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Mathematical modelling of liquid transport in swelling pharmaceutical immediate release tablets

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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

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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.

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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

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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

1/2

t− 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/

t

isthennegativeastheporositydecreasesin 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

t

0

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]
(4)

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=100␮mandDs,F2=50␮mforF1andF2,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]
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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=100␮m andDs,F2=50␮m).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]
(6)

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]
(7)

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

1/2

Rc,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]
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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

1/3

Ds

2

(A.3)

Anincreaseoftheparticleradiuscausesadecreaseofthecapillary radius:

Rc =Rc,0− Rp−D2s

=Rc,0−Ds 2

ı0+b·t

ı0

1/3 −1

.

(A.4)

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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

=

R2

c,0−

Np

N

Ds

2

2

ı0+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,Wthemaximumorequilibriumuptakeandkaconstant. 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

2

ı= 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+˛(ıı0ı

0)t

= t

˛+ 1

ı∞−ı0t

(B.8)

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Figure

Fig. ε and theswellingdisplacementthepenetration Rc are the porosity and the pore radius of the wetted medium,                     the    Thisporosity depth,definesε and�ı,the�andpore radiusLR.(b) Hydration= of�thepowder + compact�ıbythedissolution.medium
Table 1
Fig. and4. Swelling behaviour of tablets plotted in (a) the swelling profile (every 5th data point is shown) and in (b) the linear representation of the modified Schott model (Eq.(12))
Fig.  t1 peak in measured the measureddisplacement, terahertz Theingwaveformentire(Eq.simulationtheswellingand�ıthesolid  circlesdepictusing theTPI liquidingress,(tracking ofL=�+�ı  ,withtheswelling and �,measured  using thetheTPI bytracking the  t  peak
+2

References

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