembraneandmakesuggestionsforitsoptimization,a mathematicalmodelcouplingconvection,diffusionandadsorptionisdevelopedand vali-datedbycomparisonofexperimentalandtheoreticalresults.Infact,themodelprovidesa descriptionoftheconcentrationprofilefromthedonor(feed)compartmentacrossthe sev-erallayerswithdifferentpropertiestotheacceptor(dialysate)compartment.Inaddition, themodelallowstopredicttheinfluenceofvariousparameterssuchasmolecule diffu-sivity,membranethickness,presenceofconvection,contentofadsorptiveparticlesonthe fluxintensificationacrossthemembrane.Comparisonwithexperimentalmeasurements demonstratesthatthemodelisabletodescribethetransmembranemassfluxvariationover timeasafunctionofhydrodynamicconditionsandmembrane/modulegeometric param-eters.Themodelalsoillustrateshowtheproposeddouble-layermembraneconceptoffers significantbenefitsintermsoftoxinremovalincomparisontoconventionaldialysis.Asso, themainachievementofthedevelopedmodelisthatitmayserveastoolforthefurther improvementoffunctionalizedmembraneintermsoftoxinremovalandoptimizationof processconditions.

1. Introduction

Hemodialysisisalife-sustainingtreatmentthatpatientsundergowhen theirkidneysmalfunction.Eventhoughthistechniqueisconstantly beingimprovedformorethanfourdecadesitisstilloneofthemajor healthcareproblemswithhighmortalityandmorbidityofthepatients. Highmortalityratesareusuallyattributedtoincompleteremovalofthe bloodtoxinsduringthedialysistreatment(Dobreetal.,2013;Meyer

etal.,2011;Vanholderetal.,2015).Thetreatmentprovidesadequate

removalofonlythesmallwatersolublemolecules,suchasureaor

∗

Correspondingauthor.

E-mailaddress:[email protected](C.Causserand).

creatinine.However,largersolutes,referredtoasmiddlemoleculesand theprotein-boundtoxins,haveinadequateclearanceevenafterthe developmentofmorepermeablehigh-fluxhemodialyzers(Elootetal.,

2012;Luoetal.,2009).

Asthisextracorporealtreatmentisprimarydrivenbydiffusion, theuseofhighvolumesofpuredialysateliquidisnecessaryto maxi-mizetheconcentrationgradientsacrossthemembrane(Waltheretal., 2006).Largevolumesofhigh-qualitydialysatesolutionmakesuchan approachnotonlyexpensive,butalsochallengingforcountrieswith scarcewaterresources.Besides,itwasdemonstratedinvariousstudies

http://dx.doi.org/10.1016/j.cherd.2017.08.017

Nomenclature

a Specificsurfaceareaoftheadsorbent(m2_per

m3_{ofmixedmatrixmembrane)}

c Concentration(molm−3₎

d Diameter(m)

D Diffusioncoefficient(m2_s−1₎

div Divergence(–)

j Molarfluxdensity(molm−2_s−1₎

J Convectiveflowthroughthemembrane(ms−1₎

k Heterogenicadsorptionconstant(ms−1₎

K Pseudomasstransferconductance(ms−1₎

l Lengthofthefiber(m)

m Mass(kg)

M Molecularweight(gmol−1₎

N Numberoffibers(–)

Pe Pecletnumber(–)

q Quantityofadsorbedspecie(mgpermgof mem-brane)

r Rateofadsorption(molm−3_s−1₎

Re Reynoldsnumber(–)

s Sinkterm(molm−3_s−1₎

S Membranesurfacearea(m2₎

Sc Schmidtnumber(–)

Sh Sherwoodnumber(–)

t Time(s)

V Volumeoffluid(m3₎

Greekletters

ı Thicknessofthedomain(m) 1 DeterminantdefinedinEq.(5)(–)

f Thielemodulus(–)

Subscripts

0 Initialtimepoint

a Aftertheadsorptivelayer

ads Adsorption

b Beforetheadsorptivelayer

conv Convection d Dialysate diff Diffusion e External f Feed h Hydraulic

in Attheinletofparticulardomain

l Lumenofthemembrane

max Maximal

out Attheoutletofparticulardomain

s Shellsideofthemodule

t Timedifferentfrom0

(Barretoetal.,2009;Buschetal.,2010;Liabeufetal.,2011;Rajetal.,2000)

thatpoorremovalofsomeuremictoxins(i.e.b-2-microglobulin),leads tootherlong-termhemodialysisrelatedcomplications.Thus,thereis anurgentneedofnewimprovementofdialysisinordertoovercome highlighteddrawbacks.

Anovelspecificapproachtomaintainthebeneficialhighgradients acrossthedialysismembranealongwithfacilitatedtransportof ure-micsoluteswasdiscussedbyMeyeretal.(2007).There,additionof thesorbenttothedialysatesolutionresultedinsignificantincrease ofremovalratesofvariousprotein-boundtoxinsevenwithouthigh dialysateflowrates.Suchresultsdemonstratedthegreatopportunities ofcombiningthebenefitsofadsorptionandfiltration.Morerecently,

someoftheauthorsofthisworkdevelopedandprovedtheconceptof thedoublelayermixedmatrixmembranes(Pavlenkoetal.,2016;Tijink

etal.,2013,2012).Thesemembranesconsistoftwolayers:(1)amixed

matrixlayer,whereactivatedcarbonparticlesareincorporatedinside aporouspolymermatrix;(2)aparticle-freeselectivelayerproviding hemocompatibilityandselectivity.Suchdesignmakesitpossibleto gainthebenefitsoftheadsorptionanddiffusioninonestep.In partic-ular,membranesinbothflatandhollowfiberdoublelayergeometries showedanexcellentremovalofsmallandprotein-boundtoxinsfrom humanplasmagivinghighpromisefortheirfurtherdevelopmentfor dialyticapplications.

However,thecontributionofthemixedmatrixlayertotheremoval of the uremic toxins, as well as the influence of the membrane characteristicsandprocessconditions needtobequantifiedonan experimentalandon atheoreticalpoint ofview.Itisforexample important toknow howto optimize thetransferthroughamixed matrixmembrane.Totacklethisquestion,inthiswork,wedeveloped amodelwithvariousparametersandtheirphysicalrelationships,in ordertodescribetheoverallperformanceofthemembranetowards removalofsmallandmiddle-sizeduremictoxins.Theobjectiveofthis modelistoofferananalyticalandnon-numericalproblemresolution andtogivetheopportunitytooptimizethemembranepropertiesand processconditions.Theupcomingsectionsareorganizedinthe follow-ingmanner:first,theselectedassumptionsalongwithmathematical derivationofrequiredequationsaredescribedinordertoformulate themodel.Secondly,thefabricationandcharacterizationof function-alizeddouble-layermembranesispresented.Then,thepredictionof solute concentrationprofilesacrossthemembranewithuseofthe modelisdemonstrated.Finally,thedevelopedmodelisvalidatedby thecomparisonwiththeexperimentalresults.

2.

Model

development

A modelispresentedtodescribethecouplingoftransport phenomena(advection,diffusionandadsorption)by describ-ingthemixedmatrixmembranesbycompositelayers(Section

2.1).Thisonedimensionmodelallowsdeterminingthe con-centrationprofileandthemassfluxacrossthemembraneand thentheefficiencyimprovementduetotheadsorbent(Section

2.2).InalastSection2.3,thismodelforthetransferthrough themembraneiscombinedwiththemassfluxbalanceson thebloodanddialysatesideinordertodepicttheclearance variationalongdialysis.

2.1. Massbalanceinthelayerscomposingthemixed matrixmembrane

Thegeometryofthesystemwasdictatedbythenovelconcept ofbloodfiltration,whichemploysthemembranes functional-izedwithadsorptiveparticles(Tijinketal.,2012).Initsway through the membrane, atoxin present in the blood (feed stream)firstpassesthroughtheparticle-freelayer(PFL)and thenthroughthemixedmatrixlayer,whereitcanadsorbon thedispersedadsorptiveparticlesasdepictedinFig.1.The effectoftheflowparalleltothemembraneinthefeed(blood) anddialysatesidearemodelledwithboundarylayers thick-ness. Atthe end, afour-layers system (Fig. 1)describe the mixedmatrixmembrane.

Themodelofsolutetransportthroughthissystemrelies onfollowingassumptions:

1) Superficialresistancesareaccountedthroughanaveraged boundarylayerthicknessalongtheYaxis(Fig.1);

2) Steady state for the transfer is considered through the membrane;

Fig.1–Theschematicrepresentationofthesystemgeometry(dotsinthefeedrepresentthetoxinmoleculespresentinthe bloodstream).1—feedsideboundarylayer,2—particle-freemembrane,3—mixedmatrixmembrane,4—dialysate boundarylayer.

3) Cartesiancoordinatesareusedtodescribeaplanar mem-branesorhollowfiberswhentheboundarylayerthickness aresmallcomparedtothefiberlumen;

4) Fluidsinthebloodanddialysatecompartmentsare New-tonian;

5) Adsorptiveparticles are consideredtobeuniformly dis-persedinsidethepolymericmatrix;

6) Adsorptionisconsideredasfirstorderheterogeneous reac-tion. Dealing with the adsorption mechanism, such a kineticconditioncancorrespondtothesystemwherethe adsorptivecapacityislargecomparedtotheamounttobe adsorbed.

7) Thetransferinsidetheadsorptiveparticlesisnot diffusion-limited.Theparticlesaresmallenoughtoavoida diffusion-controlledadsorptioninsidetheparticles.

8) Thesoluteisconsideredsmallenoughforitsrejectionby themembranecanbeconsideredtobezero.Hence,the par-titioncoefficientisthentakenequaltoone.Inthecaseof futureapplicationsinwhichthesolutewouldnotbesmall comparedtotheporesize,themodelcouldbeeasily mod-ifiedtointegrateapartitioncoefficientbetweenthebulk andthemembraneinordertotakeintoaccounttheeffect ofporeontheselectivity.

Themasstransportofsoluteacrossthemembraneis con-ventionallydescribedbythe genericcontinuityequation in theformofEq.(1):

∂c

∂t+div (j) =±s (1)

wheresisthesinktermwhichrepresentsheretheadsorption ofthepermeatingsolute.

Asthemodelisbeingdevelopedforthesteadystate con-dition,thetimederivativeofthesoluteconcentrationisnil here.Besides,analyzingthegeometryofthemodelonemay distinguishtwotypesofregionsdependingonthe possible toxinremovalmechanisms:oneswithconvectionand diffu-siononly(parts1, 2and 4inFig. 1), andone layerwhere, inadditiontothesetwomechanisms,adsorptionalsotakes place(part3inFig.1).AtsteadystateandforCartesian coor-dinates,thetotalmolarflux,j,intheregionwithoutadsorbent

Fig.2–Schematicrepresentationofformulatedsystem withthekeyparametersforthedifferentlayersrepresented inFig.1.

(s=0)isconstant(Eq.(1))andhasadiffusiveandanadvective contributions(Eq.(2)):

j=−Ddc

dx+Jc (2)

wherecisthelocalsoluteconcentrationgradient,xisthe dis-tancealongthedirectionnormaltothemembranesurface,D

isdiffusioncoefficient,andJisconvectiveflowthroughthe membrane.

Ifassumingagivenconcentrationattheinletandthe out-let, theintegrationofEq.(2)givesthemassfluxacrossthe regionsofthesystemwithoutadsorptiveparticles.

j=J



c2−c1e Jı D



1−eJıD (3)

wherec2andc1standfortheconcentrationsattheoutletand

TheapplicationofEq.(1)totheMMM layerresultsina differentialequation(Eq.(4)),whichapartfromdiffusiveand convective terms,also accountsfor the presenceof solute adsorption: Dd 2_c dx2−J dc dx−akc=0 (4)

where a stands for specific surface area of adsorbent in m2_perm3 _{ofmixedmatrix membraneandkisheterogenic}

adsorption constant. The Eq.(4) is integrated through the MMMlayerbyconsideringasboundaryconditionsfixed con-centration,c2andc3attheinletandtheoutletofthelayers

respectively (Fig. 2). Thedetails ofthe integrationare pre-sentedintheSupplementaryinformation1.ItresultsinEq.

(5),whichdescribesthesoluteconcentrationatanydepth(x) insidetheMMMlayer:

c=c3 e

!

_Pe3 2

!

_x ı3−1

sinh

!

x1 ı3

sinh (1) −c2 ePe32 ı3x_sinh

!

1

!

x ı3−1

sinh (1) (5) where 1=

p

Pe2 3+4ϕ2 2 ; Pe3= Jı3 D3; ϕ=

q

ak

D3ı3.Pe3 is the Peclet

number(ratioofadvectionoverdiffusion)intheMMMlayer (part3inFig.1)andϕistheThielemodulusrepresentingthe ratioofthereactiononthediffusionrateinsidetheMMMlayer. ThemolarfluxbeforetheMMMlayer(jb),andafter(ja)are

determinedbywritingthemolarfluxEq.(2)attheinletx=0 andtheoutletx=ı3boundariesrespectively:

jb=c2



_{D31cosh (1)} ı3sinh (1) +D3Pe3 2ı3



−c3D3e −Pe3 2 1 ı3sinh (1) (6) ja=c2 D3e Pe3 2 1 ı3sinh (1) −c3



_{D31cosh (1)} ı3sinh (1) −D3Pe3 2ı3



(7) Finally, the solute molar flux density before this layer is greater thanthe oneafter, jb>ja; thedifferencebeing

rela-tivetotheadsorptionrateofthepermeatingspeciesinside theMMM.Summarizingtheaforementionedmodel formula-tion,theschematicdescriptionoftheentiresystemwithits keyparameters(interfaceconcentrationsandfluxes)involved ispresentedinFig.2.

2.2. Globalmassfluxacrossthemixedmatrix membrane

Themodelisestablishedbyconsideringthecontinuityofthe molarfluxbeforeandaftertheMMMlayer.Thefollowing sys-temoffiveequationswithtwounknownmolarfluxdensities

jbandja,andthreeunknownconcentrationsattheinterfaces

ofdomainsc1,c2andc3hastobesolved:



































jb=cfK1a−c1K1b; jb=c1K2a−c2K2b; jb=c2K3a−c3K3b; ja=c2K3c−c3K3d; ja=c3K4a−cdK4b; (8)

Theexpressionsforeachpseudomasstransferconductance (Ki,j)aredeterminedfromthewritingofthedifferentialmolar

balancespresentedintheprevioussection.Theexpressions fortheseconductancesaregivenintheAppendixA.

The solution for the concentration of the permeating speciesontheinterfacesadjacenttotheadsorptivelayerc2

andc3aregivenbelow:

c2= K1aK2a(K4a+K3d) cf+K3bK4d(K2a+K1b) cd (K3a(K2a+K1b) +K1bK2b) (K4a+K3d) −K3b(K2a+K1b) K3c (9) c3= K1aK2aK3ccf+(K3a(K2a+K1b) +K1bK2b) K4dcd (K3a(K2a+K1b) +K1bK2b) (K4a+K3d) −K3cK3b(K2a+K1b) (10)

CombiningtheselasttwoexpressionswithEq.(6)andEq.(7), enablesdeterminingthemolarfluxuphillanddownhillthe MMMlayer.

Thepropertiesofthelayers(solutediffusioncoefficients andthickness)andtheoperatingconditions(rateofconvective flow,adsorptionconstantandsoluteconcentrationonboth sidesofamembrane)providethepossibilitytoanalyzethe transferthroughthesystemataparticulartimeofthedialysis process.

Furthermore,themodelenablesanalyzingtheimpactof the adsorptive particles on the local solute concentration insidetheMMM.Thetransferefficiencycanbecomparedwith the pure diffusion casefor different scenariossuchas: (a) absenceofconvection(soluteremovaloccursonlyby diffu-sionandadsorption);(b)absenceofadsorption(presenceof diffusionandconvection),and(c)absenceofbothconvection andadsorption.Thebenefitofthefunctionalizedmembrane comparedtoaconventionalsinglelayermembraneofsame thicknesscanbedefinedthroughasolutetransport enhance-mentfactor(STEF):

STEF= jb jdiff

(11)

Inpractice,STEFisdefinedasaratioofmolarfluxdensity (jb)obtainedinoneoftheaforementionedscenariostothe

molarfluxdensityobtainedwhenonlydiffusivetransportis involved(jdiff).Thediffusivecaseisusedasareferenceof

clas-sicaldialysisandthusSTEFdemonstratestheintensification ofthesoluteremovalduetothecontributionofconvection or/andadsorption.

Ingeneral,membranefoulingphenomenaduetoprotein adhesiononthemembrane,aswellas,otherphenomenasuch asbloodcoagulationandthrombusinthemembranemodule canalsoplayaroleonthesolutetransferwhenworkingin realconditionswithblood.Suchphenomenawouldinducean additionalsuperficialmasstransferresistancethatcouldbe accountedforinanimprovedversionofthemodel.TheMMM arespecificallydesignedtominimisethesephenomena.The particlefreemembranelayerwhichisindirectcontactwith blood,ismadebyPES/PVPpolymerblend,alsousedinthe developmentofthecommercialdialysismembranesandhas verygoodbloodcompatibility.Infact,ourrecentstudyshowed thattheMMMcanachievesuperiorremovalofproteinbound toxinsfromhumanplasmacomparedtocommercialdialysers withoutfoulingphenomena(Pavlenkoetal.,2016)

Table1–Spinningconditionsofthefiberpreparation.

Membrane1 Membrane2 Innerlayerpumpingflow

(mL/min)

0.4 0.9 Outerlayerpumpingflow

(mL/min)

1.6 3.2 Borepumpingflow(mL/min) 2.8 2.7 Borecomposition Demiwater 5wt%PVPin

ultrapure water Airgap(cm) 10 – Pullingspeed(m/min) 3.5 7 ContentofPVPinthedope

solutionforselectivelayer (wt%)

7 10

2.3. Clearanceanddialysismodelling

Themodelpresentedintheprevioussectionhelpsto deter-mine the mass flux for given conditions of concentration acrossthemembrane.Thismassfluxdefinestheclearance foragiventimeofthedialysis.Inordertodepictthewhole dialysisprocess,themassflux(Eqs.(6),(8)and(10))hastobe solvedtogetherwithmassbalancesonthefeedcompartment andonthedialysatecompartment.Theglobal(Eqs.(12)and

(13))andthepartialmassbalance(Eqs.(14)and(15))forthe feedandthedialysatesideare:

dVf dt =−JS (12) dVd dt =JS (13) d

!

cfVf

dt =−jbS (14) d (cdVd) dt =jaS (15)

wherecfandcdstandforthesoluteconcentrationinthefeed

anddialysate,Sisthefiltrationsurfacearea,Vf andVd are

volumesoffeedanddialysate.

If consideringthat the transient characteristic time for themassfluxestablishmentisverysmall comparedtothe order ofmagnitudeofthe dialysis time, themodel can be solvedforpseudosteadystateconditions(consideringa suc-cessionofsteadystateforthe transfer).Theintegrationof Eqs.(12)–(15)combinedwiththemassfluxgiveninthe previ-oussectionenablesthedeterminationofthevariationofthe soluteconcentrationinthebloodandthedialysateandthen theclearancekinetics.

3.

Validation

experiments

3.1. Membranefabricationandcharacterization

Doublelayermixedmatrixmembraneswereprepared accord-ingtoamultistepprocedure.Fortheselectivelayer,apolymer solution was prepared by dissolving 15% polyethersulfone (PES,UltrasonE6020,BASF,Germany) and polyvinylpyrroli-done(PVP,K90,Fluka,Germany)ofdifferentquantity(details in Table 1) in ultra-pure N-methylpyrrolidone (NMP, Acros Organics,Belgium).Adsorptivelayerwasbasedonasolution containing14%PESand1.4%PVPdissolvedinNMP.Activated

Table2–Spinneretspecifications.

Membrane1 Membrane2 Innerdiameterneedle(mm) 0.16 0.26 Outerdiameterneedle(mm) 0.26 0.46 Innerdiameterfirstorifice(mm) 0.46 0.66 Outerdiameterfirstorifice(mm) 0.66 0.96 Innerdiametersecondorifice(mm) 0.86 1.66

Table3–Uremictoxins.

Type Specificproperties Mainrepresentatives Unboundsmall <500Da Urea,creatinine Unboundmiddle 500Da–60kDa Leptin,endothelin,

b2-microglobulin

Proteinbound Capableofprotein binding

Indoxylsulfate,p-cresol

Table4–Characteristicsofb2-microglobulinand a-lactalbumin.

Protein Molecularweight,kDa pI b2-Microglobulin 11.8 5.7

a-Lactalbumin 14.2 4.5

carbon(AC)withaveragediameterof27mmwasadded gradu-allytothepolymersolutiontoobtainafinalconcentrationof 60%byweight.Bothsolutionsweredegassedfor48hbefore thespinning.

Twodifferentspinningconditionswereutilizedaimingto obtainmembranesofdifferenttransportproperties.Thefirst batchofmembranes(furtherdenoted asMembrane1) was formed via immersion precipitation method at the condi-tionssummarizedinTable1.Thesecondbatchofmembranes (hereondenotedasMembrane2)wasfabricatedaccordingto theproceduredescribedbyTijinketal.(2013)withthe condi-tions,whicharealsopresentedinTable1.

Table2describesthespecificationofusedspinnerets. The membranes were cleaned by ultra-pure water to removetheremainingsolvent.Thefiberswerethendriedat 37◦_{Cfor2handsubsequentlywerefracturedinliquid}

nitro-gen.Thesamplesweredriedundervacuumat30◦_Candgold

sputteredusingaBalzersUnionSCD040sputtercoater (Oer-likonBalzers,Balzers,Liechtenstein). Finally, sampleswere imaged usingaJEOL JSM-5600LVScanningElectron Micro-scope(JEOL,Tokyo,Japan).

Dryfiberswerecutintopiecesof15cmeachandthenglued intomodules(innerdiameterof4mm)of3fiberseach,and 10cminlength.Clean-waterfluxwasthenmeasuredforeach modulewithuseoftheset-up(Fig.3fromPavlenkoetal.,2016) describedindetailinthefollowingsection.Forthis,modules wereinitiallypre-pressurizedat2barfor1handthentested attransmembranepressuresof0.5,1.0,1.5and2.0bar,with ultrapurewater.Themassofcollectedpermeatewas moni-toredcontinuously.

3.2. Adsorptionkineticsinstaticmode

Atthisstage,theknowledgeoftheadsorptivecapacityofthe carbonparticlesisanimportantissue.Experimentswere con-ductedtodeterminethekineticsofadsorptionofcreatinine and a-lactalbumin onto the membrane. These two solutes havebeenchosenasmodelstoillustratethepotentialitiesand thelimitationofmixedmatrixmembranesintheremovalof unboundsmallandmiddleuremictoxins(Table3).

Fig.3–Schematicrepresentationoftheappliedset-up(fromPavlenkoetal.,2016).

Creatinine was selected as a model of unbound small toxins group when a-lactabuminwas selected as a model compound for unbound middle-sized toxins such as b2

-microglobulin. Similarities between the molecular weight andtheIsoelectricpointofbothproteinsjustifythischoice (Table4).

Forthepurposeofkineticsofadsorptionstudy,fivepieces ofhollow-fibermembranesof5cmeachwereplacedin25mL of0.1g/Lsolutionofcreatinineora-lactalbumininphosphate buffersolution(PBS).Thesolutionwasstirredtoavoid trans-portlimitationsbetweensoluteandsurfaceofadsorptivelayer ofthemembrane.Thedecayofsoluteconcentrationwithtime wasmonitoredbysamplingof0.5mL/sampleevery3minfor furtheranalysisbyUV-spectroscopy.Thetimingofsample col-lection wasadjustedso astolimit thetotal withdrawalof initial volumeat10%.A lineardecrease ofthe solute con-centrationintheinitial stageoftheprocess wasobserved. Theinitialslope ofthe concentrationdecreaseprovides an informationontherateofadsorptionwhichmaybeusedin thecalculationoftheproductoftheadsorptionconstantand specificsurfaceareaofadsorbent(ak).

3.3. Experimentalstudyoftheclearanceofsmalland middlesizedsolutes

Theexperimentalanalysisoftheclearanceofsoluteswiththe dual-layermembraneswasconductedwithuseoftheset-up (ConvergenceB.V.,Enschede,TheNetherlands)presentedon

Fig.3.

Briefly, the experimental set-up consists of2 peristaltic pumps, 4pressure detectors,2 back-pressure valvesand 2 reservoirsforthemodelsolutions(Pavlenkoetal.,2016).All thepartsoftheset-upareconnectedviaPTFEtubingsto mem-branemodulewhichhasinletsandoutletsforthefeedand dialysatesolutionsthatarepumpedincounter-current con-figuration.Theflowratesofthefeedanddialysatesolutions arecontroledbytheperistalticpumps,whileconstantTMPis generatedbytheback-pressurevalves.Pressureinthesystem ismonitoredby4pressuredetectorswhicharelocatednear themembranemodule’sinletsandoutlets.Compartmentsfor themodulesolutionsarepositionedonthebalancestocontrol themassofthesystemovertime.

Forthevalidationexperiment,PBS-based0.1g/Lsolutions ofcreatinine(Sigma–Aldrich,France)asamodelmoleculefor the small uremic toxin; and a-lactalbumin (Sigma–Aldrich, France)asamodelcompoundforthemiddle-sizedmolecule wereprepared,andwerefurtherusedasfeedsolutions.The puresolutionofPBSwasusedasthedialysate.

Validation experiments were conducted using 50mL of thefeedanddialysatesolutions.Moreover,wefollowedthe convention that the flow rate of the dialysate has to be twiceashigh astheflowrateofthefeed stream(Henrich, 2009).Assuch,thefeedflowratewassetas5mL/min,while the dialysateflowwasequalto10mL/min.Inaddition, the feed solution wasflown insidethe hollowfiber membrane lumen,whilethedialysatewaspumpedthroughtheshellside ofthemodule.For experimentsindiffusivemode (conven-tionaldialysis),thetransmembranepressure(TMP)waskept around0bar,meaningthatthetoxinremovaldrivingforcewas generatedonlybythedifferenceofitsconcentrationacross themembrane.Incontrary,anothersetofexperimentswas conductedwithutilizationofbothdiffusionandconvection (hemodiafiltrationmode).TheretheTMPwassetat0.17baror 0.5barindifferentexperimentsinordertoinducethe convec-tivetransportthroughthemembrane.Thesepressureregimes werechoseninorderto,ononehand,avoidthenecessityto addmorefeedtothesystemduringtheexperiment,andon theotherhand,togenerateaconvectiveflowwitha notice-able impacton the overallsolute removal.Throughout the experiment,theweightofthefeedanddialysatesolutionswas continuouslymonitored.Thedecreaseofthefeed(increaseof thedialysate)masswasconsideredtobeduetothepresence ofconvection.Samplesof1mLweretakenfromboth compart-mentsatthesametimeandtheconcentrationofcreatinine and a-lactalbumin wasmeasured byUV-spectroscopy (Var-ian,Cary300ScanUV–visibleSpectrophotometer)at25◦_Cina

10mmquartzcuvetteat230nmand280nm,respectively.The increaseinsoluteconcentrationinthedialysatewasascribed tothediffusivetransport,incaseoftheexperimentconducted in diffusive mode (conventional dialysis) or to a combina-tionofdiffusionandconvectionforexperimentsperformed in hemodiafiltrationmode. In parallel, based on mismatch betweentheamountofsolutedisappearedfromthefeedside and foundinthedialysatethe quantityofsoluteadsorbed

Fig.4–SEMimagesofthedual-layerMMM(Membrane2:-a-and-b-;Membrane1:-c-,-d-and-e-). insidethe membrane was defined. Finally, the total solute

removalwasconsideredtobethesumofdiffusive/convective and adsorptive removals. Each validation experiment was conductedthreetimes.Aftertheexperiments,themodules weredriedinairandtheeffectivesurfaceareaofmembrane insidethemodulewasmeasured.

4.

Results

and

discussion

4.1. Membranefabricationandcharacterization

Fig.4demonstratesthestructureandthemorphologyofthe membranes.

Fig.4comparestheimagesoftheMembrane2(Fig.4a,b) andMembrane1(Fig.4c–e)discussedinthisstudy.Forboth membranes,onecandistinguishtheinnerparticle-freeregion witha denserskin layeron the lumen sideand the outer particle-loaded (MMM)layer. In both cases, the two layers are well attached, the MMM layerismore porous and the particle-freeinnerselectivelayer,mainlydeterminestothe

Table5–Geometricaldimensionsofthefabricated fibers.

Membrane1 Membrane2 Thicknessoftheinnerlayer,mm 21 49

ThicknessofMMM,mm 47 111 Lumendiameter,mm 450 669 Externaldiameterofthefiber,mm 586 984

masstransport resistanceofthe entiremembrane. Forthe Membrane 1, the particle freeinner hasa denseskin (see

Fig. 4-e-) andarather fingerlikepore morphologytowards theMMMlayer.FortheMembrane2,theinnerparticlefree layerhasoverallspongelikemorphologyandislessdense(see

Fig.4-b-).Forbothmembranes1and2,SEMimagessuggest that theactivatedcarbonparticles are quiteuniformly dis-tributedinsidethepolymericmatrix.Ingeneral,Membrane 1ismuchthinnerthanMembrane2.Thedimensionsofthe fibersaresummarizedinTable5.Thesevalueswerefurther

Table6–Valuesofreactionkineticparameter,ak,for varioussystems“Membrane-solute”.

Membrane1 Membrane2 Creatinine 0.152s−1 _0.082s−1

a-Lactalbumin –a _0.011s−1 a _{Wasnotdetermined,becauseMembrane1wasimpermeablefor}

a-lactalbumin.

usedforsimulationsbasedonthemasstransfermodel,which hasbeendescribedhereabove.

ThecleanwaterfluxtestsrevealedthattheMembrane1 issignificantlylesspermeable thanM2:2.5±1.1L/m2_/h/bar

vs58.4±9.3L/m2_{/h/bar(Tijinketal.,2013)respectively,}

con-sistent with the SEM observations. The integrity of the membranes was notaffected duringthe water permeance testing, meaning that the fabricated membranes possess sufficientmechanicalstrengthforthedialysisand hemodi-afiltrationexperiments.

The productof heterogeneous adsorption constant and specific surface area of activated carbon (ak) is refered to as a reaction kinetic parameter hereafter, for both mem-branetypeswasdeterminedwithrespecttocreatinineand a-lactalbumin(Table6).

AccordingtoTable6, thereactionkineticparameter,ak, parameter forthe adsorptionofcreatinine is almost twice higherforMembrane1thanforMembrane2.Sincethe mem-brane material and the adsorptive particles are the same forbothmembranes,thedifferenceinadsorptiveproperties betweenthesemembranesmay beattributedtothe differ-ent accessibility ofactivatedcarbon particles embeddedin thepolymermatrix.Inparallel,thecomparisonofadsorptive propertiesofMembrane2withrespecttocreatinineand a-lactalbumindemonstratesthat theadsorptionofthe latter issignificantly lower than the former (only 0.011s−1_{). This}

maybeattributedtoeitheradifferenceinsteric hindrance or connected to the different affinity of activated carbon particlesforcreatinineanda-lactalbumin.Finally,various per-meationtrialswithMembrane1revealedthatitiscompletely impermeablefora-lactalbumin;therefore,theexperimental estimationof“ak”forthiscasewasnotperformed.

4.2. Doublelayermembranetransfermodeling

ToperformthemodellingofthetransferthroughtheMMM, the boundary layer thicknesses (ı) have to be estimated. Sincethe validationofthe model wasperformed withthe hollow-fibermembranes placedinthe module,twosets of hydrodynamiccorrelations,forlumenandshellsides,were used.Theflowratesoffeedanddialysatestreamsweretaken from thefiltration theset-up utilizedduringthe validation experiments.Theestimationoftheconditionsinthelumen

sidewasdoneusingtwofollowingcorrelations(Yangetal., 2013): Shl=2.66Re0.25l Sc0.33l



_d h_l l



0.33 ;ıl= dh_l Shl (16)

whilethequantificationofthehydrodynamicconditionsin theshellsidewasdonebyapplyinganothersetofequations (Yangetal.,2013): Shs=1.25Re0.93s Sc0.33s



dhs l



0.93 ; dhs =

!

d2 s−Nd2e

(ds+Nde) ; ıs= dhs Shs (17)

wheredhstandsforthehydraulicdiameter,Nisthenumber

offibers,de and ds are the externaldiameter ofthe

mem-brane and of the module (shell) respectively. The selected correlationsareassumedtobeapplicable,astheywere devel-opedforhollowfibermodulesoperatedinlaminarconditions (R<2000),whileinourexperimentsRe intheshelland the lumensidewasbelow100.

Thevalidationexperimentswereconductedusingfeedand dialysateflowratesof5mL/minand10mL/min,respectively. Sincealowconcentrationofsolutewasusedforthe prepa-ration of all feed solutions and the purephosphate buffer wasusedasthedialysate,theviscosityanddensityofboth solutions were assumed to beequalto the ones ofwater: 10−3_{Pasand 1000kg/m}3 _{respectivelyfor the calculationof}

theboundarylayerthicknesses.Basedontheaforementioned hydrodynamiccorrelations,thethicknessesoftheboundary layersinthelumenandshellsidesweredetermined(Table7). Theboundary layerthickness inthe creatinine removal experiments differsincaseofMembrane1and Membrane 2. This difference is ascribed to the difference in mem-brane dimensions (see Section 4.1). Similarly, due to the differencebetweentheouterdiametersofbothmembranes, the boundary layerthickness on the shell side was found smallerforMembrane1.Similarconclusionsarefoundforthe a-lactalbuminremovalexperiments.Thelumenboundary lay-erswerefoundtobe17.9mmand23.8mm,whiletheshellside boundarylayersare87.1mmand124mmforMembrane1and Membrane2,respectively.

AccordingtodatapresentedbyShawetal.(2009)the diffu-sioncoefficientofcreatinine inwater atroomtemperature is 9×10−10_m2_{/s,whileKim et al. (2013) used the valueof}

5.31×10−10_m2_{/s.Therefore,hereweusedtheaveragevalue}

of7×10−10_m2_{/s.Tothebestofourknowledge,thediffusion}

coefficientofa-lactalbuminwasnotreportedintheliterature. Thus withuse ofEinsteinequation and moleculeeffective radius reported by Fu et al. (2005) it was calculated to be equalto1.07×10−10_m2_{/s.Finally, forthe sakeofsimplicity}

weassumedthatthesolutediffusioncoefficientisthesame intheallregionsofthesystem.

Table7–Boundarylayerthicknessesinthevalidationexperiments.

Membrane1 Membrane2 Experimentwith

creatinine

Lumenboundarylayer,mm 30.7 45.3 Shellsideboundarylayer,mm 197 245 Experimentwith

a-lactalbumin

Lumenboundarylayer,mm 17.9 23.8 Shellsideboundarylayer,mm 87.1 124

Fig.5–Soluteconcentrationprofilesacrossthemembranedependingontheremovalmechanismsinvolved.(BBLstandsfor bloodboundarylayer,PFL—particlefreelayer,MMM—mixedmatrixmembrane)(ImageA—diffusionvs

diffusion+adsorpsion;imageB—diffusion+convectionvsdiffusion+convection+adsorption).

Table8–Inputparametersappliedtoproducecreatinine concentrationprofileacrossMembrane2.

Nameofinputparameter Value Bloodboundarylayerthickness 5×10−5_m

Particle-freelayerthickness 4.9×10−5_m

MMMthickness 1.11×10−4_m

akparameter 0.082s−1

Solutediffusioncoefficient 7×10−10_m2_/s

Thicknessofdialysateboundarylayer 2.3×10−4_m

Convectiveflowratethroughthemembrane 1.1×10−6_m/sa

Soluteconcentrationinthebloodside 0.885mol/m3

Sievingcoefficient 1

a _{Experimentallymeasuredaverageconvectiveflowrateoverthe}

filtrationprocess.

Theparametersusedforoursimulationshavebeen gath-eredinTable8.Inprinciple,theseparametersrepresentthe filtrationofcreatininethroughtheMembrane2.

The predicted solute concentration profiles across the membranewithandwithoutadsorptionhavebeenplottedin

Fig.5,intheabsenceofsoluteinthedialysatestream. According to Fig. 5the presence ofadsorptive particles favorsthereductionofsoluteconcentrationinsidethe mem-brane.Themolarfluxofsolutethroughthemembraneisthen moreimportant:onecannoteasteeperconcentrationprofile inthebloodboundarylayer(BBL).Thiseffectisattributedto theadsorptionofafractionofthepermeatingspeciesbythe particlespresent inthe MMMthatisacceleratingthe mass transfer.Quantitativeexpressionofsuchphenomenaisgiven bythesolutetransferenhancementfactor(STEF)parameter providedbythemodel(Table9).

FromTable9onemayconcludethatfortheselectedset ofinputparameters,theadditionofconvectiveflowtosolely diffusion-drivensoluteremovalresultsina1.39timesgreater

Fig.6–Soluteconcentrationprofileacrossthemembrane andfeedboundarylayeratvariousratesofconvective flows.

removalrate,whileadsorptionprovidesaSTEFof2.21. More-over, the combination ofall threeremoval mechanisms is characterized byaSTEFof2.58,meaningthatcomparedto apurediffusivetransfer,thecompletesystemallowstogain morethan2.5timesaflowofsoluteoutofthefeed(blood).

Inparallel,theconcentrationinsidethemembraneaswell as the molarfluxacross the membrane dependon hydro-dynamics on the feed side. Fig. 6 shows the influence of convectionflowrate(Pecletnumber)onthesystem,where dif-fusion,convectionandadsorptionareemployedinthesolute removalfromthefeedstream.

As it can be seen in Fig. 6, at lower values of Pe the solute concentration in the MMM is significantly different fromtheoneinthebulk.However,atPe>1,whenconvection becomes dominant over diffusion, the solute is more

effi-Table9–Solutetransferaccelerationforvarioussystemconditions.

Soluteremovalmechanismsinvolved Molarfluxdensity,ja Solutetransfer

enhancementfactor Diffusion 1.41×10−6_molm−2_s−1 ₁a

Diffusionandadsorption 3.12×10−6_molm−2_s−1 _2.21

Diffusionandconvection 1.95×10−6_molm−2_s−1 _1.39

Diffusion,convection,andadsorption 3.63×10−6_molm−2_s−1 _2.58 a _{Takenasreference.}

Fig.7–Resultsofvalidationexperiment1vstheoretical prediction(continuouslinesarepredictedevolutionof soluteremoval,whiledotsrepresenttheexperimental results).

cientlytransferredtowardstheadsorptivelayer,andthebulk andmembraneconcentrationsareclosetoeachother.Inorder tokeepaPelargerthan1,theconvectiveflowof1.4×10−5_m/s

orgreaterhastobegenerated.Ontheotherhand,an exces-siveconvectionrateleadstothereducedimpactofadsorption onthesoluteremoval.Therefore,athighPeonedoesnottake advantageofthepresenceoftheadsorptivecapacityofthe membrane.

4.3. Experimentalresultsvsmodelling

Afirstexperimentwasconductedinthediffusivemodewith useofcreatininesolutionandMembrane1.Theexperimental resultsarecomparedtothemodelinFig.7.

AccordingtoFig.7thediffusivecurveisideallypredicting theremovalofcreatininebydiffusion.Bothexperimentaland theoreticalresultsdemonstratethatduringafour-hour exper-imentatgivenconditions,onemayachievediffusiveremoval of0.6mgofcreatinineinourconditions.Theslightdeviation ofdiffusivecurvefromlinearityisaresultofdecreaseinsolute concentrationinthefeedandincreaseinthedialysate,which leadstothereductionofthedrivingforceacrossthe mem-brane.Incontrary,theadsorptioncurvefitstheexperimental dataonlyintheearlystageoftheprocess.Thisobservation may beexplainedbythe saturationofadsorptiveparticles, whichsuggeststhepresenceoflimitingmembraneadsorption capacity.Inourexperimentthiseffectstartstobenoticeable afterhalfanhourofexperimentandultimately reachinga membraneadsorptivecapacityof0.34mgor,moreprecisely, 26.27mg ofcreatinine per 1mgof membrane. Therefore, a modificationofthemodelwiththenewfittedinput parame-ter“Membranelimitingadsorptioncapacity”(MLAC)wasdone aimingtoaccountfortheactivatedcarbonparticlessaturation (Fig.8).Inordertointroducethisparameterinthemodel,the followingconditionswereapplied:

(

q<qmax, r=akc q≥qmax, r=0

(18)

Henceuntiltheamountofadsorbedspecies(q)isinferior tothemaximal capacity(qmax),the adsorptionmechanism

isactiveand ishappeningaccording tothepreviously dis-cussedassumptions.Oncethemaximalmembranecapacity

Fig.8–Resultsofvalidationexperiment1vstheoretical predictionaccountingsaturationoftheadsorbent.

hasbeenreached,theadsorptionisnolongerpossibleand therefore,theadsorptiveremovalmechanismisswitchedoff bythemodel.TheresultofMLACapplicationforthe experi-ment1ispresentedinFig.8.

According toFig. 8,the removalofcreatinine reaches a plateau. Thiscould beduetosaturationofparticles and/or limitationsinaccessibilityofthecarbonadsorptionsitesby thesoluteunderdiffusiveconditions.Infact,ourrecentwork hasshowedthatunderconvectiveconditionstheadsorption andremovalcansignificantlyincrease(Tijinketal.,2013)

Theinsertionofthelimitingadsorptioncapacity parame-terenablesaratherprecisepredictionoftheoutcomeofthe experimentalcreatinineremoval.Finally,theanalysisshowed thattheadsorptiveparticlesinthemembraneenhance the transfer ofcreatinine by 93% within the first 45min. Such observationprovidesthenumericaljustificationofthe benefi-cialuseofdual-layermembraneswithincorporatedparticles ofactivatedcarbon.

Avalidationexperimentwithuseofcreatininesolutionand Membrane1wasalsoconductedinhemodiafiltrationmode (applying TMP).However,duetotightstructureofthe skin-layerintheparticlefreemembranetheconvectionwaslimited evenathighpressures,thereforetheconvectivetermhada negligible contributionto masstransfer duringthe experi-ment.Consequently,theresultofthisexperimentwasalmost identicaltotheonewithdiffusivetransportonly(resultsnot shown).

In addition, due to poor permeability of a-lactalbumin throughMembrane1anditspoordiffusivitythrough Mem-brane 2 during the diffusive experiment, no noticeable concentrationreductioninthefeedsidewasobserved. There-fore,theresultsofthesetrialsarealsonotpresentedinthis paperaswell.

Inordertoevaluatethevalidityofthemodelwithrespect tomiddlesizedsolutes,asecondexperimentwasconducted withMembrane2anda-lactalbuminatTMPof0.5bar, result-ing intheconvectiveflow of3.38×10−6_{m/s.Theresultsof}

thesecondexperimentandtheircomparisonwiththemodel predictionsmodelareshowninFig.9.

AsshowninFig.9,theexperimentalandcomputeddata forthediffusive+convectiveremovalofa-lactalbuminarein verygoodagreement.Thankstoasignificantconvectionrate, theremovalof2mgofa-lactalbuminwasachievedwithinfour hoursinourexperimentalconditions.Inaddition,the adsorp-tiveremovalof0.47mgwasobservedafterhalfanhour,and didnotchangethroughouttheremainingtime.This

observa-Fig.9–Resultsofvalidationexperiment2vstheoretical prediction.

tionindicatesthat,alikethefirstexperiment,thesaturationof adsorptiveparticleswasachievedwithintheearlystageofthe process.Thefastsaturationmaybeattributedtothepresence ofconvection,whichenabledfacilitatedtransportofprotein towardstheactivatedcarbonparticles.Thishypothesiswas supported bythe analysis, which demonstrated that since convectionismuchfasterthandiffusionofa-lactalbumin,it provided8.4timessolutetransportenhancementinthe begin-ning ofthe process.Inthe sametime, thepresenceofthe adsorptivelayerresultedin71%ofproteinremoval facilita-tion.Finally,the cumulativeeffectofMMMandconvection enabled 9.4timesgreater soluteremoval rateatthe initial momentoftheexperiment(t=0)thanonemayexpectfrom solelydiffusion-basedprocessgiventheexperimental condi-tionsusedinthepresentresearch.

Sinceduringexperiment2thesaturationoftheMMMwas achieved,wemodifiedthemodelwiththe“Membranelimiting adsorptioncapacity”parameterasbefore.Basedonthe exper-imentalresulttheinputvalueof8.1mgofa-lactalbuminper mgofmembranewastaken.Inaddition,thecontentof acti-vatedcarboninsideMembrane2waspreviouslyreportedto be53%(Tijinketal.,2012).Therefore,theMLACvaluemaybe alsopresentedas15.2mgofa-lactalbuminpermgofactivated carboninsidethemembrane.Thisvalueisclosetothe exper-imentallymeasuredadsorptivecapacityofactivatedcarbon powderof15.9mgofa-lactalbuminpermgofactivatedcarbon. The comparison between validation experiment 2 and modifiedmodelisshowninFig.10.

AccordingtoFig.10,thesaturationoftheadsorbentwas reachedafter30minofexperiment,andtheremainingtime, a-lactalbuminwasremovedfromthefeedmainlybyconvection andpartiallybydiffusion.

ThusthemodelrevealsthattheMMMlayerisefficientin theearlystagesoftheprocess,significantlyincreasingtherate ofremoval.Beyondthatpoint,correspondingtothesaturation oftheadsorbents,therateofremovalisruledbythediffusion andconvectionasforaclassicalhemodiafiltrationprocess.

ObtainedMLACvalueforMembrane2maybeviewedfrom a different perspective. The module consisting of 3 fibers of10cmeachand lumendiameterof669mm(Table5) pro-videdthetotalfiltrationarea of0.567×10−3_m2_,allowedan

adsorptiveremovalof0.47mgofa-lactalbumin.Considering thecommonhemodialyserwithaneffectivesurfaceareaof 1.5m2_{,theproportionalscale-upofourmodulewouldresult}

in 1242mg removal of a-lactalbumin, which significantly

Fig.10–Resultsofvalidationexperiment2vstheoretical predictionaccountingsaturationoftheadsorbent.

exceedsthedesiredb2-microglobulinremovalof350–700mg

persession(basedonthriceweeklytreatment)(Druekeand Massy, 2009). As such, even without further optimization (reduction oflumendiameter, tuningofsieving properties, selectionofproperoperatingconditionsetc),thismembrane maybeeffectivelyusedfortheremovalmiddle-sizeduremic toxins.

5.

Conclusion

Theconceptofdoublelayermembranesaimingthe improve-mentofremovalofbloodtoxinsisapromisingadvancement ofdialysistreatment.Inthepresentstudy,wepresentedthe model whichallowsmorein-depthanalysisofinterplayof threesoluteremovalmechanisms:diffusion,convection,and adsorption.Themodeldemonstrated thesolute concentra-tionprofileacrossthemembraneandquantifiedthetransfer improvementinducedbytheadsorptionlayerinsidethe mem-brane. The model was validated via comparison of model predictionswithoutcomeofexperimental testingof home-madedoublelayermixedmatrixmembranes.Thedeveloped model provides an accurate agreement with diffusive and convectiveremovalsobtainedexperimentallyforbothsmall andmiddlesizeduremictoxins.Moreover,evenarather sim-plistic approachinthe modelling ofadsorptive removalof toxins(first-orderreaction)providesapossibilitytoevaluate theamountofadsorbedspeciesattheearlystagesof treat-mentprocess.Thedevelopedmodelmaybefurtherapplied intheoptimizationofdoublelayermembranepropertiesand theprocessconditions.

Acknowledgements

Authors would liketo acknowledge theMarie Skłodowska-Curiefoundation(ProjectBIOART:grantno.316690, EU-FP7-PEOPLE-ITN-2012)forthefinancialsupportofthisproject.

The authors are grateful to Ilaria Geremia (Bioartificial organsgroup,departmentofBiomaterialsscienceand tech-nology,UniversityofTwente)forprovidingtheSEMimagesof theduallayerMembrane1.

Appendix

A.

Thecompleteexpressionsforpseudomasstransfer conduc-tances(Ki,j),whichareusedinSection2.2,havethefollowing

form: K1a= JePe1 ePe1−1 (A.1) K1b= J ePe1−1; (A.2) K2a= JePe2 ePe2−1; (A.3) K2b= J ePe2−1; (A.4) K3a= D3

p

Pe2₃+4ϕ2_cosh(

p

Pe2 3+4ϕ2 2 ) 2ı3sinh

 p

_Pe2 3+4ϕ2 2



+ D3Pe3 2ı3 ; (A.5) K3b= D3e− Pe3 2

p

Pe2 3+4ϕ2 2ı3sinh

 p

Pe2 3+4ϕ2 2



; (A.6) K3c= D3e Pe3 2

p

Pe2 3+4ϕ2 2ı3sinh

 p

Pe2 3+4ϕ2 2



(A.7) K3d= D3

p

Pe2₃+4ϕ2_cosh(

p

Pe2 3+4ϕ2 2 )) 2ı3sinh

 p

Pe2₃+4ϕ2 2



− D3Pe3 2ı3 (A.8) K4a= Je Pe4 ePe4−1; (A.9) K4b= J ePe4−1 (A.10)

Appendix

B.

Supplementary

data

Supplementary data associated with this

arti-cle can be found, in the online version, at

http://dx.doi.org/10.1016/j.cherd.2017.08.017.

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