2013有限剂量皮肤质量平衡包括横向部分：比较试验,药代动力学模型和扩散模型

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JournalofControlledRelease

journalhomepage:www.elsevier.com/locate/jconrel

Finitedoseskinmassbalanceincludingthelateralpart:Comparisonbetweenexperiment,pharmacokineticmodelinganddiffusionmodels

D.Selzera,1,T.Hahna,1,A.Naegelb,1,M.Heisigb,K.H.Kostkac,C.M.Lehra,d,D.Neumanne,U.F.Schaefera,G.Wittumb,⁎

aBiopharmaceuticsandPharmaceuticalTechnology,SaarlandUniversity,CampusA41,66123Saarbruecken,GermanyGoethe-CenterforScientificComputing,Goethe-University,Kettenhofweg139,60325FrankfurtamMain,GermanycDepartmentofPlasticandHandSurgery,Caritaskrankenhaus,Heeresstr.49,66822Lebach,GermanydDepartmentofDrugDelivery(DDEL),Helmholtz-InstituteforPharmaceuticalResearchSaarland(HIPS)HelmholtzCenterforInfectionResearch(HZI),SaarlandUniversity,CampusA41,66123Saarbruecken,GermanyeScientificConsilience,StarterzentrumA12,SaarlandUniversity,66123Saarbruecken,Germany

barticleinfoabstract

Thisworkinvestigatesinvitrofinitedoseskinabsorptionofthemodelcompoundsflufenamicacidandcaffeineexperimentallyandmathematically.Themassbalanceindifferentskincompartments(donor,stratumcorneum(SC),deeperskinlayers(DSL),lateralskinpartsandacceptor)isanalyzedasafunctionoftime.Forbothsub-stanceshighamountswerefoundinthelateralskincompartmentafter6hofincubation,whichemphasizesnottoelidethesepartsinthemodeling.Here,threedifferentmathematicalmodelswereinvestigatedandtestedwiththeexperimentaldata:apharmacokineticmodel(PK),adetailedmicroscopictwo-dimensionaldiffusionmodel(MICRO)andamacroscopichomogenizeddiffusionmodel(MACRO).WhilethePKmodelwasfittedtotheexperimentaldata,theMICROandtheMACROmodelsemployedinputparametersderivedfrominfinitedosestudiestopredicttheunderlyingdiffusionprocess.Allmodelscouldsatisfyinglypredictordescribetheex-perimentaldata.ThePKmodelandMACROmodelalsofeaturethelateralparts.

©2012ElsevierB.V.Allrightsreserved.

Articlehistory:

Received19July2012Accepted12October2012

Availableonline22October2012Keywords:

SkinpenetrationSkinabsorption

HomogenizeddiffusionmodelMathematicalmodelingPredictionmassprofiles

1.Introduction

Asdeliveryofdrugstotheskinisgainingmorerelevancenowa-days,thepharmaceuticalindustryisinterestedinevaluatingtheab-sorptionofactiveentitiestotheskin.Moreover,thecosmeticindustryisalsointerestedinariskassessment,sincethesystemicavailabilityoftopicallyappliedcosmeticsubstancesshouldnormallybeavoided.

Theexperimentalevaluationisusuallyperformedininfinitedoseskinabsorptionexperiments.Intheseexperiments,thedoseontheapplicationsiteandtheamountofsubstanceinthedonorarelargeandeffectivelydonotvarywithtime.Unfortunately,oneobviousdrawbackoftheseexperimentsisthattheyhaveoftenlittleincommonwithrealisticinvivoexposurescenarios,whenonlyasmallamountofatopicalformulationisevenlydistributedontheskinsurface.Corre-spondingly,theOECDguideline428[1]andtheguidancedocument28[2],definefinitedoseexperimentsfortheskinbyanapplieddose

⁎Correspondingauthorat:Goethe-CenterforScientificComputing,Goethe-University,Kettenhofweg139,D-60325FrankfurtamMain,Germany.Tel.:+496979825259;fax:+496979825268.

E-mailaddress:wittum@gcsc.uni-frankfurt.de(G.Wittum).1Theseauthorscontributedequally.0168-3659/$–seefrontmatter©2012ElsevierB.V.Allrightsreserved.http://dx.doi.org/10.1016/j.jconrel.2012.10.009

of10μl/cm2orless.Duetothelimitedamountofsubstance,thetran-sientcharactermayleadtoasubstantialdeviationfromthesteady-statesituationthatischaracteristicforaninfinitedosesetting.Inthiscontext,inparticulartheroleofthelateralcompartmentinexperimen-talstudieshasrecentlygainedsomeattention[3–5].

Asskinabsorptionexperimentsaregenerallytimeconsuming,thedevelopmentofsuitablemathematicaltoolsanddescriptionsisequallyimportant(i)forinvestigatingtheabsorptionbehaviorand(ii)forpredictionsofthesubstanceabsorption,whichcancontributetoreducethenumberofexperimentssignificantly.Forthefinitedosesituation,differentmodelshavebeendeveloped.Mostofthesemodelssolelyfocusonthepermeationprofilesofadrugthroughaskinmembrane[6,7]anddonotconsiderthedrugamountineachskinlayer.Tothebestofourknowledgetheeffectoflateraldiffusionhasalsonotbeeninvestigatedyet.

Theaimofthepresentstudyistoinvestigatethetransientprocessofinvitroskinabsorptionunderfinitedoseconditionsbothexperi-mentallyandfromamodelingperspective.Theworkinparticularshedssomelightontheinfluenceofalateralcompartmentonmassbalanceprofiles.Asafirststep,massbalanceprofilesforthedifferentskinlayerswereestablishedexperimentally.Afterincubationwithoneofthetwomodeldrugsflufenamicacid(FFA)orcaffeine(CAF),theskincompartmentswereseparatedcarefullybyahighlystan-dardizedmethodfortape-strippingthestratumcorneum(SC)[8,9]

120D.Selzeretal./JournalofControlledRelease165(2013)119–128

andsubsequentseparationoftheDSLcompartmentfromthelateralcompartment(seeSupplementaryInformation).Informationabouttheabsorptionbehaviorofthesubstancetotheskincanbegainedbyquantifyingthedrugamountineachcompartment.Thisallowsdetecting,forexample,theformationofadrugdepotintheupperskinlayers[10,11],whichmayplayanimportantrole,forcontrolledre-leasesystemssuchasnano-sizedsystemsappliedtotheskin[12,13].

Onthemodelingside,weappliedthreedifferentmathematicalmodels:thepharmacokineticmodel(PK)wasdevelopeddescribingthemassprofilesofthedrugineachskincompartmentovertime.ThenoveltyofthismodelwastheseparationoftheskinintonotonlytheSCandthedeeperskinlayers(DSL),butalsointoanaddi-tionallateralskinpart.Thiscompartmentisusuallyneglectedduringmathematicaldescriptionsoftheexperimentaldata.Tosimulatethemassprofilesinthedifferentcompartments,amicroscopicdetaileddiffusionmodel(MICRO)previouslydevelopedforinfinitedose[14]wasadaptedtothefinitedosescenario.Theinputparametersforthismodelwerepreviouslyderivedfromtheinfinitedoseexperi-ments[15],andthesameparameterswereusedforthefinitedosecase.Finally,ahomogenizeddiffusionmodel(MACRO)wasusedtopredictthemassprofilesonalargerscalethanthedetaileddiffusionmodel,includingthelateralcompartmentintothemodel.

2.Materialsandmethods2.1.Skin

Excisedhumanfull-thicknessskinwasobtainedfromthreefemaleCaucasiandonorsundergoingabdominalplasticsurgery.ThestudywasapprovedbytheEthicalCommitteeoftheAerztekammerdesSaarlandesno.204/08.Theskinwastreatedaccordingto[8]andstoredinthefreezerat−26°Cforamaximumofsixmonths.Carewastakentostrictlyavoidrepeatedfreezingandthawingoftheskin.Ithasbeenshownthatthistreatmentdoesnotimpairthebarri-erfunctionoftheskin[16–19].

2.2.Skinabsorptionexperiments

A25mmpunchofthefrozenskinwastakenonthedayoftheex-perimentandallowedtothaw.Aftercleaningtheskinwithdistilledwateranddryingitwithcottonballs,theskinwasmountedinaFranzdiffusioncell(FD-C)withtheepidermissideup.Theskinwasequilibratedwiththereceptorsolution(SoerensenphosphatebufferpH7.4)foronehour.Previousstudieshaveshownthatafterthistimetheskinhydrationisinequilibriumstate.Afterwards,25μlofthedonorsolutionwasappliedtotheskin.ThedonorformulationwasspreadevenlyovertheincubationareabymeansofaTeflonpunchwithadiameterof14mm[20].Afterremovalofthepunch,aTeflondiskwithadiameterof14mmwasplacedontothedonortopreventevaporationduringincubation.Thepunchwaslaterrinsedoffandtheactualamountofthedonorformulationremainingontheskinwasdetermined.Inallexperiments,afinitedosewasachievedaccordingto[1,2].Afterapplicationofthedonor,thedonorcompartmentwassealedwithparafilmandaluminumfoil.Thewholesetupwasincubatedat32±1°C.ThedonorconsistedoftherespectivedrugdissolvedinSoerensenphosphatebufferpH7.4inconcentrationsof1mg/mlforflufenamicacid(FFA),and1mg/mland12.5mg/mlforcaffeine(CAF).Informationabouttheisolationofthedifferentskincompartments(stretchingoftheskin,tape-strippingoftheSCandcryo-sectioningoftheDSL)andtheestimationofthestretchingamountfordatacorrectionisprovidedintheSupplemen-taryInformation.Forallexperimentssinkconditionsintheacceptorweremaintained,i.e.,theconcentrationneverexceeded10%ofthesaturationconcentrationofthedrug.

2.3.Massbalance

Afterincubation,allcompartmentsofthesetupwerethoroughlyexaminedfordrugcontent,namelythedonorsolutionremainingontheskinsurface(donor),thetape-strips(SC),thecryo-cuts(DSL),andthelateralskinparts.FFAwasextractedusing0.05Nsodiumhy-droxidesolutionandCAFwithphosphatebufferpH2.6for2h.Thesamplesfromtheacceptorweremeasuredwithoutfurtherdilutionsteps.FFAandCAFwerequantifiedbymeansofHPLCaccordingto[15]withanadjustedwavelengthof262nmforCAF.Theobtainedtotalrecoveryrangedfrom83.8%to96.8%forFFA(LLOQof50ng/ml)andfrom87.9%to107.3%forCAF(LLOQof30ng/ml)forallexperiments.

2.4.Pharmacokineticmodeling

Pharmacokinetic(PK)modelingwasappliedtoanalyzethemassprofilesinthedifferentcompartmentsforFFAandCAFfinitedoseex-periments.Themodeltreatsthedifferentskinlayersaswell-stirredcompartments.Thisyieldsinasetofordinarydifferentialequations(ODEs)describingthemassineachofthecompartments.Forthemassfluxbetweenthecompartmentsfirst-orderkineticswereas-sumed.Asanextensionoftheclassicalmodels,alateralcompartmentwasaddedexplicitly(Fig.1).Basedontheexperimentalconditionswhereonlyacomposedlateralpartwasaddressedthetransportcon-stantsk5andk6(seeTableS1)inthismodelwereusedtorepresentanaveragedtransportbetweentheSCandDSLandthelateralcom-partment.Thisreducedsetofunknownsalsoyieldedmorestablefittingresults.

SincethePKmodelisafittingmodelonlyandexperimentalwise,onlytheaccumulatedmassinsidethewholelateralpartwasavail-able,ahighernumberofdegreesoffreedom(morerateconstantsandmasssplitforSCandDSLinsidethelateralpart)yieldedinstablemodels.

ThesetofODEswasnumericallyintegratedandfitted(nonlinearleastsquaresregression)toexperimentaldatausingPython2.7.2usingtheSciPypackage[21].Therootmeansquaredeviation(RMSD)wasusedasanindicatorforaccuracy.Agridsearchwasappliedtofindappropriatestartingvaluesinordertoavoidstickinginalocalminimum.2.5.Detailed(microscopic)diffusionmodel

Analternativetransportmodelisadiffusionprocessinanidealized,two-dimensionalmodelmembrane.Thishaspreviouslybeendescribedforaninfinitedosesetting[14,15],andisnowextendedtoafinitedosescenario.Thecorepartofthemodelisabrick-and-mortarlikestructure,whichrepresentstheSC.Onboththetopandthebottomsideofthisidealizedmembranetwoadditionalcompartmentswereadded,(cf.Fig.2).Thesecompartmentsmodelthedonorchamber(DON)ofthediffusioncellandthedeeperskinlayers(DSL),respectively.TheheightsforDONandDSLcompartmentsarehDON=0.1mmandhDSL=3mm.ThesubstancemassintheacceptorcompartmentisdefinedasthemassthatisleavingthesystemattheboundaryΓSINKathDSL=3mm.

Fig.1.Pharmacokineticmodeldevelopedtoanalyzedrugdistributionbetweenvariousrelevantcompartmentsintheskin.Here,k1tok7arefirst-orderrateconstants.

D.Selzeretal./JournalofControlledRelease165(2013)119–128121

InterfacehDON= 0.1 mm Donor ΩΓDON/LIP

DON wCOR= 30 μm Lipid layers hΩLIP

COR= 1 μm CorneocytesΩCOR

δLIP = 0.1 μm InterfaceΓCOR/LIP hDeeper skin layers DSL= 3 mm Ω DSLInterface

ΓDSL/LIP

ΓSINKFig.2.Microscopicdiffusionmodel(modifiedfrom[14]):dimensions(left)anddo-mainsandinterfaces(right).ThebottomboundaryΓSINKprovidesaperfectsink.Diffu-sionandpartitioncoefficientsarelistedinTable1.

InitiallynosubstanceispresentinSCandDSL,andadefinedconcentra-tion,whichmatchestheexperimentalsetup,isprovidedinthedonor.

Themodelequationisgivenby[22]∂Þþ∂u󰀂þ∂hi

tðkux½−kD∂xy−kD∂yu¼0

wherethesolutionu=u(x,y,t)isdefinedbyasetofdiffusionandparti-tioncoefficients,whichareassumedtobepiecewiseconstantforanyphaseA∈{DON,COR,LIP,DSL}.Definingtheconcentrationc:=ku,thefunctionskareuniquelydefineduptoaconstantfactorbytheinterfaceconditionKA/BcB=cA.Inthiscasethefunctionuisassumedtobecontin-uousontheinterfacebetweentwophasesA,B∈{DON,SC,DSL},A≠B.Bydefinition,fluxesacrosstheinterfacearepreserved.TheboundaryΓSINKprovidesaperfectsink,naturalno-fluxconditionsareimposedonallremainingexteriorboundaries.

Similarmodelshavebeendiscussedintheliterature,e.g.[23,24].However,thefinitedosemodelemployedinthisworkistothebestofourknowledgethefirstmodelconsideringthemicroscopicstruc-tureofthestratumcorneum.Alateralcompartmentisnotincludedinthemodel,sincemodelingadiffusioncellwithadiameterof15mminafullresolutionmodelwithacorneocytewidthof30μmleadstounreasonablyexpensivecomputations.

Themodelreliesonavarietyofdiffusionandpartitioncoefficients,whichareprovidedinTable1.Exceptforthediffusioncoefficientinthedonor,DDON,allparametersweretakenfromthepreviouslypub-lishedinfinitedoseexperiments[14,15].

ArelationshiptopredictthediffusioncoefficientDDONinanaqueoussolutionfromthesolutemolecularweightwasproposedbyAndersonetal.[25]withD0

−n

DON¼DDON•MW

ð1Þ

Table1

Inputparametersforsimulationofskinabsorptionwith2Ddiffusionmodel[14,15].ParameterFFACAFDDON[cm2/h]2.47×10−22.92×10−2DLIP[cm2/h]1.10×10−42.10×10−4DCOR[cm2/h]5.10×10−71.40×10−7DDSL[cm2/h]

4.90×10−32.30×10−3KLIP/DON20.322.15KCOR/LIP0.212.22KDSL/LIP0.1

0.08

whereDDONisthediffusioncoefficientinacertainmedium,nandD0DONareconstantsassociatedwiththemediumatadefinedtemper-ature,andMWthemolecularweightinDa.Using37substancesandtheircorrespondingmolecularweight(gatheredfromthePubChemdatabase[26])anddiffusioncoefficientinwaterat27°C[27]thepreviousequationwasfittedtothelog-transformeddatausingalin-earmodel.ThisresultedinDCAFDon=2.92×10−2cm2/hwitha95%CIof

[2.77,3.08]×10−2cm2/handDFFADon=2.47×10−2cm2/hwitha95%

CIof[2.31,2.65]×10−2cm2/hrespectively.2.6.Homogenized(macroscopic)diffusionmodel

Asanalternative,weconsiderahomogenized(macroscopic)diffusionmodel(MACRO).Thismodelconsidersanaxisymmetricsetupincylindri-calcoordinates.Asketchoftheunderlyingmodelgeometry,whichin-cludesalateralcompartment,isdepictedinFig.3.

Onthisgeometryweseekacontinuoussolutionu=u(r,z,t)ofthemodelequation

∂ÂtðrkuÞþ∂r−rD󰀁rr∂ruÃþ∂Âz−rD󰀁Ã

zz∂z

u¼0ð2Þ

inthenewspatialvariablesr,zforthelateral(r)andtransversal(z)directionrespectively.Thisapproachissimilarto[28,29],butinaddi-tion,theradiusrisusedasascalingfactorwhichreflectstheincreaseinvolumeintheouterpartsofthediffusioncell.Thediffusioncoeffi-cientsDrrandDzzarethediagonalentriesofananisotropicdiffusiontensor,whichallowtheinclusionofthestructureofthecorneocytesimplicitly.

Diffusionintheisundirected,i.e.,D

󰀁donor(DON)andinthedeeperskinlayers(DSL)

DON;rr¼D󰀁DON;zz¼kDONDDONandD󰀁DSL;rr¼D󰀁DSL;zz¼kDSLDDSL.DiffusionintheSCismodeledbycoefficientsD󰀁SC;ii∶¼DLIPaiiðξÞ

ð3aÞ

withthenon-dimensionalfactor󰀄

∞󰀅󰀄󰀅

αðξÞ¼α0

α0ii−αii1−α0

iiξiiiiþÀα∞ÁÀ0Áð3bÞ

ii−1þ1−αiiξ

representingtheanisotropyasafunctionoftheeffectivecorneocyte

diffusivityξ¼

DCOR

DK

COR=LIP

ð4Þ

LIPandoftwogeometrydependentconstantsa0iiandα∞ii.Representative

valuesforlateral(a0rr=1.76×10−1,a∞rr=3.51×103)andtransver-sal(a0rr=7.35×10−4,a∞rr=6.51×103)anisotropieswereselectedaccordingtothetheoreticalfoundationsin[30].

Fig.3.Homogenizeddiffusionmodel:themodelgeometryisacross-sectionthroughhalfadiffusioncell(notdrawntoscale).

122D.Selzeretal./JournalofControlledRelease165(2013)119–128

FortheMACROmodel,twodifferentstrategieswereapplied:intheMACRO(1)model,allparametersexceptforthediffusioncoefficientsintheSC,DSC,rrandDSC,zz,aretakenfrom[14].Thecoefficientsaredeter-minedfrom(Eq.(3a))byaleastsquaresregressionanalysisforthediffu-sivityξ∶=exp(θ1).ThisbasicdataisprovidedinTable2a.

Inordertoimproveonthisfurther,amodelwiththreeregressionparameters,referredtoasMACRO(3),wasapplied.Inadditiontothepreviouslymentionedparameterξ,DSLdiffusivityandpartitioncoef-ficientswereparameterizedby:DDSL:¼DDSLexpðθ2ÞKSC=DON:¼KSC=DONexpðθ3ÞKSC=DSL:¼KSC=DSL=KSC=DONKSC=DON

Inthismodel,allparametersexceptforDDONdependonthere-gressionparametersθ1,θ2andθ3.

2.7.Modeloverviewandparameterestimates

AcomparisonofthethreedifferentmathematicalmodelsisprovidedinTable3a.ForthePKandbothMACROmodelssomeparameterswereoptimizedusinganon-linearregressionanalysistoobtainabetterfittotheexperimentaldata.Intheseparameterestimations,whichwereperformedbyaGauss–Newtonsolver,thestandarddeviation(sumofsquaredresiduals/numberofdegreesoffreedom)wasusedastheobjective.3.Results

3.1.Experimentaldata

TheFFAfinitedoseexperimentsresultedinarelativelyfastdecreaseofdruginthedonor.Rightafter15minthedrugamountinthedonorwasreducedtoabout60%oftheapplieddose(Fig.4A).Afterthisquickdonorreduction,thedonorcontinuedtodepletemoreslowly,reachingabout20%remaininginthedonorcompartmentafter6hofincubation.IntheSCabout30%oftheap-plieddosewasfoundafter15minandaplateauwasreachedandmaintaineduntil2hafterapplication(Fig.4B).Afterwards,theFFAamountdecreasedto10–20%withhighvariabilityatthe4hand6hdatapoints.ThedrugamountintheDSLincreasedconstantly,reachingaplateauatapproximately50%after4hofincubation(Fig.4C).NoFFAcouldbequantifiedintheacceptorcompartmentuntil6hofincubationduetoanalyticalreasons(Fig.4D).Thedrugamountdeterminedinthelateralskinpartappearedtoincreaseafterlongerincubation(Fig.4E).However,thedataisaffectedby

Ã

Ã

ÃÃ

highvariability,especiallyforlatertimepoints,atendencycompara-bletotheothercompartments.

Inthe1mg/mlCAFexperiments,thedruginthedonordidnotde-creaseasfastasforFFA.After5min,stillmorethan85%oftheinitialdosecouldbefound,decreasingslowlyandreachingabout60%remainingontheskinafter6h(Fig.5A).Atthelastmeasuringtimepoint(12h),stillmorethan40%ofthedrugwasfoundinthedonor.IntheSC,aplateauatabout25%wasreachedafter2handmaintaineduntil6h,thenslowlyincreasingtoabout35%attheendofincubationafter12h(Fig.5B).TheamountofdrugintheDSLremainedverylowuntil6hduetothevaluesclosetothelowerlimitofquantificationandthenincreasedtoabout10%at12h(Fig.5C).Onlyafterthelongestincubationat12h,CAFwasfoundintheacceptor(Fig.5D).However,thevariationwasquitehighduetotheconcentrationvaluesclosetotheLLOQ.TheCAFamountinthelateralcompartmentslowlyincreased,reachingabout10%oftheapplieddoseafter12hofincubation(Fig.5E).

Forthe12.5mg/mlexperiments,thecourseoftherelativedrugamountinthedonorwassimilartothe1mg/mldata(Fig.6A).IntheSC,aplateaubetween10%and15%wasreachedquicklyandmaintaineduntil6hafterapplication(Fig.6B).TheamountofCAFintheDSLincreasedslowly,reachingabout10%after6h(Fig.6C).After6hofincubationabout5%oftheappliedCAFwasfoundintheacceptorphase(Fig.6D).NoCAFwasdetectedatearliertimepoints.InthelateralcompartmenttheCAFamountincreasedtoabout15%after6h(Fig.6E).

Forthe12.5mg/mland1mg/mlCAFdatadifferencesinrelativeamountintheSC(Figs.5Dand6D)andDSL(Figs.5Cand6D)wereobserved.6hafterapplicationSClevelsat1mg/mlwereapproxi-mately1.5timestherelativeamountofthe12.5mg/mlexperimentsandDSLlevelsat12.5mg/mlwereapproximately6timestherelativeamountofthe1mg/mlexperiments.Thedonordepletedroughly10%fasterinthe1mg/mlCAFexperiments.3.2.Resultsfordifferentmodels

TheoutcomeofallthethreemodelsforFFAandCAFisdepictedinFigs.4–6.Asummaryofstandarddeviationsforallmodelsisaddi-tionallyprovidedinTable3b.Thedeviationfromtheobjectiveisfoundtobereasonablysmallinallcases.

TheMICROandMACROmodelsslightlyoverpredicttheamountintheSCforthe12.5mg/mlCAFexperimentsandslightlyoverpredicttheamountintheDSLforthe1mg/mlCAFexperiments.ForthePKmodeltherateconstantsfoundbyregressionandthecorrespondingstandarderrorsarereportedinTableS1.ForFFAitwasobviouslynotpossibletofittheeffluxrateconstantk7,whichwasthussettozero.TheoptimizeddiffusivitiesfortheMACRO(1)modelareshowninTable2a.TheoptimizedparametersfortheMACRO(3)model,indicatedbyan*,arelistedinTable2b.Thecorresponding

Table2

Inputparametersforsimulationswiththehomogenizeddiffusionmodel.a)FortheMACRO(1)modelwithoneregressionparameterallparametersexceptforDSCaretakenfrom[14].b)FortheMACRO(3)modelwiththreeregressionparametersallparametersexceptforDDONwereoptimizedaccordingtotheparameterizationprovidedinthetext.Theop-timizedparametersarelabeledwithanasterisk.TheregressionparametersandstandarderrorsforbothmodelsarereportedinTableS2aandbrespectively.Parameter

FFA

CAF(1mg/ml)2.92×10−23.70×10−51.69×10−74.7027.002.92×10−23.70×10−51.54×10−71.87×10−35.7733.13

CAF(12.5mg/ml)2.92×10−23.70×10−52.29×10−74.7027.002.92×10−23.73×10−55.97×10−710.72×10−32.5714.75

ReferenceEq.(1)Eq.(3a)Eq.(3a)[14][14]Eq.(1)Eq.(3a)Eq.(3a)

a)HomogenizedMACRO(1)modelwithoneregressionparameter.DDON[cm2/h]2.47×10−2Ã2󰀁D[cm/h]2.00×10−5SC;rr󰀁ÃD[cm2/h]10.43×10−7SC;zzKSC/DON5.88KSC/DSL3.00

b)HomogenizedMACRO(3)modelwiththreeregressionparameters.DDON[cm2/h]2.47×10−2Ã2󰀁DSC;rr[cm/h]1.98×10−5Ã2󰀁DSC;zz[cm/h]7.72×10−72DDSL*[cm/h]3.26×10−3KSC*/DON8.54KSC*/DSL4.36

D.Selzeretal./JournalofControlledRelease165(2013)119–128

Table3

Overviewofmodelfeaturesandcomparisonofgoodnessoffit.

PK

a)GeneralinformationPrimarypurpose

TypeofinputparametersNumberofinputparametersNumberofregressionparametersTimedependentSpatialresolutionLateralcompartmentComputationalcost

Fit

Rateconstants6forFFA/7forCAF6forFFA/7forCAFYes

No,only

compartmentModeledLow

MACRO(3)

MACRO(1)

MICRO

123

Diffusionandpartitioncoeff.6+geometry3

Yes

Macroscopic

SeparateforSCandDSLIntermediate

Diffusionandpartitioncoeff.6+geometry1

Yes

Macroscopic

SeparateforSCandDSLIntermediate

Prediction

Diffusionandpartitioncoeff.7+geometryNone

Yes

UptocellularstructuresNotmodeledHigh

b)Regressionresults:standarddeviationandnumberofindependentvariables(inparentheses)FFA4.82(24)5.23(27)CAF(1mg/ml)3.26(13)4.32(17)CAF(12.5mg/ml)3.56(13)3.99(17)

a6.61(29)

5.03(19)5.94(19)

8.47(24)a5.76(16)a5.27(16)aUnfittedmodel,w/olateralcompartment.Valuesareprovidedforcomparisonbetweenmodelsonly.

regressioncoefficientsarereportedinTableS2aandbofthesupple-mentaryinformationrespectively.4.Discussion

Basedontheexperimentaldesignandtheanatomicalconditions,weinvestigatedthemassbalanceprofilesforin-vitrofinitedoseskindiffusionexperiments.Fullthicknessskinwasseparatedandanalyzedindifferentcompartments(donor,SC,DSL,lateralcompartment,accep-tor)fortwosubstancesinaqueoussolution:FFA(1mg/ml)andCAF(1mg/mland12.5mg/ml).Thisexperimentalsetupwasthenrepre-sentedindifferentmathematicalmodels.

ExperimentalmassprofilesforFFAclearlyshowafinitedosebehav-iorwithastrongdepletionofthedonorandadepletionofmassintheSCafterreachingaplateauinthecourseoftheexperiment(Fig.4A,B).Incontrast,CAFexperimentsshowmoreinfinitedose-likeproper-tieswithasignificantlyslowerdepletionofthedonorandaconstantmassplateauintheSCoversampletime(Figs.5A,Band6A,B).Thisisinagreementwiththeexperimentsfromtheinfinitedosescenario,showingaslowerpenetrationofCAFincomparisontoFFA[15].Onereasonforthesedifferencesmightbeadisparityinthermodynamicac-tivityduetothehighersolubilityofCAFinthedonormediumincom-parisontoFFA.MassprofilesofCAFintheSCareclearlydifferentforthedifferentconcentrations:forCAFat1mg/mlabout25to35%ofthedrugcanbefoundintheSC,whereasforthehigherconcentrationat12.5mg/mlthevaluesreachonlyupto15%.OneexplanationforthedifferentprofilesisasaturationeffectintheSC,allowingonlyacer-tainCAFamounttobetakenupbytheSC,e.g.reversibleproteinbindingtokeratin.BindinginSCwasshownrecentlyfortheophylline,whichhasaverysimilarstructuretoCAF[31–33].

ThiseffecthasconsequencesfortheamountofsubstanceintheDSLovertime,showingamuchhigheramountfor12.5mg/ml(11%)incomparisonto1mg/ml(2%)after6hofincubation.Surpris-ingly,forbothconcentrationsnearlythesamerelativeamountwasfoundinthelateralcompartment.

Inthelateralskinparts,morethan10%forCAFat12.5mg/mlafter6hofincubation(Fig.5E),wasdetected.Thisprovidesevidencethatthelateralpartsdependingonsubstanceandexperimentalsetupmayplayacertainroleintheskinabsorptionprocess.Therefore,thelater-alcompartmentshouldnottobeelidedinmodelingfinitedoseskinabsorptionstudies.Moststudiesinvestigatingfinitedoseexperi-mentsdonotdiscussthelateraldrugamountfoundintheirstudiesbutlateraldiffusionofe.g.benzylalcoholintotheclampedpartsofthediffusioncellhasbeenassumedpreviously[34].InagreementwiththeresultsfromGeeetal.[3],whichdeterminedthelateraltrans-portofCAF,hydrocortisone,andibuprofen,thelipophilicFFAexhibitedalowertendencytomovelaterallythanthehydrophilicCAF.Fromthemicroscopicpictures(Fig.S3)takenbothfromuntreatedskinandfromskinsqueezedinanFD-Cfor20h,nosignificantdifferencewasfoundfortheSCthickness(p=0.218).Thedermis,however,appearedmuchdenserthanbefore.Thegeometryandphysiologyoftheskinhaveclearlychangedanditisassumedthatthisalteredconfigurationmayre-ducethepossibilityofusingmeasureddiffusivityandpartitionbehaviorfromfullyhydratedskinexperiments.

WiththePK,MICROandMACROmodelsthisstudyemployedthreedifferentcomputationalmodelsservingverydifferentpurposestodescribeandpredicttheexperimentalsetup.

IncontrasttoSetaetal.[35],theexperimentalresolutionoftheepidermisisdividedinlipophilic(SC)andhydrophilicparts(viableepidermis),whichismorereasonablefromaphysiologicalpointofviewinouropinion.ThesameholdstruefortheworksbyKastingetal.[24,34]whichfocusedonvolatilecompoundsandthusdidonlysupplymassbalanceinformationabouttheskinasawholetis-sue[36,37]and,inafurtherstudy,abouttheepidermisanddermis[38].Forthemodelingpart,theaforementionedworksbelongtotheclassofone-dimensionaldiffusionmodels,e.g.,[39–42,23].Theprimaryfocusintheseworkswasonabsorption,i.e.,massaccumulationintheacceptor,however.Noneoftheabovecitedmodelsdoconsiderlateraldiffusiontopartsoftheskinoutsidetheincubatedapplicationarea.

Toexplainthemasstransportbetweendifferentcompartmentsandtoevaluatethisinfluenceofthelateralcompartmentfurther,aneasytouseextendedpharmacokineticmodel(PK)wasconstructed.Inthisap-proachthewholeexperimentalsetupisdescribedintermsofcompart-ments.Interestinglyenough,theclassofPKmodelsrecentlyregainedsomeattention:theworkbyDaviesetal.[43],e.g.,showedtobeeffi-cientwithtwoskincompartmentsalready.OverviewarticlesrelatingtooneortwocompartmentPKmodelscanbefoundintheliterature[44–46].PKmodelsusingfirst-orderrateconstantsweretypicallyusedtomonitortheinvivosituation[42,47]andalotofeffortwasspenttryingtorelaterateconstantstophysicochemicalpropertiesofthediffusantandtofitamodeltoplasma-levelcurvesorinvitroperme-ationprofiles[48,49].Incontrasttothepreviousmentionedwork,alat-eralcompartmentwasaddedtofullydescribetheexperimentalsetting.

Although,whencomparedtotheMICROmodel,thisapproachdoesonlyprovideaverylimitedspatialresolution,themodelfeaturesalowcomputationalcomplexity.Inordertoprovideadditionalinsightofalleffectsoccurringduringincubation,modelparametersweredeter-minedbyparameterfittingwithrespecttotheexperimentaldata.Toreducethenumberofconstantsanaveragedtransport(rateconstantsk5andk6)betweenSCandlateralcompartmentaswellasbetweenDSLandlateralcompartment,wasassumed.Unsurprisingly,thisyieldsexcellentagreementtotheexperimentaldata.FortheCAF

124D.Selzeretal./JournalofControlledRelease165(2013)119–128

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Fig.4.FFAmassprofilesovertime:experimentaldata,PKfitting,detaileddiffusionsimulationandhomogenizeddiffusionmodels.A:Massprofileofthedonorcompartment.B:MassprofileoftheSCcompartment.C:MassprofileoftheDSLcompartment.D:Massprofileoftheacceptorcompartment(diffusionmodelwithmassleavingthesystem).E:Massprofileofthelateralcompartment(diffusionmodeldatanotavailable).

experimentstheratiosofrateconstantsk3andk4(SCandDSL)andk5andk6(SC/DSLandlateralcompartment)respectivelyareinagreementwiththeexperimentalfindings,showingafastertransportfromtheSCtotheDSLforthe12.5mg/mlexperimentsandafastertrans-portfromtheSC/DSLtothelateralcompartmentforthe1mg/mlexperiments.

Afterasuccessfuldescriptionoftheexperimentalsetupadetaileddiffusionmodel(MICRO)drivenbyinputparametersgatheredfromin-finitedoseexperimentswasapplied.Ithasbeenshown,thatthisisrea-sonableevenforfinitedosekinetics[7,35],whereasestimationfromfinitedoseexperimentsmayshowgreatintra-individualvariability

[40].Incontrasttoone-dimensionalmodelscitedabove,theworkathandreliesonamicroscopictwo-dimensionalgeometry.Thismodelisthusmeanttobepredictive,asallparametersareinprincipledirectlyrelatedtophysicochemicalpropertiesofthesubstance.

AnoutoftheboxuseoftheMICROmodelprovidedgoodpredic-tionsofthemassprofilesineachcompartmentforbothdrugsovertheincubationtimesof6hor12h:whiletheagreementswiththeexperimentareexcellentforFFA,someminordifferencesarefoundforCAF.AtlowdonorCAFconcentration,e.g.,themodelproducesanoverestimateofDSLconcentration,whereasathighdonorCAFconcentration,anoverestimateofSCconcentrationisobtained.Due

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Fig.5.CAF(1mg/ml)massprofilesovertime:experimentaldata,PKfitting,detaileddiffusionsimulationandhomogenizeddiffusionmodels.A:Massprofileofthedonorcompart-ment.B:MassprofileoftheSCcompartment.C:MassprofileoftheDSLcompartment.D:Massprofileoftheacceptorcompartment(diffusionmodelwithmassleavingthesystem).E:Massprofileofthelateralcompartment(diffusionmodeldatanotavailable).

tothedifferingexperimentalresults,thisfindingcouldbeexpected.ExplanationsfortheoverestimationcouldbealateraldiffusionintheDSLandasaturationeffectintheSC.Furthermoresomeminordifferencesintheacceptorcompartmentcanbeeasilyexplainedbyanalyticalreasons,e.g.concentrationbelowtheLLOQ.

Thegeneraldrawbackofthemodelishowevertheabsenceofalateralcompartment.Asexplainedabove,thisisduetotheveryhighresolution.Alsonote,thatalthoughaparameteroptimizationispossibleinprinciple,itwasnotappliedhere,becauseofthehighcomputationalcomplexity.Assomeauthorssuggest,thehighresolutionPKmodels,e.g.,[7],pro-videanimprovedspatialresolutioninz-direction.Inthiscase,eachcom-partmentrepresentsasinglelayeroftissueinSCandDSLrespectively.Asthiscorrespondstoasetofdiscretizedpartialdifferentialequations,wepursueadifferentapproachwiththeclassofMACROmodels.

ThehomogenizedMACROmodelsserveasacompromisebetweenthepreviouslymentionedmodels:incontrasttotheMICROmodelthecellsintheSCmembranearenotresolvedinfulldetail,butsophisticat-edlyrepresentedinanaveraged(homogenized)way.Thisallowsrepresentingadiffusioncellwithrealisticdimensionsona1cmscale

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Fig.6.CAF(12.5mg/ml)massprofilesovertime:experimentaldata,PKfitting,detaileddiffusionsimulationandhomogenizeddiffusionmodels.A:Massprofileofthedonorcom-partment.B:MassprofileoftheSCcompartment.C:MassprofileoftheDSLcompartment.D:Massprofileoftheacceptorcompartment(diffusionmodelwithmassleavingthesystem).E:Massprofileofthelateralcompartment(diffusionmodeldatanotavailable).

(cf.Fig.2).LikeinthePKmodelitisthuspossibletostudytheinfluenceoflateraldiffusionwiththeMACROmodelsaswell.Inaddition,thesemodelsinheritthedesirablefeaturesofpredictionandparameterinter-pretationfromtheMICROmodel.Notehowever,thattheyetunknownvariationinphysiologyandtheinvolvedalterationofthediffusionandpartitionparametersattheboundarytothelateralcompartmenthavenotbeenincludedinthecomputationalmodel.

DuetothereducedcomputationalcomplexityoftheMACROmodels,thesemodelsarealsosuitableforparameteroptimizationonareasonablecomputationaltimescale.Whenitcomestoaccuracy,notsurprisingly,theMACROmodelsrangebetweenthePKmodel,whichwasoptimizedbyparameterfitting,andtheMICROmodel,whichhasapredictivecharacter.TheMACRO(1)modelbeingopti-mizedbasedonasingleparameterproducesslightlyweakerresultsthantheMACRO(3)model,whichisobtainedbyoptimizingthreeparameters.AsindicatedinTable2,theabsolutevaluesofthecorre-spondingdiffusionandpartitioncoefficientsarequitecomparablebe-tweenbothmodelshowever.

󰀁SC;zz,Theapparentdiffusioncoefficientsintransversaldirection,D

arethreeordersofmagnitudesmallerthanthecoefficientsforlipid

󰀁SC;zzareintherangediffusionDLIP.ForCAF,theresultingvaluesforD

󰀁SC;rrareonereportedin[14].Similarly,thelateralcoefficientsD

D.Selzeretal./JournalofControlledRelease165(2013)119–128127

orderofmagnitudesmallerthanDLIP.FortheinterpretationofDDSLitisimportanttoconsiderthatthisvaluemaybeafflictedwithsomeerror,asaconstantthicknessof3mmwasassumed.

AsshowninEq.(3a),theSCdiffusioncoefficientsD

󰀁SC;zzandD󰀁SC;rrarecoupled.Thishasanotherimportantconsequencefortheinter-pretationofthecontributionoflateraldiffusion.InparticularwhenconsideringCAFat1mg/ml,overshootingoccursfortheamountinDSL(Fig.5C),whereasundershootingoccursfortheamountinLAT(Fig.5E).Lesspronounced,thesamephenomenonalsooccursforCAFat12.5mg/ml(Fig.6CandE).ThisillustratesthatthemodeldoesnotpermittoacceleratediffusionintothelateralpartoftheSCandtoslowdowndiffusionintotheDSLatthesametime.TheexactoppositeholdstrueforthePKmodel,inwhichanappropriatechoiceofrateconstantsk5andk6mayhaveacompensatoryeffect.Thiscompletelyallowsforreflectingtheexperimentallyobservedlateraldiffusion.IncontrasttothePKmodel,bothMACROmodelsdistin-guishbetweenthelateralpartoftheSCandoftheDSLcompartment.Moreprecisely,bothMACROmodelsindeedsupportthehypothesisthatlateraldiffusionintheSCissubordinateandmainlyoccursintheDSL(Fig.S4a–c).Duetothelargercross-sectionalareaoftheDSLcomparedtotheSCthisseemsreasonable.5.Conclusion

TheexperimentalsetupoffinitedoseskinabsorptionexperimentsinaFranzdiffusioncellcanbethoroughlyinvestigatedbymeansofthepresentedmathematicalmodels.Allexplainand,toacertainex-tent,alsopredicttheexperimentaldata.In-vitromasstransporttothelateralpartsforthetransientcaseshouldbeconsideredinordertogetoptimalrecoveryexperimentallyandpropermassbalanceinthemathematicalmodels.

ThePKmodelissuitablefordescribingthewholesetupandgivesinformationaboutthediffusionprocessalsointothelateralskinparts.Itincludessomestructuralinformation,butisprimarilybasedonfittedparameters(rateconstants)thataredifficulttointerpret.Thisapproachmayexpandtheknowledgeaboutthewholeabsorp-tionprocessinaFranzdiffusioncell.

Reflectingadifferentphilosophy,thedetailedMICROdiffusionmodelpredictsthediffusionofFFAandCAFintothedifferentskincompartmentsinafinitedosesetupreasonably.Itisatransferofaninfinitedosemodeltothefinitedosescenarioanddoesnotrelyonfittedparameters.Duetothecomputationalcomplexityitcurrentlydoesnotincludealateralcompartmenthowever.

Asacompromise,thehomogenizedMACROdiffusionmodelhastheadvantageofdescribingtheexperimentalsetupmoreaccuratelywhilekeepingtheadvantagesofadiffusionmodel.Thedomainsepa-rationintheSC(corneocytesandlipidchannel)isreflectedinanav-eragedsense.Thus,simulationofthedrugtransportintothedifferentcompartmentsincludingthelateralpartispossible.AppendixA.Supplementarydata

Supplementarydatatothisarticlecanbefoundonlineathttp://dx.doi.org/10.1016/j.jconrel.2012.10.009.References

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