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ModelinginFreeze-Drying:Past,CurrentState,andFuturePerspec;ves

MichaelJ.PikalSchoolofPharmacy

UniversityofConnec;cut

Seealso:TchessalovS,NuluS,LatshawIID,DassuD.“AnIndustryPerspec;veontheApplica;onofModelingtoLyophiliza;onProcessScaleUpandTransfer,Am.Pharm.Rev.,20(2)2017pp52-58.

EarlyWorkbyMarcusKarel(book,1975)

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Mo;va;onforModelingStudies•  GuideFormula;onandProcessOp;miza;onEfforts:

Helpdesignandinterpretexperiments–  Provideabe^erunderstandingoftheimpactofprocessandformula;onvaria;onsoncycle;meandproducttemperaturehistory•  effectsofvaria;onincollapsetemperature,productconcentra;on,fillvolume

•  effectsofvaria;oninchamberpressureandshelftemperature

•  AssistinTroubleShoo;ngProblems–  Quan;fytheeffectsofheattransfervaria;onsonproducttemperaturehistory•  containereffects•  vialposi;oneffects

–  Quan;fytheeffectsoffreezingvaria;onsoncycle;meandproducttemperaturehistory

–  “Explain”originofproductdefects(i.e.,collapse,degrada;on,...)

SteadyStateModels•  Advantages:

–  veryquickly,andeasily,caninves;gateicetemperatureanddrying;meforprimarydrying•  withminimalsuitablemassandheattransferinputdata

•  Limita;ons–  Cannotprovideinforma;onduringperiodsofshelftemperatureincrease(i.e.,duringnon-steadystate)

–  Cannotprovideinforma;ononresidualmoisturein“dry”layer.

–  Isnormallylimitedtoone-dimensionalproblems(i.e.,slabgeometry)•  cannotinves;gateimpactofspacialheterogeneityinheattransferormaterialproper;es.

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TheNon-SteadyStateModelinTwoDimensions

(threedimensionswithanaxisofsymmetry)

•  Baseduponasetofcoupleddifferen;alequa;ons(M.J.Millman,A.I.Liapis,andJ.M.Marchello,AIChEJ.31,1594-1604(1985)

–  conserva;onofmass–  conserva;onofenergy–  inputdataformassandheattransfercoefficients–  flexibleboundarycondi;ons

•  allowsavarietyofproblemstobestudied

•  UsesFiniteElementAnalysis–  allowsextensionto2-D&studyofcomplexgeome;es

•  Employsa“ModularSoiwarePackage”(Passage™/FreezeDrying,H.AkayandW.J.Mascarenhas,TechnalysisInc.,Indianapolis,IN)

–  foreaseandflexibilityofuse

SomeNon-SteadyModelsforSublima;onandDesorp;onDrying1.Ananalysisofthelyophiliza;onprocessusingasorp;on-sublima;onmodelandvariousopera;onalpolicies-1985By:Millman,M.J.;Liapis,A.I.;Marchello,1985.J.M.AIChEJournal,31(10),1594-1604Thefreeze-dryingprocessisstudiedundervariousoperaQonalpoliciesthroughtheuseofasorpQon-sublimaQonmodel.…2.TheNon-SteadyStateModelingofFreezeDrying:In-ProcessProductTemperatureandMoistureContentMappingandPharmaceu;calProductQualityApplica;ons”-2005(denoted“PassageFD”)By:PikalM,MascarenhasW,AkayH,CardonS,BhugraC,JameelF,andRambhatlaS.,(2005)Pharm.Develop.&Technol,10(1),17-32.Anearlierworkfromtheselaboratoriespresentedanon-steadystate,two-dimensionalmodel…TheearliertreatmentfocusedonthemathemaQcaldetailsofthefiniteelementformulaQon...TheobjecQveofthecurrentstudyistoprovidethephysicalraQonalforthechoiceofboundarycondiQons,tovalidatethemodelbycomparisonofcalculatedresultswithexperimentaldata,and…3.Developmentofsimplifiedmodelsforthefreeze-dryingprocessandinves;ga;onoftheop;malopera;ngcondi;ons-2008By:Velardi,SalvatoreA.;Barresi,AntonelloA.(2008)ChemicalEngineeringResearchandDesign,86(1).9-22ThispaperisfocusedonthemodelingoftheprimarydryingofalyophilizaQonprocessinvials.Adetailedmono-dimensionalmodeltakingintoaccountmassandenergybalancesinthedriedlayerandatthesublimaQnginterface,energybalanceinthefrozenlayerandalongthevialwallissetup…

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TheNonsteadyStateModelingofFreezeDrying:In-ProcessProductTemperatureandMoistureContentMappingandPharmaceu;calProductQualityApplica;ons,Pikalet.al.-PassageFreezeDrying

SampleDatafromPassage(5%Sucrose)

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SimpleSteadyStateHeatandMassTransferTheoryisUsefulinProcessDesignfor

PrimaryDrying

• “What if” calculations, for impact of variation in shelf temperature and chamber pressure! •  Design Space Evaluation – Scale-Up Calculations – Robustness Testing (edge of failure)

• As accurate as experiment!

SimpleSteadyStateHeatandMassTransferTheory

Mass Transfer : dmdt

= ApP0 T( ) − Pc( )

ˆ R ps

; lnP0 =−6144.96

T+ 24.01849

Heat Transfer : dQdt

= Av ⋅ Kv Pc( ) ⋅ Ts − T − ΔT( ); ΔT → function of dm/dt

Coupling : dQdt

= ΔH s ⋅dmdt

ΔH s Ap / Av( ) ⋅ P0 T( ) − Pc( )ˆ R ps

- K v Ts − T − ΔT( )= 0

• One Equation, one unknown (T): Solve for T, get dm/dt. and then calculate drying time- Basis of the “Lyo-Calculator”

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ExperimentandCalcula;onsAgreeWell:Blue=Exp.,Red=Calc.

Product 5% w/w

Vial Fill cc

Shelf, interior, °C

Pc, Torr

1° Drying Time, hr

Shelf Surface,Ts

Mean Tp

Max Tp

PVP W5816 8 -5 0.1 25.8 -9.6 -27.8 -25.3 26.9 -9.9 -27.3 -24.6 Mannitol W5816 8 -5 0.1 33.4 -8.6 -22.4 -20.2 34.8 -8.9 -22.9 -18.5 Mannitol W5816 8 +15 0.1 19.2 +6.2 -17.0 -14.2 19.1 +8.0 -17.0 -11.8 Mannitol W5816 8 +15 0.4 14.0 +5.7 -13.0 -11.9 15.8 +6.6 -11.8 -8.0 Mannitol 5303 20 +15 0.4 19.2 +6.1 -14.5 -12.8 19.0 +8.1 -13.5 -9.7

OLD METHODOLGY:M. Pikal, PDA Journal, 39, 115-138 (1985)

Now available as “Lyo-Calculator”

TheLyoCalculator:Simula;onofPrimaryDryingonExcelSteady State Modeling of Primary Drying for Aqueous Systems

Temperature Historythis calculation divides up the product into 10 "slices" and evaluates the temperature as a function of drying time and gives the total drying time, in hr.

to run: 1. Input data in red           2. Go to Tools, Macro, and then run Macro FreezeDry    

lmax Ap R0 A1 A2 Vfill C, g/g ∆mH2O0.909838469 5.97 1.4 16 0 5 0.05 4.833333333

10^4Kv Av/Ap Ap Vfill, mL C ∆mH2O Ts, °C Pc ∆m

9.6 1.200 5.97 5 0.05 4.833333333 283.2 0.15 0.483333333Major Output of Calculations

n l, cm R T Ts,°C Ts,K P0 Goal Seek Fn dm/dt ∆T tdry, hr t°C, sub t°C bottom Hours0 0 1.4 235.8008284 -40 233.2 0.129821137 4.27E-05 -0.0860 -0.30 0 -37.4 -37.7 0.010.090983847 2.85574155 251.4771747 10 283.2 0.658957292 5.05E-05 1.0640 3.37 0.99 -21.7 -18.3 1.020.181967694 4.311483099 254.2444671 10 283.2 0.859734351 5.87E-05 0.9828 2.77 0.47 -18.9 -16.2 1.530.272951541 5.767224649 256.2918212 10 283.2 1.042835084 -1.39E-07 0.9242 2.28 0.51 -16.9 -14.6 2.040.363935387 7.222966199 257.9209051 10 283.2 1.213347738 -3.12E-07 0.8789 1.85 0.54 -15.2 -13.4 2.550.454919234 8.678707748 259.2753982 10 283.2 1.374168027 -6.23E-07 0.8421 1.48 0.56 -13.9 -12.4 3.160.545903081 10.1344493 260.4358025 10 283.2 1.527219979 -1.10E-06 0.8113 1.14 0.58 -12.7 -11.6 3.770.636886928 11.59019085 261.451712 10 283.2 1.673857126 -1.74E-06 0.7849 0.83 0.61 -11.7 -10.9 4.380.727870775 13.0459324 262.3558692 10 283.2 1.815080504 -2.53E-06 0.7620 0.54 0.62 -10.8 -10.3 4.990.818854622 14.50167395 263.1710346 10 283.2 1.951658587 -3.41E-06 0.7417 0.26 0.64 -10.0 -9.7 5.5

100.909838469 15.9574155 263.9136749 10 283.2 2.08419866 -4.33E-06 0.7236 0.00 0.66 -9.2 -9.2 6.20.7663 6.2 -16.2

Total mean

Symbols Definition

10^4Kv vial heat transfer coefficient in cal s-1 cm-2 K-1

Av cross sectional area, sq cm, of outside of vial

Apcross sectional area, sq cm, of inside of vial (product area)

Vfill vial fill volume, mL

CConcentration of solute, in volume fraction

Ts, K shelf surface temperature in KPc Chamber pressure set point, in TorrR0, A1 and A2 Constants in the empirical relationship describing

the product resistance vs. cake thickness

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Roleof“DesignofExperiments”(DOE)inPrimaryDryingDesign

•  Virtually, no role at all –  DOE is useful when mechanistic understanding is

poor –  The physics of primary drying is well understood (i.e. “Lyo-Calculator)

•  General statistics dogma: DOE is an efficient way to generate a “response surface” (or Design Space) –  Not true for freeze drying in general, and is very

inefficient for primary drying.

DOE:Box-BehnkenDesignIndependent variables: (3) chamber Pressure, shelf

temperature, Ice nucleation temperature Responses:(3) 1° drying time (hr), mean product temp. maximum product temp in 1° drying, sublimation rate

• 15 freeze drying experiments, average 2 days per experiment---> 30 days run time • Physics Driven: do runs in green, 4 runs--> 8 days

Exp. # Pattern Pchamber T shelf Ice Nucl. Temp 1° dry hr Tp mean Tp(max) mean dm/dt1 0.4 -5 -12.5 33.9 -18.7 -15.1 0.2282 0.25 -5 -5 31.4 -21.3 -17.2 0.2463 /++0 0.4 15 -12.5 16 -13.8 -8.2 0.4834 0.1 -5 -12.5 35.5 -23.6 -18.6 0.2185 /000 0.25 5 -12.5 22.1 -17.8 -12.6 0.3506 0.25 15 -20 17.3 -14.4 -8.2 0.4477 0.1 15 -12.5 19.4 -18.4 -11.8 0.3988 0.25 -5 -20 35 -19.9 -15.6 0.2219 0.1 5 -20 26.4 -19.8 -13.8 0.29310 /000 0.25 5 -12.5 22.4 -18.3 -13.4 0.34511 /+0+ 0.4 5 -5 21.1 -16.8 -12.3 0.36612 0.4 5 -20 23 -15.2 -10.4 0.33613 /000 0.25 5 -12.5 21.2 -17.2 -12 0.36514 /0++ 0.25 15 -5 16.1 -16.5 -10.6 0.48015 0.1 5 -5 24.1 -21.9 -16.1 0.321

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UsingPhysics•  Vial Heat Transfer Coefficients

–  Previously determined for all vials used by company, vs. Pressure-3 days required for each vial type

•  Dry Layer Resistance- 4 experiments! –  Unique to formulation and ice nucleation temperature –  Need runs at three ice nucleation temperatures

•  See GREEN on previous slide –  Rp evaluated from MTM data and/or –  cycle product temperatures-excel spreadsheet available

–  Prudent to do one of the runs that give high product temperature to compare with center point temp.-necessary if running close to or above Tc due to Rp(T) in this region, good to do additional run at another T as well to better determine Rp(T). •  Provides two replicate runs for Rp @ center point ice nucl. •  Provides validation of calculations in extreme case

–  Resistance normally independent of temperature, but not near collapse temperature!

• Total Run Time of 8 days, save 22 d, $66MM

Evalua;onofDesignSpace

•  FromExperiment–  EvaluateproductResistance,Rp

•  Maybeasfunc;onofproducttemperature–  EvaluateVialHeatTransferCoefficient,Kv

•  Centervialsandedgevials•  Asafunc;onofchamberpressure

–  Evaluatelimitsimposedbyequipment•  Minimumpressurecontrollableasafunc;onofsublima;onrate(i.e.,

pressuredependenceofchokedflowwTDLAS)

•  UseSteadyStateHeatandMassTransferTheory(LyoCalculator)toevaluate:–  Massfluxandproducttemperatureasafunc;onofchamber

pressureandshelftemperature

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TheCalcula;ons•  Sublimation Rate at fixed Product Temperature

vs Chamber Pressure: simple calc. –  P0, Ap, Rp+Rs all constantà linear dm/dt vs Pc

•  LyoCalculator for Effect of chamber pressure on sublimation rate and product temperature at fixed Ts –  Sublimation rate vs Pressure characterizes proximity

to dryer “overload”-”choked flow curve” •  Potential for dryer overload greatest early in primary drying

(best to use dm/dt early) –  Product Temperature vs Pressure (at fixed Ts) defines

product limit (Too Hot). •  Potential for too hot maximum near end of primary drying (w

constant shelf temperature and constant pressure). Best to use product temperature at end of primary drying.

dmdt

=Ap P0 − Pc( )R̂p + Rs

A“DesignSpace”:Schema;cSearles&Nail,BioPharmInterna;onal,21(1),2008

Equipment Limitation (i.e.,choked flow curve)-will be discussed later

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ScaleUpofDesignSpace:ValuesofKvforScale-Up

•  Center Vials (hexagonal pack)

•  Edge Vials: Calculation of effect more complex –  Need Wall Temperatures from OQ Data!

104Kv Mfg( )−104Kv Lab( ) = Av eMfg − eLab( ), cal/cm2s K−1

eMfg = emissivity in Mfg, typically ≈ 0.3eLab = emissivity in Lab, typically ≈ 0.65

ΔKvE = Kv Edge( )−Kv Center( )104ΔKvE ≈1.5(Typical Lab), 104ΔKvE ≈1.2−1.5(Mfg, same wall temperature as Lab)* *Exact value depends on shelf spacing and wall emissivity

ExcelSpreadsheetforScale-UpofEdgeVialEffect

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TheScale-UpFactorforKv:Valuesof104∆KvEforrepresenta;veFormula;ons/Processes

• Scale Up Adjustments Small, particularly for mannitol and Protein Formulation (protein rich)• Assumes same wall temperatures in mfg as in lab

0

0.5

1

1.5

2

2.5

3

3.5

Mannitol Protein Formulation Sucrose

10^4

∆K

vE

Laboratory Manufacturing, edge trays Maufacturing, no trays

SecondaryDrying:ExcelBased

•  Aseriesofsteadystatecalcula;onsofheatandmasstransferinthetwohalvesofthedrylayer.

•  Drying;meisdividedintoalargenumberof;mesteps,whereineach;mestepsteadystatecondi;onsprevail.– Waterdesorp;onisothermandmasstransfercoefficientdataarerequired.

•  Temporalvaria;onsofaveragemoisturecontentsandproducttemperaturesareoutputsandarecomparedwithexperiment.

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TheExcelInputandOutput

ComparisonofExcelCalcula;onswithPassageandExperimentalValues

Shows comparison of Excel based calculated results with PASSAGE calculations and experimental studies with time (dCw/dt) based kg data. Procedure for Evaluation of kg and all input information provided were same as in the PASSAGE calculations

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Comparison of Calculations with Experiment using “Universal” values for E and kg0. Log plot for <Cw> vs time showing comparison of theoretical (lines) and experimental results (symbols) for different drying runs: (1) Ice Nucleation Temperatures: (a) -5 °C, (b) -7 °C, (c) -10 °C; (2) Secondary Drying Temperatures: (a) 25 °C, (b) 40 °C, (c) 50 °C; and (3) Sucrose Concentrations: (a) 5%, (b) 10%, (c) 15%

1(a) 1(b) 1(c)

2(a) 2(b) 2(c)

3(a) 3(b) 3(c)

ComparisonofExcelwithExperiment

(a) (b)

SomeExampleApplica;onsfor5%Sucroseramprate0.2°C/minute

Effectoftemperature-;mesequences Differencebetweentopandbo^om

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ModelingofFreezing

•  Simplemodeloffreezeconcentra;onvs;meiseasy•  Predic;onoficecrystalsizeandheterogeneitymuchmoredifficult,

•  But…–  Someprogressmadewithcomplexmodelsforbothvialsandsyringes.

ModelingFreezeConcentra;on:AmorphousSoluteM.J.Pikal,notpublishedinjournal

t°C % H2O(l)

6050403020100-30

-20

-10

0

10

0

10

20

30

40

50

60

70

80

t°C%H2O(l)

Water Removal During Freezing: Moxalactam di-Sodium

Minutes

-30°C Shelf

Tg'

•FixIceNuclea;ontemperature•Simplepseudosteadystatewith,dQ/dt=AvKv(P=1atm).(Ts-Tp)•Iceformedduringini;alfreezeby∆mice

.∆Hf=Cp(soln)(Tf-Tn)•Postnuclea;on,producttempandsolncomposi;on(&iceformed)byassumingequilibriumfreezing(followliquiduscurve)

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DetailedFreezinginVials-1NakagawaK,HoeotA,VessotS,AndrieuJ.Modelingoffreezingstrepduringfreezedryingofdrugsinvials.AIChEJournal.2007.

53(5):1362:1372

•  Experimental–  7mmfilldepth–  Controlledicenuclea;on

(ultrasound)(-7C)•  Withsiliconoiltoallowultrasound

towork

•  Model–  Fixicenuclea;ontemperature–  Fourier’s2ndLawforcooling

–  Rateoficenuclea;onassumeddirectlypropor;onaltodegreeofsuper-cooling

–  GrowthratedeterminedbyHeattransfer

SizeiceCrystalsvsTnCoolingcurves

TheGeometry

DetailedFreezinginVials-2MorphologyandsizeIcecrystalsnotUniform

ExperimentalView,Tn=-7.4°HeterogeneousandSmallericecrystalsatlowerTn

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FreezinginSyringes:DataExperimentalStudyandModelingofFreeze-DryinginSyringeConfiguraQon.PartI:FreezingStep

A.Hoeot,1J.Andrieu,1S.Vessot,1E.Shalaev,2L.A.Gatlin,2andS.Rickees2DryingTechnology,27:40–48,2009

•  RadialFreezing(shelf-40°)–  Experimentalobserva;on

•  homogeneousfreezingradiallyandtoptobo^om•  Variableicenuclea;ontemperature,smallericecrystalswithlowericenuclea;ontemperature(meanicecrystalsize28µm)– Meanicenuclea;ontemperature≈-12°C

FreezinginSyringes:Model•  Doesnotmodelnuclea;on:nuclea;ontemperaturesbasedonexperiment•  Icegeneraterateen;relycontrolledbyheattransfer•  Frac;onoficeformeddeterminedbyproducttemperature

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FutureNeeds•  Developeasiertousemodelsoffreezing

–  buts;llgivingusefulaccuracy-Excelbased?•  Clarifyroleofconvec;oninheattransferduring1°drying•  Addi;onalapplica;onsforSteadyStateModelsin1°Drying

DesignSpaceDetermina;on–  Includetheimpactofnaturalvaria;ons–  Includevariableshelftemperatureindesignspacedetermina;on

•  Publishdatabasesforparametersto“feedthemodels”–  KvandRp

•  Kvbygravimetric(orTDLASforaverageKv)•  RpbyMTMoranalysisofcycledata:ScaleuprequiresSSAonmfgandLab

–  MfgDryer“behavior”duringrepresenta;vedryingruns•  Characterizemore“Types”ofproduc;onDryersbyCFD•  Educate(convince?)theFDAthatmodelingcanaddvalue!!