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
Any correspondence concerning this service should be sent to the repository
administrator:
staff-oatao@listes-diff.inp-toulouse.fr
O
pen
A
rchive
T
OULOUSE
A
rchive
O
uverte (
OATAO
)
OATAO is an open access repository that collects the work of Toulouse researchers and
makes it freely available over the web where possible.
This is an author-deposited version published in :
http://oatao.univ-toulouse.fr/
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
aaLaboratoiredeGénieChimique,UniversitédeToulouse,CNRS,INPT,UPS,Toulouse,France
bBioartificialOrgans,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:caussera@chimie.ups-tlse.fr(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(m2per
m3ofmixedmatrixmembrane)
c Concentration(molm−3)
d Diameter(m)
D Diffusioncoefficient(m2s−1)
div Divergence(–)
j Molarfluxdensity(molm−2s−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−3s−1)
Re Reynoldsnumber(–)
s Sinkterm(molm−3s−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 2c dx2−J dc dx−akc=0 (4)
where a stands for specific surface area of adsorbent in m2perm3 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 ı3xsinh!
1!
x ı3−1 sinh (1) (5) where 1=p
Pe2 3+4ϕ2 2 ; Pe3= Jı3 D3; ϕ=q
akD3ı3.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; thedifferencebeingrela-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 hl l 0.33 ;ıl= dhl 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: 10−3Pasand 1000kg/m3 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 9×10−10m2/s,whileKim et al. (2013) used the valueof
5.31×10−10m2/s.Therefore,hereweusedtheaveragevalue
of7×10−10m2/s.Tothebestofourknowledge,thediffusion
coefficientofa-lactalbuminwasnotreportedintheliterature. Thus withuse ofEinsteinequation and moleculeeffective radius reported by Fu et al. (2005) it was calculated to be equalto1.07×10−10m2/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 5×10−5m
Particle-freelayerthickness 4.9×10−5m
MMMthickness 1.11×10−4m
akparameter 0.082s−1
Solutediffusioncoefficient 7×10−10m2/s
Thicknessofdialysateboundarylayer 2.3×10−4m
Convectiveflowratethroughthemembrane 1.1×10−6m/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 1.41×10−6molm−2s−1 1a
Diffusionandadsorption 3.12×10−6molm−2s−1 2.21
Diffusionandconvection 1.95×10−6molm−2s−1 1.39
Diffusion,convection,andadsorption 3.63×10−6molm−2s−1 2.58 a Takenasreference.
Fig.7–Resultsofvalidationexperiment1vstheoretical prediction(continuouslinesarepredictedevolutionof soluteremoval,whiledotsrepresenttheexperimental results).
cientlytransferredtowardstheadsorptivelayer,andthebulk andmembraneconcentrationsareclosetoeachother.Inorder tokeepaPelargerthan1,theconvectiveflowof1.4×10−5m/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 of3.38×10−6m/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 of0.567×10−3m2,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+4ϕ2cosh(p
Pe2 3+4ϕ2 2 ) 2ı3sinhp
Pe2 3+4ϕ2 2 + D3Pe3 2ı3 ; (A.5) K3b= D3e− Pe3 2p
Pe2 3+4ϕ2 2ı3sinhp
Pe2 3+4ϕ2 2 ; (A.6) K3c= D3e Pe3 2p
Pe2 3+4ϕ2 2ı3sinhp
Pe2 3+4ϕ2 2 (A.7) K3d= D3p
Pe23+4ϕ2cosh(p
Pe2 3+4ϕ2 2 )) 2ı3sinhp
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.
References
Barreto,F.C.,Barreto,D.V.,Liabeuf,S.,Meert,N.,Glorieux,G., Temmar,M.,Choukroun,G.,Vanholder,R.,Massy,Z.A.,2009.
Serumindoxylsulfateisassociatedwithvasculardiseaseand mortalityinchronickidneydiseasepatients.Clin.J.Am.Soc. Nephrol.4,1551–1558.
Busch,M.,Franke,S.,Ruster,C.,Wolf,G.,2010.Advanced glycationend-productsandthekidney.Eur.J.Clin.Invest.40, 742–755.
Dobre,M.,Meyer,T.W.,Hostetter,T.H.,2013.Searchingforuremic toxins.Clin.J.Am.Soc.Nephrol.8,322–327.
Drueke,T.B.,Massy,Z.A.,2009.Beta2-microglobulin.Semin.Dial. 22,378–380.
Eloot,S.,VanBiesen,W.,Vanholder,R.,2012.Asadbutforgotten truth:thestoryofslow-movingsolutesinfasthemodialysis. Semin.Dial.25,505–509.
Fu,B.M.,Adamson,R.H.,Curry,F.R.,2005.Determinationof microvesselpermeabilityandtissuediffusioncoefficientof solutesbylaserscanningconfocalmicroscopy.J.Biomech. Eng.127,270–278.
Henrich,W.L.,2009.PrinciplesandPracticeofDialysis,fourthed. LippincottWilliams&Wilkins.
Kim,J.C.,Cruz,D.,Garzotto,F.,Kaushik,M.,Teixeria,C.,Baldwin, M.,Baldwin,I.,Nalesso,F.,Kim,J.H.,Kang,E.,Kim,H.C., Ronco,C.,2013.Effectsofdialysateflowconfigurationsin continuousrenalreplacementtherapyonsoluteremoval: computationalmodeling.BloodPurif.35,106–111.
Liabeuf,S.,Drüeke,T.B.,Massy,Z.A.,2011.Protein-bounduremic toxins:newinsightfromclinicalstudies.Toxins3,911–919.
Luo,F.J.,Patel,K.P.,Marquez,I.O.,Plummer,N.S.,Hostetter,T.H., Meyer,T.W.,2009.Effectofincreasingdialyzermasstransfer areacoefficientanddialysateflowonclearanceof
protein-boundsolutes:apilotcrossovertrial.Am.J.Kidney Dis.53,1042–1049.
Meyer,T.W.,Peattie,J.W.,Miller,J.D.,Dinh,D.C.,Recht,N.S., Walther,J.L.,Hostetter,T.H.,2007.Increasingtheclearanceof protein-boundsolutesbyadditionofasorbenttothe dialysate.J.Am.Soc.Nephrol.18,868–874.
Meyer,T.W.,Sirich,T.L.,Hostetter,T.H.,2011.Dialysiscannotbe dosed.Semin.Dial.24,471–479.
Pavlenko,D.,vanGeffen,E.,vanSteenbergen,M.J.,Glorieux,G., Vanholder,R.,Gerritsen,K.G.F.,Stamatialis,D.,2016.New low-fluxmixedmatrixmembranesthatoffersuperiorremoval ofprotein-boundtoxinsfromhumanplasma.Sci.Rep.6,1–9.
Raj,D.S.,Choudhury,D.,Welbourne,T.C.,Levi,M.,2000.
Advancedglycationendproducts:aNephrologist’s perspective.Am.J.KidneyDis.35,365–380.
Shaw,R.A.,Rigatto,C.,Reslerova,M.,Ying,S.L.,Man,A.,Schattka, B.,Battrell,C.F.,Matthewson,J.,Mansfield,C.,2009.Toward point-of-carediagnosticmetabolicfingerprinting:
quantificationofplasmacreatininebyinfraredspectroscopy ofmicrofluidic-preprocessedsamples.Analyst134,1224–1231.
Tijink,M.S.,Wester,M.,Glorieux,G.,Gerritsen,K.G.,Sun,J., Swart,P.C.,Borneman,Z.,Wessling,M.,Vanholder,R.,Joles, J.A.,Stamatialis,D.,2013.Mixedmatrixhollowfiber
membranesforremovalofprotein-boundtoxinsfromhuman plasma.Biomaterials34,7819–7828.
Tijink,M.S.,Wester,M.,Sun,J.,Saris,A.,Bolhuis-Versteeg,L.A., Saiful,S.,Joles,J.A.,Borneman,Z.,Wessling,M.,Stamatialis, D.F.,2012.Anovelapproachforbloodpurification:
mixed-matrixmembranescombiningdiffusionand adsorptioninonestep.ActaBiomater.8,2279–2287.
Vanholder,R.,Glorieux,G.,Eloot,S.,2015.Onceuponatimein dialysis:thelastdaysofKt/V?KidneyInt.88,460–465.
Walther,J.L.,Bartlett,D.W.,Chew,W.,Robertson,C.R.,Hostetter, T.H.,Meyer,T.W.,2006.Downloadablecomputermodelsfor renalreplacementtherapy.KidneyInt.69,1056–1063.
Yang,X.,Wang,R.,Fane,A.G.,Tang,C.Y.,Wenten,I.G.,2013.
Membranemoduledesignanddynamicshear-induced techniquestoenhanceliquidseparationbyhollowfiber modules:areview.Desalin.WaterTreat.51,3604–3627.