ContentslistsavailableatScienceDirect
International
Journal
of
Pharmaceutics
j o u r n al ho me p a g e :w w w . e l s e v i e r . c o m / l o c a t e / i j p h a r m
Mathematical
modelling
of
liquid
transport
in
swelling
pharmaceutical
immediate
release
tablets
Daniel
Markl
a,
Samy
Yassin
a,
D.
Ian
Wilson
a,
Daniel
J.
Goodwin
b,
Andrew
Anderson
b,
J.
Axel
Zeitler
a,∗aDepartmentofChemicalEngineeringandBiotechnology,UniversityofCambridge,WestCambridgeSite,PhilippaFawcettDrive,CambridgeCB30AS,UK
bGlaxoSmithKline,NewFrontiersSciencePark,Harlow,EssexCM195AW,UK
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received30January2017
Receivedinrevisedform8March2017 Accepted6April2017
Availableonline8April2017
Keywords:
Powdercompacts Tabletdisintegration Terahertzpulsedimaging Liquidpenetration Swelling Porosity
a
b
s
t
r
a
c
t
Oraldosageformsareanintegralpartofmodernhealthcareandaccountforthemajorityofdrugdelivery systems.Traditionallytheanalysisofthedissolutionbehaviourofadosageformisusedasthekey parametertoassesstheperformanceofadrugproduct.However,understandingthemechanismsof disintegrationisofcriticalimportancetoimprovethequalityofdrugdeliverysystems.Thedisintegration performanceisprimarilyimpactedbythehydrationandsubsequentswellingofthepowdercompact. Herewecompareliquidingressandswellingdataobtainedusingterahertzpulsedimaging(TPI)toaset ofmathematicalmodels.Theinterlinkbetweenhydrationkineticsandswellingisdescribedbyamodel basedonDarcy’slawandamodifiedswellingmodelbasedonthatofSchott.Ournewmodelincludes theevolutionofporosity,poresizeandpermeabilityasafunctionofhydrationtime.Resultsobtained fromtwosetsofsamplespreparedfrompuremicro-crystallinecellulose(MCC)indicateacleardifference inhydrationandswellingforsamplesofdifferentporositiesandparticlesizes,whicharecapturedby themodel.Couplinganovelimagingtechnique,suchasTPI,andmathematicalmodelsallowsbetter understandingofhydrationandswellingandeventuallytabletdisintegration.
©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Oraldosageforms,suchastablets,areoneofthemostwidely useddrugdeliverysystemsfortreatingpatientstoday.Thereare strictregulationsonhowthesetabletsshouldperform:immediate releaseformulationsshouldreleasetheirentireactive pharmaceu-ticalingredient(API)within10minofingestion(Guptaetal.,2006; Battuetal.,2007;CorveleynandRemon,1997),whileforsustained releasetabletformulationstherequirementistoreleasetheirfull APIcontentwithin24h(Davisetal.,1986).Furthermore,generics manufacturershavetopassbio-equivalencystudieswithrespect totheoriginalproductsonwhich thegeneric productis based (Sarantopoulosetal.,1995).Twokeypre-requisitesarerequired toachievethis:(1)acomprehensiveunderstandingofthe excip-ientsusedandhowtheywillaffecttabletdisintegrationaswell asdissolution;and(2)reliableandaccurateanalyticaltechniques forassessingtabletdissolution,disintegrationandAPIreleaseand theunderlyingmechanisms(Lin, 1988;Yu, 2008).Progresshas
∗Correspondingauthor.
E-mailaddress:jaz22@cam.ac.uk(J.A.Zeitler).
URL:http://thz.ceb.cam.ac.uk(J.A.Zeitler).
beenmadeexperimentallytounderstandhydrationphenomena andAPIreleasekineticsobservedintabletsandthusrelatethemto thechoiceofexcipientandresultingtabletmicrostructureusing destructive (Segale et al., 2010) and non-destructive analytical techniques(Battuetal.,2007;Bietal.,1999;Chenetal.,2010; Kazarian and van der Weerd, 2008; Kazarian and Chan, 2006; Rajabi-Siahboomietal.,1996).
Magneticresonanceimaging(MRI)hasbeenusedextensively toinvestigatethehydrationandswellingprocessesinsustained releasedformulations(Chenetal.,2010;FyfeandBlazek,1997; Rajabi-Siahboomietal.,1996,1994).ForexampleMRIhasshown that multiple phases of hydration co-exist in hydroxypropyl-methylcellulose(HPMC)compacts:viz.adryphaseatthecoreof thetablet,followedbyaglassyphase,aswollenglassyphaseand finallyagelphaseindirectcontactwiththedissolutionmedium (Chenetal.,2010;Zhangetal.,2011,2013).Morerecentlyitwas shownthatterahertzpulsedimaging(TPI)canbeusedtomeasure theliquidingressandtheswellingofrapiddisintegratingtablets simultaneously(Yassinetal.,2015a).
Alongsidetheneedforgoodanalyticaltechniquestoassessthe tabletperformance,thereisaneedfortheoreticalmodels predict-ingtabletdisintegration,dissolutionandAPIreleasetorationalise formulationdesign(SiepmannandSiepmann,2013;Colomboetal.,
http://dx.doi.org/10.1016/j.ijpharm.2017.04.015
1992b,a).Itwasobservedthattherateofliquiduptakeintotheir polymermatrixfollowedthesameprofileastheswelling;this indi-catesthatbothprocesseswerecontrolledbythetransportofthe solventintothepolymer matrix (Schott, 1992b,a).Compacts of samplesthatcontainedalargeamountofcross-linkedpolymeror micro-crystallinepolymerbothtypicallyexhibittwosubsequent phasesofswelling:initialswellingduetoliquidpenetrationand secondaryswellingduetothedisentanglementanddiffusionof thepolymersintothehydratingsolution(Schott,1992a,b).
Inthis workwe demonstratehowmathematicalmodelscan
beusedtogivedetailed insightintoliquidingressandswelling dynamics.Themodelfortheliquidingressinaswellingtabletis basedonDarcy’slawand iscomparedtosimulationsusingthe well-knownWashburnequation.Theswelling ismodelledby a Schottmodel,whichwasmodifiedtoenablethecomparisonwith theexperimentaldataandtodescribethereductioninporesize. Themodelsarecalibratedandcomparedtoexperimentaldatafrom puremicro-crystallinecellulose(MCC)powdercompactsmeasured byTPIduringhydrationinaflowcell.
2. Theory
2.1. Liquidpenetrationinarigidporousmedium
Oneofthekeyprocessesinvolvedinthedisintegrationofatablet isliquidpenetrationdrivenbycapillaryforces.Capillarytransport isanimportantfieldofresearchduetoitsnumerousapplications, suchasinpetroleumengineering,inhydrology(e.g.,movement ofgroundwater),inconsumerproducts(e.g.,markerpens,candle wicksandsponges)orinplants(e.g.,transportofwaterfromthe rootstothetips).Thefirstanalyticalsolutionforcapillaryriseasa functionoftimewasderivedbyLucasandWashburn(1921).They describedtheflowthroughporousmediabyconsideringthepore spaceasabundleofcapillarytubesofvaryingdiameterembedded inthesolidmatrix(seeFig.1).
The location of the liquid front, L, acrossthe entire porous mediumcanthusbeexpressedbythewell-knownWashburn equa-tionas
L=
Re
cos
2 t, (1)
whereRe=R2h/Rc,0(withRhasthehydraulicradiusandRc,0asthe poreradiusofthedrymaterial)isthemeaneffectiveporeradius,
isthesurfacetensionoftheliquid,isthecontactangleandis theviscosityofthefluid.
Furthermore,thetotalvolumetricfluxofNewtonianliquidsin anisotropicporousmediumcanbedescribedbyDarcy’slaw:
q=−K
P
L . (2)
Fig.1. Approximationoftheporousstructureofapowdercompact.(a)Theactual tabletconsistingofforgedparticlesformingchannelsthroughthepowdercompact. (b)Onlytheeffectivechannels(withaneffectiveporesize)areconsideredinthe model.(c)Cylindricalchannelsareapproximatedfromtheporestructure.Theinset highlightsthecapillaryactioninacylindricalchannel.Eachchannelischaracterised byitsliquidheightLanditsporeradiusRc,iwithaninitialporeradiusRc,0.Thecontact
angleisafluid/matrixpropertywhichisassumedtobeconstantforallchannels.
PisthepressuredifferenceandKistheintrinsicpermeability, whichisrelated tothepropertiesoftheporouspharmaceutical powdercompact.Thepressuredifferencecanbedescribedbythe Young–Laplaceequation(P=Pc=cos/Rc,0)consideringonlythe capillarypressure,Pc,andneglectingtheeffectofgravity.
Assumingthatthenetmassflowinandoutoftheporoussystem mustbezero(whereinflowsarenegativeandoutflowsarepositive) inarigidporoussystem,thecontinuityequationforincompressible fluidscanbeexpressedas
∇
q=0. (3)InordertocalculateL(t)fromEq.(2)oneneedstoaccountfor thefactthatonlyafractionofthevolumeisavailablefortheliquid flow.Therefore,dividingqbyε0yieldsanequationfortheliquid frontdependingontimeintheformof
L=
K4cos
ε0Rc,0 t.
(4)
Adetailedderivationofthisequationwaspreviouslyprovidedby Masoodietal.(2007).
2.2. Liquidpenetrationinaswellingporousmedium
Swellingofaporousmediumcausesachangeintheintrinsic permeability,porosityandporeradius.Therefore,the aforemen-tionedmodelsarenolongervalidandonehastoconsidertheaffect ofswellingonliquidpenetration.SchuchardtandBerg(1991) pre-sentedamodelwheretheyassumedalineardecreasewithtimeof theporeradiusinthewettedareaoftheporousmedium(a compos-iteofcelluloseandsuperabsorbentfibres).TheyconsideredRhas
theeffectivehydrodynamicradiusinthewettedregionbehindthe advancingliquidfront,whichchangesovertime.Rc,0isthecapillary radiusinthedrymaterialandthusitisthelengthscalerelatedto theliquidmeniscus.Rhisapproximatedasalinearfunctionoftime
Fig.2. Schematicofswellingandtheingressoffluid/solventinthepowdercompact.Thetabletisheldtightlyarounditscircumferencebyasampleholder.(a)Drypowder compactdefiningtheinitialtabletthicknessı0,thetabletdiameterw,theporosityε0andtheporeradiusRc,0.(b)Hydrationofthepowdercompactbythedissolutionmedium.
Thisdefinestheswellingı,displacementandthepenetrationdepthL=+ı.εandRcaretheporosityandtheporeradiusofthewettedmedium,respectively.
poreradiusconstriction.Theyusedthisassumptionandmodified theWashburnequationtogive
L=
Rc,0cos
2
1/2t− a
Rc,0t
2+ a2 3R2
c,0
t3
1/2. (5)
Ifthereductionrateconstanta=0,thenEq.(5)yieldsEq.(1). MasoodiandPillai(2010)presentedamodelforthewickingand swellingofpaperbasedonDarcy’slawandunderconsiderationof thedynamicchangeofporosity.Sinceswellingisaneffectofliquid absorptionbythepolymer,thecontinuityequationaspresentedin Eq.(3)hastobemodifiedto
∇
q=−S−∂
ε∂
t. (6)Anegativeterm(i.e.sinkterm)ontherightsideoftheequation correspondstoabsorptionofliquidbythematrixandthusit disap-pearsfromtheporespace,whereasapositiveterm(i.e.sourceterm) createsliquidintheporespace.Shereistherateofliquid absorp-tionbythesolidmatrixandispositiveforaswellingmaterial.
∂
ε/∂
tisthennegativeastheporositydecreasesin thewetted region. MasoodiandPillai(2010)proposedalinearrelationbetweenSand thetimederivativeoftheratiobetweenthesolidandthetotal vol-umeofthematrix.Wefurtherassumethattherateofincreaseof thedryparticles’volumeisequaltothevolumetricrateofliquid absorptionbysolidmatrix.Thiscanbeexpressedas
S=
∂
εs∂
t =−∂
ε∂
t, (7)usingtherelationbetweentheporosityεandthesolidmatrixratio
εs,i.e.εs=1−ε.Thecontinuityequationthenhasthesameform
asEq.(3).Themodelforliquidpenetrationinaswellingporous
mediumbecomes
L=
2Pc
ε0
t0
K(t)dt. (8)
Adetailedderivationofthemodel(henceforthreferredtoas modi-fiedDarcy’slawapproach)isgiveninMasoodiandPillai(2010). The intrinsic permeability Kchanges withtime. Thereare sev-eralestablishedempiricalandtheoreticalmodelsintheliterature
forestimatingthepermeability.Inthisstudyweusedamodified
Carman–Kozenyequationwith
K=Ds2180c ε
3
(1−ε)2, (9)
with Ds as the mean diameter of the powder particles. It is well-knownthattheCarman–Kozenyequationoverestimatesthe permeabilityofporousbeds(Roughetal.,2002),whichiscorrected forherebytheconstantc.
2.3. Modellingswellingpharmaceuticaltablets
Thissectionpresentsthenewmodeltodescribeliquidingressin swellingpharmaceuticaltablets.Thetabletismodelledasa cylin-derwithinitialthicknessı0anddiameterw.Duetothegeometry ofourexperimentalsetup,wherethesampleisheldtightlyaround itscircumference(seeFig.2),thetabletcanonlyswellintheaxial direction,causinganincreaseofthetabletthicknessı.Therefore,w
andthusthecross-sectionarea,At,ofthetabletdonotchange dur-ingthehydrationprocess.Asinpreviousmodels,wealsodescribe theporesascapillarieswithinitialradiusRc,0.
Thetabletswells,increasinginheightandcausingthe capil-laryradius,Rc,todecrease(Fig.3).Weproposethatthefractional increaseinvolumeofthewettedpowdercompactisequaltothe
Fig.3. Schematicoftheswellingoftheparticles.Theparticleradius,Rp,directly impactsthecapillaryradiusbyRc=Rc,0−(Rp−Rp,0).Thestripedandthegreyareas
[image:3.646.146.462.54.236.2] [image:3.646.311.563.548.711.2]Vc,0 =R2
c,0 ı0 =
Vt,0R2
c,0
withVcandVtasthecapillaryandthetabletvolumeinthewetted
areaandusingı= Vt
Vt,0ı0.Thiscanthenbeusedtodeterminethe
porosityasafunctionofthecapillaryradiusby
ε= Vc
Vt =
Vc,0VVt
t,0 R2
c R2 c,0
Vt =ε0
R2
c
R2
c,0
. (11)
Themodeldescribestheswellingprocessintermsofthe con-strictionoftheporesinthetabletmatrixastheyfillupwithwater duringhydration.Oncetheporesarefilledwithwaterastressis exertedonthesolid bridgesbetweentheMCCparticles bythe pressurefromthepenetratingfluid.Atthesametimestored elas-ticpotentialenergythatoriginatesfromtheaxialcompactionis releasedwhentheMCCparticles contactwater. Thisrelaxation processisrelativelyrapid(atimescaleofseconds)andcausesthe particlestoswell.Therefore,theswellingprocessinducedbythe liquidingressisinstantaneousandwilllastuntilthetabletisfully hydrated.
Asecond,much slower,phaseof swellingfollowstheinitial hydrationifthetabletisstillintactatthispoint,e.g.in formula-tionswherenodisintegrantispresent.Thissecondswellingprocess appearstobeasymptoticin natureand canbe modelledusing theSchott model(Schott,1992b,a), whichwasmodifiedto(see AppendixB)
ı= t
˛+ 1
ı∞−ı0t
. (12)
˛isamaterialconstantthatcanbedeterminedexperimentallyby linearregressionandı∞representsthemaximumthicknessofthe tablet.˛isrelatedtotheinitialrateofswellingby
lim
t→0
d
dt(ı)=
1
˛. (13)
Wecanreplacebby1/˛inEq.(10).Thisrepresentsthe connec-tionbetweenthemodifiedSchottmodelandtheliquidpenetration describedbythemodifiedDarcy’slawapproach.Inlinewiththe timeusedinEq.(10),wedefinedt=0swhenthetabletwasexposed totheliquid.
Moreover,theswelling ofthetablet providesaconsiderable amountofinformationregardingthestrainthatthetabletis under-goingduringhydration.Theresultingstrainfromtabletswelling issometimesreferredtoasaswellingpressure(Bussemeretal., 2003).Understandingthemechanismsofswellingandhow this impactstabletdisintegrationallowsformulatorstoassessthe phys-icalaspectsoftabletdisintegrationandhowthechoiceofexcipients andtheirphysicalattributes(particlesize,particleporosity,etc.) impactsdisintegration.Byfullyunderstandingthemechanismof
1.Theporestructurecanbeapproximatedbycylindricaltubes. 2.Theabsorptionrateofliquidisequaltothe(volumetric)swelling
rateofthesolidmatrix.
3.Thepermeabilityisisotropicovertheentirewettedareaandit canbedescribedbyamodifiedCarman–Kozenyequation. 4.Theporosityisuniformacrosstheentirewettedvolume. 5.Theeffectofgravityonliquidpenetrationisnegligible.
3. Experimental
3.1. Materialsandcompressionoftabletcompacts
Flat-faced, round tablet compacts of 10mm diameter were compressedusingacompactionsimulator(PhoenixCalibrations, Phoenix,AZ,USA).Thedevicewasinstrumentedtomeasurethe forcesofcompactionthroughoutthecompressioncycle.Powder wasloadedinto thedies usinga hopper and volumetric filling wasperformedtopreparetwotabletsetswithtargetporosities ofε0=0.10 (labelledasF1;3 repeats)and 0.15 (labelledasF2;
6repeats)andatargetthicknessof1.5mm.Thetabletsusedin thisstudywerepartofthosepresentedinYassinetal.(2015a). Forthispreviousworktheauthorsusedonlyasubsetofthe pre-paredtablets.Thecompactionprocedureand theporositiesare thesame.Theporosityoftwo tabletsofeachsetwasmeasured by terahertz time-domain spectroscopy. It was also calculated fromthevolumeofatabletandtheweightedaverageofthetrue densityaswellasbythetabletmassandthevolumemeasured usingheliumpycnometry.Theaverageporositiesfromthethree methods were10.45±0.30% and14.73±0.63%for the10% and 15%porositytablets,respectively.Themeanparticlesizediffered, withDs,F1=100mandDs,F2=50mforF1andF2,respectively. ThesamplesconsistofMCCusingagradewhich alloweddirect compaction (Avicel® PH-102for F1and Avicel® PH-101 for F2, FMCBioPolymers,Philadelphia,PA,USA).TheF2tabletswereonly reportedinthesupportinginformationofYassinetal.(2015a),and itisimportanttohighlightthatthosetabletswereproducedusinga differentgrade(Avicel®PH-101)thanthosepresentedinthemain manuscript.Theexperimentswereconductedwithpurewateras theliquid.Thepropertiesofthetabletsandtheliquidforeachset arelistedinTable1.
3.2. Liquidpenetrationandswellingmeasurementsusing terahertzpulsedimaging(TPI)
[image:4.646.301.554.92.161.2]Fig.4. Swellingbehaviouroftabletsplottedin(a)theswellingprofile(every5thdatapointisshown)andin(b)thelinearrepresentationofthemodifiedSchottmodel(Eq. (12)).TheaverageprofilesofsetF1(blacktriangles)andofF2(greycircles)wereusedtoestimate˛andı∞,whicharerequiredtosimulatethemodifiedSchottmodel(solid anddashedline).Theshadedareasindicatethestandarddeviationcalculatedfromtherespectiverepeats.
removenoiseandtoenhancethefeaturesintheterahertz wave-form(Yassin etal.,2015b;MacPherson,2013;Zeitler andShen, 2013).
Thefeaturesobservedinthedeconvolvedwaveformsaredue tothedifferencesinrefractiveindexbetweenthedifferentphases inthetabletduringhydration:aninitialpeak(t1)inthereflected time-domainwaveformoftheterahertzpulseoriginatesfromthe tablet surfaceand is due to thedifferencebetweenthe refrac-tiveindexofair(n=1)andthetabletmixture(typically1<n<2). Bytrackingt1 withtimetheswellingatthefrontsurfaceofthe tabletcanbequantified.In thedrytablet thechangein refrac-tiveindexfromthetabletmaterialtoairat thebacksurfaceof thetabletresultsinanegativeresponseontheterahertz wave-form(t2)(Yassinetal.,2015a).Oncethetabletcomesintocontact withwaterthet2responseinvertsintoapositivepeakduetothe inherentlyhigherrefractiveindexofwater (n≈2.1at1THzand 20◦C)(Zelsmann,1995)comparedtoairorpolymerandthispeak isreferredtoast3.Bymonitoringthepositionoft3withtimeitis possibletoextractthevelocityofthepenetratingwaterfrontwithin thetablet.Consequently,thesedataallowthemeasurementofthe swelling,ı,andthedisplacement,,ofthewaterfrontrelative totheoriginaltabletthicknessı0asillustratedinFig.2.Thetotal liquidingressintheporousmedium,L,canthenbecalculatedby
L=+ı.
4. Resultsanddiscussion
Inthefollowingsectiontheexperimentalresultswillbe com-paredtodifferentmathematicalmodels.Resultsfortwodifferent tabletsets(F1andF2)arepresented,whichvaryintermsofporosity (ε0,F1=0.10andε0,F2=0.15)andmeanparticlesize(Ds,F1=100m andDs,F2=50m).ByrelatingtheTPImeasurementstomodelsof swellingandliquidingresswegainabetterunderstandingofthe physicaltransformationsthatoccurwithinthetablet.This discus-sionwillbesplitintotwosections:(1)applicationofthemodified Schottmodeltodescribetabletswelling,and(2)applicationofthe Darcyapproachtomodeltheliquidpenetrationofswellingporous media.
4.1. ApplicationofthemodifiedSchottmodel
Asoutlinedabove,theSchottmodeldescribesaswellingprocess thatresultsfromtheunravellingofthepolymermacromolecules after diffusion into solvent upon hydration (Schott, 1992a).To determinetheparameters˛andı∞itismostconvenientto rear-rangeEq.(12)inalinearformt/ı=˛+t/(ı∞−ı0),whereAand 1/(ı∞−ı0)canbeextractedfromtheyinterceptandthe gradi-entofthelinearfit,respectively(Fig.4).Duringextractionofthe
˛andı∞ parameterscareistakentorestrictthefittothedata
wherenocracksforminthetabletasthissignificantlyaffectsthe swellingbehaviour.TheparametersobtainedarelistedinTable2. Acleardifferencebetweenthetwodatasetsisevident.Thetablet setswitha higherporosityandsmallerparticlesizeswellmore towardstheendoftheexperiment,asindicatedbyasmaller gra-dient(seeFig.4).Using thevalues of˛andı∞it ispossibleto comparetheswellingbehaviourofindividualtabletsandthe aver-ageswellingprofile(Fig.5).TheSchottmodelapproximatesthe swellingbehaviourquitewell(adjustedR2=0.999andR2=0.998 forF1andF2,respectively),butitdoesnotfollowtheexperimental swellingresultsofindividualsamplesperfectly.Thisisattributed totheinherentvariationsinthetabletmicrostructureafter com-paction.
TheswellingprofilesthatweremeasuredbyTPIfromthese sam-plescanbeseparatedintotwophasesofswelling,thetransition betweenwhichisindicatedbytheverticaldashedlineinFig.5. Thetransitiontimeofthetwophaseswaschosenasthetimewhen theswellingratedroppedsignificantly,whichisapproximatelythe timewhenthetabletwasfullyhydrated.Thefirstphaseofswelling isdominatedbyfluidingressintothedrytabletmatrix.Oncethe initialliquidpenetrationceasestherateofswellingdecreases.All threeprofilesshowthatthemagnitudeofswellinginthesecond phaseislessthaninthefirstphase(ı=0.3mmvs.ı=0.1mm). Thereasonwhyweobservethefirstphaseofpenetrationinduced swellinginsuchapronouncedwaycanbeattributedtothe hygro-scopicityofMCC.Asaresultliquidpenetrationisfacilitatedand hencetherewillbeasharpchangeincapillarypressureaswater beginstofillthepores.
Thelastdatapointoftheliquidingress,,measuredbyTPI occursbetween1.2and1.4mm,which indicatesthatthewater fronthadalmostreachedthefrontfaceofthetablet.However,the reasonthatwecannotresolvethefinalstageofthehydration pro-cessistwofold:(i)duetotheoptics(i.e.thefocuswassettothe centreofthetablet)andtheresolutionlimitoftheterahertzsystem used;(ii)duetospreadingoftheliquidwithincreasing penetra-tiondepthcausinganindistinctliquidfront.Therefore,thepeaks intheterahertzwaveformoriginatingfromthewaterfront,t3,and fromthefrontfaceofthetablet,t1,cannotbedifferentiatedatthis point.
[image:5.646.139.472.55.182.2]TheliquidingresswasdescribedbytheWashburnequation(Eq. (1)),theDarcylawapproach(Eq.(4))andthemodifiedDarcylaw approach(Eq.(8)).TheWashburnequationrequiresanestimateof theeffectiveporeradiusReandfortheothertwomodelswehave toknowtheinitialporeradius,Rc,0,and thescalingfactorcfor thepermeability.ThevaluesofRe,Rc,0andcwerefittedusingthe averageliquidingressprofileoftheF1tablets.Fig.6comparesthe
Fig.5.ComparisonoftabletswellinginindividualtabletsanalysedusingTPI(black circles)andcalculatedswellingusingtheSchottmodel(solidlines)where(a),(b) and(c)areexperimentalrepeatsoftabletsofF1.Thesolidpointsareexperimental dataandthesolidlinesaretheresultsusingthemodelgeneratedfromtheaverage swellingprofileofallthreesamples.Thebluetrianglesrepresenttheliquidingress, ,measuredusingtheTPIbytrackingthet3peakofthetime-domainwaveform.The dashedverticalredlinemarksthetransitionfrompenetrationcontrolledswelling (stageI)topolymerabsorptioncontrolledswelling(stageII),whenthedrymatrix isfullyhydrated.(Forinterpretationofthereferencestocolourinthisfigurelegend, thereaderisreferredtothewebversionofthearticle.)
Darcy’slawapproachesyieldthesamepenetrationkineticsdueto theirsimilarityandthefittingoftheparameters.
InordertousethemodelforthepredictionofF2samples,we havetoadapttheporeradius.WeproposetoadaptRc,0basedon theporosityratio(area-basedporosity)betweenthetwodifferent sets
ε0,F1
ε0,F2 =
NF1R2c,0,F1 /At
NF2R2c,0,F2 /At =
NF1R2c,0,F1
NF2R2c,0,F2
, (14)
[image:6.646.313.540.305.636.2]Fig.7. LiquidingressofF1(blacktriangles)andF2(greysolidcircles)samplestogetherwiththerespectivesimulations(solidanddashedline)usingthemodifiedDarcy’s lawapproach(Eq.(8)).(a)showsthesimulationfortheentireexperimentaltimeand(b)providesfurtherdetailoftheinitialhydrationphase.
NF1 andNF2 are thenumberof capillaries insample F1and F2 respectively.Wecanderiveanequationfortheporeradiusoftablet setF2asafunctionoftheporosities,thenumberofpores,andthe poreradiusoftabletsetF1:
Rc,0,F2=
NF2
NF1
ε0,F2
ε0,F1
1/2Rc,0,F1 (15)
Weestimatedtheratioofthenumberofcapillariesfromtheratio ofthenumberofparticlesinF1andF2.Thenumberofparticles,Np,
canbeapproximatedbyNp=At/(R2p )withRp=Ds/2.Theratioof
thenumberofcapillariesinF1andinF2canthusbeexpressedas
NF2/NF1=Np,F2/Np,F1=R2p,F1/R2p,F2=4.Eq.(15)canbeappliedto asamplewithadifferentporositythanthatusedforthefitting. Con-sequently,theparametersrequiredfortheDarcy’slawapproach wereestimatedfromthedatafromtheF1tablets.Theporeradius wasadaptedfortheF2tabletsbasedonEq.(15),whereascwas keptconstantforallsamples.Theparametersaresummarisedin Table2.InterestinglytheonsetofliquidingressisfasterfortheF1 powdercompacts(Fig.7).Thiscanbeattributedtothelarger par-ticlesizeofthetabletsetF1,whichcausealargerswellinginthe initialphase.However,thetotalswellingislargerforthesmaller particles(ı∞is1.96mmand2.23mmforF1andF2tabletsandsee Fig.4)andthustheliquidcanpenetrateconsiderablyfurtherinto thepowdercompact.
Itisinterestingtonotethatthereisaconsistentdiscrepancy betweentheswellingmodelandtheexperimentaldatainthe ini-tial2s.Themodelpredictsafasteronsetofswellingcomparedto thatweobservedusingTPI.Webelievethatthisisduetothe exper-imentalsetup:atthebeginningoftheexperimenttheflowcellis filledwithliquidfrombottomtotopandthusthetabletisalso wet-tedfrombottomtotop.Therefore,ittakesafewsecondsforthe entiretabletfacetobewetted,whichslowsdowntheinitialliquid ingress.Improvementoftheexperimentalsetupisthesubjectof furtherresearch.
ThereisonlyaslightdifferencebetweentheWashburn equa-tion/Darcy’slaw and themodified Darcy’slaw approach atthe beginningoftheexperiment,whereastheydiffernoticeablyand the experimental data towards the end. This is not surprising astheWashburnequationandtheDarcy’slawapproachdonot consideranychangeinporositywhichleadstoa constant pres-sureand constantpermeability.Theperformance ofthemodels canbequantifiedfortheF1tabletsbytheadjustedR2,whichis 0.994and0.982forthemod.Darcy’slawapproachandthe Wash-burnequation/Darcy’slaw,respectively.Theliquidpenetrationfor shorttime-scalesdependslinearlyonthetimeandthusit devi-atesfromthe√tbehaviourasassumedintheWashburnequation and in both Darcy’s lawapproaches. Such kineticswere previ-ouslydiscussedby Ridgway andGane (2002),where theliquid absorptionchangedfromaninertial Bosanquetregime(L∝t)to
theviscousLaplace–Poiseuilleflowregime(L∝√t).Therefore,the performanceofthemodelsforshort-timescalesmaybeimproved byconsideringtheBosanquetregime.Thereisalsotheeffectofthe initialaccelerationassociatedwiththeformationofthemeniscus, whichtheWashburnequationdoesnotconsider.
Moreover,theporosity,poreradiusandpermeabilityundergoa significantchangeashighlightedinFig.8.Itisclearthatthe aver-ageporeradiusdecreasesastheindividualparticleswells,which causesadecreaseoftheporosity.Asthepermeabilityisalsoa func-tionoftheporeradius,itchangesovertimeaswell.Thedynamic changesarelesspronouncedforthetabletsetwiththelarger poros-ity(ε0=0.15),whichisprimarilyduetothesmallerparticlesizeof theMCCparticles.Thenumberofparticlescompactedislargerfor smallerparticlesinordertoreachthesametabletsolidfractionas
Fig.8. SimulationresultsusingthemodifiedDarcy’slawapproachofporeradius ratio,porosityratioandpermeabilityforthetwodifferenttabletsets,F1(ε0=0.10,
solidline)andF2(ε0=0.15,dashedline).Rc,0(ε0)istheinitialporeradius(porosity)
[image:7.646.135.474.54.184.2]ityissensitivetothemicrostructureofthetablets.Oneadditional factorwhichisexpectedtoplayasignificantroleistheformation of(micro)crackswithinthetabletsduringdisintegrationandthe subsequentcrackpropagationduringswelling.Thisresultsinthe buildupofconsiderablestrainintheparticlematrix.Thestrainis thedrivingforceforcrackformationinthetabletstructureandonce acrackhasformedtheporositywithinthetabletmatrixincreases. Asaresultliquidpenetrationisfacilitated,thetabletswells fur-therandthecrackpropagatesthroughthematrixdrivenbythe additionalswellingofthematrix.Therewerenonoticeable dis-continuitiesintheexperimentaldatawhich wouldindicatethe presenceofmacroscopiccracks.
5. Conclusions
Wehaveshownthatitispossibletodescribetheliquidingress andtheswellinginpowdercompactsusingmathematicalmodels. Byapplyingthesemodelsitispossibletogain insightsintothe mechanismofliquidingressandswellingintabletsupon hydra-tion.Themodelsfurtherallowtheanalysisofthereductionofpore radius,dynamicchangesinporosityaswellasinpermeability.
Thedrawbackofthemathematicalmodeldescribedinthisstudy isthatitcombinesatheoreticalmodelwithanempiricallyderived model(themodifiedSchottmodel)andthereforereliesonhaving tocalibrateforAandı∞priortousingthemodelforpredictions. Itwouldbehighlydesirabletodevelopmodelsbasedonthe phys-icalpropertiesofthepowderanddrytabletsalone.Suchamodel wouldreducetheamountofworkneededtounderstandtheimpact ofsubtlechangesinformulationontheproductperformance.The presentedmodeldoesnot takeintoaccount fracturecontrolled swelling,whichmayplayanimportantroleduringdisintegration ofotherformulations.Moreresearchisnecessarytodescribethe formationandpropagationofcracksanditsimpactonthe disinte-gration.
Moreover,themodelspresentedinthisstudyrequirean aver-ageporeradius,whichwasestimatedfromexperimentaldata.In futurekeycharacteristicsofthemicrostructure,suchasthepore radius,ofporouspowdercompactscanbemeasuredusingX-ray computedmicrotomography(Al-RaoushandWillson,2005)ora nitrogenadsorptionmethod(Westermarcketal.,1999).Itcould becalculated bya theoreticalmodel based ontheparticlesize distribution(Nordströmetal.,2013).Thiswill,however,require thedevelopmentofarelationbetweenameasuredporeradiusor poresizedistributionandtheeffectiveporesizedrivingtheliquid ingress.Everymethodologyusedtomeasuretheporesizeaswellas theporosityhastobecriticallyassessedastheresultsfromdifferent methodsmightdifferconsiderablydependingontheunderlying physicalmeasurementprinciple.Furthermore,thepresented mod-elsdescribethemicrostructureonlybytheporosityandanaverage
S.Y.wouldliketothanktheEPSRCforastudentship.Additionaldata relatedtothispublicationareavailableattheCambridgeUniversity repository(https://doi.org/10.17863/CAM.9101).
AppendixA. Poreradiusreduction
Thevolumeratioforawettedtabletisgivenby
Vt
Vt,0 =
ı(t)At
ı0At =
ı0+b·t
ı0
(A.1)
andforasinglewettedparticle
Vp
Vp,0 =
3 4 R3p
3 4 R3p
= R 3
p
R3
p,0
, (A.2)
whereAtisthesurfaceareaofthetablet,Rpistheparticleradiusat timet,Rp,0=Ds/2istheinitialparticleradius,andbistheswelling rate.Thetabletcanonlyswellinitsaxialdirectionasitistightlyheld arounditscircumference.Theswellingofanindividualparticlecan swellinall3directionscausingthereductionofthecapillarytube radius.Althoughtheparticlesatthetabletbandwillberestricted bythesampleholdertoswellintheradialdirection,thiswillhavea negligibleimpactontheswellingbehaviouroftheparticlesatthe tabletcentrewhereweperformedtheTPImeasurements.Using Eqs.(A.1)and(A.2)wecanderiveanexpressionforthechangeof thewettedparticleradiusasafunctionoftime:
R3
p
(Ds/2)3
= ı0+b·t
ı0
Rp=
ı0+b·t
ı0
1/3Ds
2
(A.3)
Anincreaseoftheparticleradiuscausesadecreaseofthecapillary radius:
Rc =Rc,0− Rp−D2s
=Rc,0−Ds 2
ı0+b·t
ı0
1/3 −1.
(A.4)
cross-sectionareaoftheparticlesplustheareaofthecapillariesis constant.Thiscanbeexpressedby
NpR2p+NR2c=NpR2p,0+NRc,20
NR2
c =NpR2p,0+NR2c,0−NpR2p
NR2
c =Np
R2
p,0−R2p
+NR2
c,0
R2
c−R2c,0=
Np
N
R2p,0−R2p
,
(A.5)
whereNpisthenumberofparticlesperunitareaandNisthe
num-berofcapillarytubes.Thisleadstothefollowingexpressionforthe capillaryradiusasafunctionoftheparticlesize
Rc =
R2
c,0+
Np
N
R2p,0−R2p
= R2c,0−
Np
N
Ds
2
2ı0+b·t
ı0
2/3 −1.
(A.6)
SinceEq.(A.6)introducesanotherfittingparameter(Np/N),weused Eq.(A.4)inthisstudy.
AppendixB. ModifiedSchottmodel
TheSchott modelrepresentedas asecond-order equationis definedas
dW
dt =k(W∞−W)
2
(B.1)
withWtheliquiduptakeingramofsolutionpergramofmedium attimet,W∞themaximumorequilibriumuptakeandkaconstant. SolvingEq.(B.1)yieldsthewateruptakedependencyontime:
W= t
˛+ t
W∞
(B.2)
˛=1/(kW∞)isaconstantandcorrespondstothereciprocalofthe initialswellingrate.Inordertoapplythismodeltotheone dimen-sionalswellingcase inthisstudy,weneedtosubstitute Wand
W∞:
W=Atıl
Atı0s =
ı
ı0
l
s (B.3)
W∞= At(ı∞−ı0)l
Atı0s =
ı∞−ı0
ı0
l
s (B.4)
ıistheextentofswellingandisgivenby
ı=ı−ı0. (B.5)
Atisthesurfaceareaofthetabletface.landsarethedensities
oftheliquidand thesolid matrix,respectively.ı∞ is the maxi-mumorequilibriumthicknessofthetablet,whichisrelatedtothe maximumliquiduptakeW∞.CombiningEqs.(B.1),(B.3)and(B.4), yields 1 ı0 l s dı
dt =k
ı∞−ı0
ı0 l s− ı ı0 l s
2ı= k
(ı∞−ı0)2l
st
k(ı∞−ı0)l
st+ı0
(B.6)
Eq.(B.6)canbewrittenintheformoftheSchottmodel(Eq.(B.2)) bydefining
k= ı0
(ı∞−ı0)2˛
s
l.
(B.7)
ThemodifiedSchottmodelisthen
ı=
ı0 ˛t
ı0+˛(ı∞ı0−ı
0)t
= t
˛+ 1
ı∞−ı0t
(B.8)
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