Study of Consolidation Parameters: Diffusion, Dissolution and Pharmacokinetic Parameters

1. Diffusion parameters
2. Dissolution parameters
3. Pharmacokinetic parameters
4. Heckel plots,
5. Similarity factors – f2 and f1,
6. Higuchi and peppas plot,
7. Concept and significance of Linearity, Standard deviation , chi square test,
student T-test , ANOVA test.

1. Modern Pharmaceutics
Dr. Kailas Mali
Professor in Pharmaceutics,
Adarsh College of Pharmacy, Vita
Study of Consolidation
Parameters

2. 1.Diffusion parameters
2.Dissolution parameters
3.Pharmacokinetic parameters
4.Heckelplots,
5.Similarity factors –f2 and f1,
6.Higuchi and peppasplot,
7.Concept and significance of Linearity, Standard deviation , chi square test,
student T-test , ANOVA test.
Contents

3. ●
Diffusionofdrugfromdosageform
Diffusion Parameters
Diffusion
●
Itisaprocessofthe masstransferofthe
individualmoleculeofasubstancebrought
aboutbyrandommolecularmotion
associatedwithadrivingforcelike
concentrationgradient(highertolower
concentration).
●
Freediffusionofthesubstancethrough
liquids,solidsandthemembranesareof
specialinterestindesigningofadosage
form.
●
Studyingdiffusionalparameterswillhelpus
tounderstandpermeationanddistributionof
drugmoleculesinlivingsystems.

4. ●
Drugreleaseformreservoirandmatrix.
Diffusion Parameters
ApplicationsofDiffusion
●
Thereleaseofdrugsfromdosageformsis
diffusioncontrolled(SRandCRProducts).
●
Molecularweightofpolymerscanbe
estimatedfromdiffusionprocess.
●
Absorptionofdrugsfromvariousroutescan
beunderstoodandpredictedformthe
principlesofdiffusion.
●
Thediffusionofdrugsintoto tissuesand
theirexcretionthroughkidneys canbe
anticipatedthroughdiffusionstudies.
●
Principlesofdiffusioncanbeusedasinvitro
modelsfordrugproteinbindingstudies.

5. ●
Thechangeinmasstransfer alsooccurs
simultaneouslywithrespecttodistance.
●
Fick’sfirstlawstatesthattheflux(therate
ofmasstransferacrossaunitsurfacearea
ofbarrier)isdirectlyproportionaltothe
concentrationgradient.
●
Where,dCischangeinconcentrationof
material,g/cm
3
;Disdiffusioncoefficientof
apenetrant,cm
2
/sec;dxischangein
distance,cm.
●
Disaffectedbytheconcentration,
temperature,pressure,solventpropertyand
chemicalnatureofdiffusant
Diffusion Parameters
Fick’sFirstLaw
●
Indiffusion,moleculesgettransportedfrom
onecompartmenttoanotheroveraperiodof
time-ie,rateofmasstransfer(dm/dt).Thisis
expressedasflux.
●
Flux(J)isequaltotherateofmasstransfer
acrossaunitsurfaceareaofabarrier.
●
Where,
dMischangeinthemassofmaterial,g
Sisbarriersurfacearea,cm
2
dtischangeintime,sec.
dt
dM
S
J
1
dx
dC
DJ

6. ●
Ifdiffusionistheratedeterminingstep,then
wecanuseFicksfirstlawofdiffusionto
describetheoverallprocess
Diffusion Parameters
●
Thenegativesignintherightsidetermin
equationsignifiesadecreaseinthe
concentration.Butfluxisalwaysapositive
quantity,becauseitincreasescontinuously
duringprocess.Thedxisperpendiculartothe
surfaceofthebarrier.
●
Combiningabovetwoequations:
●
Equationrepresentstherateofmasstransfer
asperFicksfirstlaw.
dx
dC
DS
dt
dM


7. ●
Lateron,NernstandBurnershowedthatkis
acompositeconstantbeingproportionalto
thediffusioncoefficient,Dandthesurface
areaofthedissolvingbody,A.Thusthe
modifiedequationiscalledastheNernst
andBurnerequation:
●
Where,histhethicknessoftheboundary
layer,Aisthesurfaceareaofdissolving
solid,Kw/oisthepartitioncoefficientof
drugandVisthevolumeofthedissolution
medium.
Diffusion Parameters
DiffusionlimitedmodelorFilmtheory
●
Thefirstdissolutionexperimentwere
conductedbytheNoyesandWhitneyand
foundthatthedissolutionrate(dc/dt),isa
linearfunctionofthedifferencebetweenthe
bulkconcentrationattimetandthe
saturationsolubility:
●
Where,kisthedissolutionrateconstant.
)(
bs
CCk
dt
dc

)(
/
bs
OW
CC
Vh
DAK
dt
dc


8. ●
Plotofconcentrationversusdistancefrom
solidsurface
Diffusion Parameters
VariablesinDiffusionProcess
●
Surfacearea(A):Surfaceareapergramofa
soliddrugcanbechangedbyalteringparticle
size.
●
Diffusionlayerthickness(h):Invitroitis
determinedbytheagitationinthebulk
solution.Invivonocontroloverthis
parameter.Thicknessofstagnantlayercan
bereducedwhendrugdissolvesinreactive
medium.Weaklybasicdruginanacidic
mediumdissolvesatfasterratethanneutral
mediumbecauseofreductioninthicknessof
stagnantlayer.

9. Drugsolubility(Cs)
●
Itisanotherdeterminantofdissolutionrate.
AsCsincreasesdissolutionratealso
increases.
Partitioncoefficient(Kw/o)
●
Describesthedrug’ssolubilitybetweentwo
phases(e.g.,lipid/water).
●
Determineshowwelladrugpenetrateslipid
membranes(importantfororaland
transdermaldelivery).
●
Highwatertooilpartitionincreasesdrug
dissolution.
Diffusion Parameters
Diffusioncoefficient(D)
●
Representstherateatwhichadrugdiffuses
throughamedium.
●
Itdependonthetemperature,sizeof
molecule,theviscosityofthemediumand
mediumproperties.
●
Increasingtheviscosityofmediumwill
decreasethediffusioncoefficientandthus
dissolutionrate.
●
TypicallyfollowsFick’sFirstandSecond
LawsofDiffusion.

10. SinkConditions
●
Ifthedrugisrapidlyremovedfromthe
receivingside(e.g.,bybloodflow),it
maintainsaconcentrationgradientfor
continuousdiffusion.
DrugReleaseMechanisms
●
Matrixdiffusion:Drugdiffusesfroma
polymermatrix
●
Reservoirdiffusion:Drugdiffusesfroma
coresurroundedbyarate-controlling
membrane.
●
Osmoticandswelling-controlled:Water
influxcontrolsdrugreleaseviadiffusion.
Diffusion Parameters
Concentrationinbulksolution(Cb)
●
InvivoCbislowduetosinkconditionswhich
increasesrateofdissolution.Invitroaddition
ofsolventsandincreaseinvolumeof
dissolutionmediumincreasestherateof
dissolution.
Permeability(P)
●
Ameasureofhoweasilyadrugcrossesa
biologicalmembrane.
●
Influencedbydiffusioncoefficient,
membranethickness,andpartition
coefficient.

11. FormulationFactors
●
Polymertype:Affectsdrugdiffusionin
matrix-basedsystems.
●
Membranethickness:Thickermembranes
slowdrugrelease.
●
pH&Ionization:Non-ionizeddrugsdiffuse
moreeasilythroughlipidmembranes.
BiologicalFactors
●
Bloodflow:Highbloodflowmaintainsa
concentrationgradient.
●
Enzymaticdegradation:Somedrugs
degradebeforediffusing.
Diffusion Parameters
Factorsaffectingdiffusion
PhysicochemicalFactors
●
Molecularsize:Largermoleculesdiffuse
slower.
●
Solubility:Poorlysolubledrugshaveslower
diffusion.
●
Lipophilicity:Affectsmembranepermeability
(higherlipophilicity=fasterdiffusion).

12. ●
Tostudythedissolutionfroma planar
systemhavingahomogenousmatrix ,the
relationobtainedis:
●
Where,Qistheamountofdrugreleasedin
timetperunitarea,Cisthedruginitial
concentration,Csisthedrugsolubilityin
matrixmediaandDisthediffusivityofthe
drugmoleculesinthematrixsubstance.
Higuchi Equation
●
Higuchidevelopedtheoreticalmodelsto
studythereleaseofwatersolubleandlow
solubledrugsincorporatedinsemi-solid
and/orsolidmatrix.
●
Cumulative%drugreleasevs.squarerootof
time.
●
Drugreleasethroughdiffusionmechanism
(Fickiandiffusionmechanism)
●
Mathematicalexpressionswereobtainedfor
drugparticlesdispersedinauniformmatrix
behavingasthediffusionmedia.

13. ●
Forthecaseinwhichthedrugisdissolved
fromasaturatedsolution(whereCoisthe
solutionconcentration)dispersedina
porousmatrix.
●
Where,Qistheamountofdrugreleasedin
timetbysurfaceunity,εisthematrix
porosity,C
0
isthesolutionconcentration
dispersedinaporousmatrixandDthe
diffusionconstantofthedrugmoleculesin
thatliquid.
Higuchi Equation
●
Planarorsphericalsystemshavingagranular
(heterogeneous)matrix,wherethedrug
concentrationinthematrixislowerthanits
solubilityandthereleaseoccursthrough
poresinthematrixobtainedrelationis:
●
Where,QistheamountofDRintimetby
surfaceunity,Cistheinitialconcentrationof
thedrug,εisthematrixporosity,τisthe
tortuosityfactorofthecapillarysystem,C
S
isthedrugsolubilityinthematrix/excipient
mediaandDthediffusionconstantofthe
drugmoleculesinthatliquid.

14. Higuchi Plot
Higuchi Equation
●
AccordingtosimplifiedHiguchimodel
equationisasfollows:
●
Where,K
H
istheHiguchidissolutionconstant.
●
Higuchidescribesdrugreleaseasdiffusion
processbasedintheFick’slaw,squareroot
timedependent.
●
Thisrelationcanbeusedtodescribethedrug
dissolutionfromseveraltypesofmodified
releasepharmaceuticalsystems.

15. Applications
TheHiguchiequationiscommonlyusedto
model:
●
Transdermaldrugdeliverysystems
●
Matrixtablets
●
Microspheresorimplantswithdispersed
drug
●
Othercontrolled-releasesystemswhere
diffusionistheprimarymechanism
Higuchi Equation
Assumptions
●
Thedrugisuniformlydispersedinthematrix.
●
Thematrixisinsolubleanddoesnotswellor
erodeduringthereleaseprocess.
●
DrugreleaseiscontrolledbyFickiandiffusion
(i.e.,thedrugdiffusesoutofthematrix
followingaconcentrationgradient).
●
Thedrugconcentrationinthematrixismuch
higherthanitssolubilityinthematrix.

16. ModifiedHiguchiModels
Formorecomplexsystems,modifiedversions
oftheHiguchiequationhavebeendeveloped,
suchas:
●
Higuchi-Modelwitherosion:Accountsfor
matrixerosion.
●
Higuchi-Modelwithswelling:Incorporates
polymerswellingeffects.
●
Non-Fickiandiffusionmodels:Forsystems
wherediffusiondeviatesfromFickian
behavior.
Higuchi Equation
Limitations
●
Theequationassumesperfectsink
conditions(i.e.,thedrugconcentrationinthe
surroundingmediumisnegligiblecompared
tothematrix).
●
Itisnotapplicabletosystemswhereswelling,
erosion,orothermechanismsdominatedrug
release.
●
Themodelisvalidonlyfortheinitial60%of
drugrelease(forsystemswherethedrug
concentrationismuchhigherthanits
solubility).

17. ●
Effectofagitation
●
Effectofdissolutionfluid
●
InfluenceofpHofdissolutionfluid
●
Effectofsurfacetensionofthedissolution
medium
●
Effectofviscosityofthedissolution
medium
●
Effectofthepresenceofunreactiveand
reactiveadditivesinthedissolution
medium.
●
Volumeofdissolutionmediumandsink
conditions
●
Deaerationofthedissolutionmedium
●
Effectoftemponthedissolutionmedium
Dissolution Parameters
●
Dissolutionistheprocessinwhichasolid
substancesolubilizesinagivensolvent.
●
Masstransferfromthesolidsurfacetothe
liquidphase.

18. ●
Dissolutiontestsusinghigh-speedagitation
maylackdiscriminativevalueandcanyield
misleadingresults.
●
Thelowestvalue(25rpm)ischaracteristic
forsuspensions.
●
Incompendialmethodsagitationspeedis
relativelylow.
●
Forthebasketmethod100rpmandfor
paddle50-75rpm.
Dissolution Parameters
Effectofagitation
●
Therelationshipbetweentheintensityof
agitationandtherateofdissolutiondiffers
considerablyaccordingtothetypeof
agitationused,degreeoflaminarand
turbulentflowinthesystem,theshapeand
designofthestirrer,andthephysicochemical
propertiesofthesolid.
●
Speedofagitationgeneratesaflowthat
continuouslychangestheliquid/solid
interfacebetweensolventanddrug.
●
Tosustainareproduciblelaminarflow,which
isessentialforgainingreliableresults,
agitationshouldbemaintainedatarelatively
lowrate.

19. ●
Dissolutionfluidwithexample
Dissolution Parameters
EffectofDissolutionFluid
●
Theselectionofapropermediumfor
dissolutiontestingdependslargelyonthe
physicochemicalpropertiesofthedrug.
●
Themediatypicallyusedindissolution
studiesincludeacidicsolutions,buffers,
surfactants,andsurfactantswithacidor
buffers.
●
Surfaceactiveagentsareusedindissolution
testmethodstoimprovethesolubilityor
wettabilityofadrug.
ExampleDissolution Fluid
Ampicillin CapsuleWater
Azithromycin CapsuleBuffers
PiroxicamCapsuleSimulated gastric fluid
Cimetidine tabletHClsolution

20. ●
Fortabletscontainingactiveingredients,
whosesolubilityisindependentofpH,the
dissolutionratedoesnotvaryconsiderably
withchangesinpHofthedissolution
mediumunlesstheycontaincertain
excipientsthatareinfluencedbypH.
●
Sodiumbicarbonate,magnesiumcarbonate,
calciumcarbonatepromotesdisintegration
oftabletinacidicmediumbyproducinggas.
Dissolution Parameters
InfluenceofpHofdissolutionfluid
●
VariationsinpHexertthegreatesteffectin
termsofdrugsolubility.
●
Forweakacids,therateofdissolution
increaseswithincreasingpH,whereas,for
weakbases,therateofdissolutionincreases
withdecreasingpH.
●
pHofstomachis~2.AcetylsalicylicacidpKa
is3.5.

21. ●
TheadditionofsurfactantbelowtheCMC
canincreasesignificantlythedissolution
ratebecauseofbetterpenetrationofthe
solventintothetabletresultingingreater
availabilityofthedrugsurface.
●
Dissolutiondataforbenzocaineindifferent
concentrationsofpolysorbate80.
Dissolution Parameters
Effectofsurfacetensionofthedissolutionfluid
●
Surfacetensionshowsasignificanteffecton
thedissolutionrateofdrugsandtheirrelease
ratefromsoliddosageforms.
●
Surfactantsandwettingagentslowerthe
contactangleand,consequently,improve
penetrationbythedissolutionmedium.
●
Theincorporationofsurface-activeagentsin
thedissolutionmediumisexpectedto
enhancethedissolutionrateofapoorly
solubledruginsoliddosageformsby
decreasingtheinterfacialtensionandmicelle
formation.

22. ●
Relationshipofviscositytodissolutionrate
ofbenzoicacidinaqueousmethylcellulose
solutionsat25°.
Dissolution Parameters
Effectofviscosityofthedissolutionfluid
●
Incaseofdiffusion-controlleddissolution
processes,itwouldbeexpectedthatthe
dissolutionratedecreaseswithanincreasein
viscosity.
●
Inthecaseofinterfacial-controlled
dissolutionprocesses,however,viscosity
shouldhavelittleeffect.
●
TheStokes-Einsteinequationdescribes
diffusioncoefficient,D,asafunctionof
viscosity,
D=µkT
●
Where,TisTemp.,µisthemobility(velocity
ataforceofonedyne);kistheBoltzmann
constant.

23. Volumeofdissolutionmediumandsink
condition
●
Thesuitablevolumeofthedissolution
mediumdependsmainlyonthesolubilityof
thedrugintheselectedfluid.
●
Ifthedrugispoorlysolubleinwater,a
reasonablylargeamountoffluidshouldbe
used.
●
Tominimizetheeffectoftheconcentration
gradientandmaintainsinkconditions,the
concentrationofthedrugshouldnotexceed
10–15%ofitsmaximumsolubilityinthe
dissolutionmediumselected.Volume
generally500ml,900mland1000mlused.
Dissolution Parameters
Effectofthepresenceofunreactiveand
reactiveadditivesinthedissolutionmedium
●
Whenneutralioniccompoundssuchas
SodiumChlorideandSodiumSulfateor
nonionicorganiccompoundssuchas
Dextrosewereaddedtothedissolution
mediumthebenzoicacidsolubilitywas
directlydependentonitssolubilityina
particularsolvent.
●
Whencertainbuffersorbaseswereaddedto
theaqueoussolvent,anincreaseinthe
dissolutionratewasobserved.

24. ●
Thisinhibitswettingandlowersthe
dissolutionrate.
●
Somedrugproductsareknowntobe
tremendouslysensitivetodissolvedgas,the
presenceofairbubblesshouldbeexpected
toincreasethemeasurementuncertaintyin
dissolutiontesting.
●
InUSPApparatus2,releasedairbubbles
depositonthepaddleshaft,thereleaseof
airbubblesaltersthehydrodynamicsofthe
systembychangingthefluidflow
characteristicsinthedissolutionvessel.
Dissolution Parameters
Deaerationofthedissolutionmedium
●
Thepresenceofdissolvedairorothergases
inthedissolutionmediummayimpactthe
dissolutionrateofcertainformulationsand
leadtovariableandunreliableresults.
●
Solubleairindistilledwatercansignificantly
loweritspHandasaresult,affecttherateof
dissolutionofpH-sensitivedrugs.
●
Anothersevereeffectisthetendencyofthe
dissolvedairtobereleasedfromthemedium
informofatinyairbubble.
●
Thesebubblescollectatthesurfaceofthe
dosageformtherebyactingasahydrophobic
barrierbetweensolventandsolidsurface.

25. ●
Foradissolvedmolecule,thediffusion
coefficient,D,dependsonthetemperature
T,accordingtotheStokesequation:
●
WherekistheBoltzmannconstantand6πηr
istheStokesforceforasphericalmolecule
(ηistheviscosityincgsorpoiseunits,andr
istheradiusofthemolecule).
Dissolution Parameters
Effectoftemperatureofthedissolutionmedium
●
Becausedrugsolubilityistemperature-
dependent,carefultemperaturecontrol
duringthedissolutionprocessisvery
importantandshouldbemaintainedwithin
0.5°.
●
Generally,atemperatureof37°Cisalways
maintainedduringdissolution
determinations.
●
Theeffectoftemperaturevariationsofthe
dissolutionmediumdependsmainlyonthe
temperature/solubilitycurvesofthedrugand
excipientsintheformulation.

26. ●
Releaseprofilebydiminishingsurfaceof
thedrugparticlesduringthedissolution.
●
ItgivesErosionreleasemechanism.
●
Appliestopharmaceuticaldosageforms
suchastablets,wherethedissolution
occursinplanesthatareparalleltothedrug
surfaceifthetabletdimensionsdiminish
proportionallyinsuchamannerthatthe
initialgeometricalformkeepsconstantall
thetime.
Hixson and CrowellsCube Root Law
●
Particleregularareaisproportionaltothe
cubicrootofitsvolumeandequationis
expressasfollows:
●
Where,W
0
istheinitialamountofdruginthe
pharmaceuticaldosageform.W
t
isthe
amountofdrugremainingasasolidstateat
timet.K
S
istheconstantincorporationthe
surfacevolumerelation.
●
Cuberootofdrug%remaininginmatrixvs.
time
●
Drugreleaseislimitedbythedissolutionrate
oftheparticles,andnotbydiffusionthrough
thepolymermatrix.

27. ●
Peakplasmaconcentration(C
max
)
●
Timeofpeakconcentration(t
max
)
●
Areaunderthecurve(AUC)
Pharmacokinetic Parameters
●
Pharmacokineticsisdefinedasthekinetics
ofdrugabsorption,distribution,metabolism
andexcretionandtheirrelationshipwith
pharmacologic,therapeuticortoxicologic
responseinmansandanimals.

28. ●
Plasmaconcentrationtimeprofileindicating
C
max
Pharmacokinetic Parameters
Peakplasmaconcentration(C
max
)
●
Itisthemaximumplasmadrugconcentration
obtainedafteroraladministrationofdrug.
●
Thepointofmaximumconcentrationofdrug
inplasmaisknownasthepeakandthe
concentrationofdrugatpeakisknownas
peakplasmaconcentration.
●
Cmaxisexpressedinµg/mlormg/L.
●
Itisrelatedwithintensityofaction.

29. ●
Plasmaconcentrationtimeprofileindicating
t
max
Pharmacokinetic Parameters
TimeofPeakConcentration(t
max
)
●
Itisthetimerequiredtoreachmaximumdrug
concentrationinplasmaafterextravascular
drugadministration.
●
Itisusefulinestimatingtherateof
absorption.
●
Itisexpressedinhours.
●
Onsettimeandonsetofactiondependson
timeofpeakconcentration.
●
Parameterisofparticularimportancein
assessingtheefficacyofdrugsusedtotreat
acuteconditionslikepainandinsomnia.

30. ●
Plasmaconcentrationtimeprofileindicating
AUC
Pharmacokinetic Parameters
AreaUnderCurve(AUC)
●
Theareaundertheplasmadrug
concentrationversustimecurve.
●
AUCisameasurementoftheextentofdrug
bioavailability.
●
TheAUCreflectsthetotalamountofactive
drugthatreachesthesystemiccirculation.
●
AUCisexpressedinmcg*hour/ml.
●
Itisimportantforthedrugsthatare
administeredrepetitivelyforthetreatmentof
chronicconditionslikeasthmaorepilepsy.

31. Disadvantages
●
Thevaluesoff1andf2aresensitivetothe
numberofdissolutiontimepointsused.
●
Iftestandreferenceformulationsare
interchanged,f2isunchangedbutf1isnot,
yetdifferencesbetweenthetwomean
profilesremainthesame.
●
Thebasisofthecriteriafordecidingthe
differenceorsimilaritybetweendissolution
profilesisunclear.
Similarity Factor
Modelindependentanalysis(f1&f2Factor)
●
Mostpopularmethodsrecommendedforuse
inanumberofFDAguidancedocuments.
●
Differencefactor(f1)
●
Similarityfactor(f2)
Advantages
●
Easytocompute.
●
Provideasinglenumbertodescribethe
comparisonofdissolutionprofiledata.

32. Similarityfactor(f2)
●
Logarithmictransformationofthesum-
squarederrorofdifferencesbetweenthe
testandthereferenceproductsoveralltime
points.
●
Where,Rjisthepercentageofdissolved
productforareferencebatchattimepointt,
Tjisthepercentageofdissolvedproductfor
thetestbatch,Nisthenumberoftime
pointsandwjisanoptionalweightfactor,If
eachtimepointisweightedequallythenit
meanswj=1.
Similarity Factor
Differencefactor(f1)
●
Thedifferencefactor(f1)measuresthe
percenterrorbetweentwocurvesoverall
timepoints
●
Where,Rjisthepercentageofdissolved
productforareferencebatchattimepointt,
Tjisthepercentageofdissolvedproductfor
thetestbatch,Nisthenumberoftimepoints.

33. ●
Comparison
Similarity Factor
●
Forrapiddissolvingproducts,thatmay
dissolve85%in15minutes,comparisonof
dissolutionprofilesisnotmandatory
Similarity factorDifference factor
It is inversely
proportional to the
average squared
difference between the
two profiles with
emphasis on the
larger difference
among all the time
points
It is proportional to the
average difference
between the two
profiles
Measures similarityMeasures the
closeness between
the two profiles
InferenceF2F1
Dissolution profilesare similar1000
Similarity or equivalence of two
profiles
>50< 15

34. Applications
●
NewANDAsubmission
●
Filingvariationtotheoriginalsubmission
CBEfilings
●
Formulachanges
●
Towaivetherequirementofbioequivalence
forsmallerstrengthsofformulation
●
Sitetransfer
●
Scaleupofthelots
●
Regulatorysubmission
Similarity Factor
DataStructureandstepstofollow
●
Thismodelindependentmethodismost
suitableforthedissolutionprofile
comparisonswhenthreetofourormore
dissolutiontimepointsareavailable.
●
Determinethedissolutionprofileoftwo
products(12unitseach)oftest(post-
change)andreference(pre-change)products
●
Usingmeandissolutionvaluesfromboth
curvesateachtimeinterval,calculatethe
differencefactorandsimilarityfactor.
●
Forcurvestobeconsideredsimilar,iff1
valuescloseto0andf2valuescloseto100.

35. ●
Thismodelisusedtoanalyzethereleaseof
dosageformswhenreleasemechanismis
notwellknownorwhenmorethanonetype
ofreleasephenomenacouldbeinvolved.
Korsmeyer–PeppasPlot
●
Itasimple,semiempiricalmodel,relating
exponentiallythedrugreleasetotheelapsed
time(t):
●
Accordingtothismodel,drugdissolutionis
functionoftheexponentoftime,definedas
‘n’intheformula.
●
Where,nisindicativeofdrugrelease
mechanism;tistime,kisrateconstant;M
0
/M

isfractionalreleaseofdrug.

36. ●
Innonlinearitisnotpossibletodetermine
othervariablevalue.
●
Inlinearcaseifwehaveonevariablevalue
itspossibletodetermineothervariable
value.
Linearity
●
Linearityisamathematicalrelationship
betweentwovariablesquantities(theymay
besameunit),whicharedirectlyproportional
toeachother.
●
Graphicallyitrepresentsastraightlinewhen
plottedagainsteachother.

37. ●
Thecloserristozero,theweakerthelinear
relationship.
●
Inpositivecorrelation,thevaluesofboth
variablestendtoincreasetogether.
●
Negativecorrelation,thevaluesofone
variabletendtoincreasewhenthevaluesof
theothervariabledecrease.
●
Thevalues1and-1bothrepresent"perfect"
correlations,positiveandnegative
respectively.
●
Twoperfectlycorrelatedvariableschange
togetheratafixedrate.Theyhavealinear
relationship;whenplottedonascatterplot,
alldatapointscanbeconnectedwitha
straightline.
Linearity
●
Thestrengthanddirectionofthelinear
relationshipbetweentwovariablequantities
istermedascorrelationcoefficient(r)(-1to
1).
●
Coefficientofdetermination(r2)rangesform
0-1.Itdenoteslinearityofline/twovariables.

38. Linear Equation
A linear equation in two variables can be
written as:
Y=mX+C
Where:
Y= dependent variable,
x= independent variable,
m= slope (rate of change),
C= y-intercept (value ofywhenx=0).
Linearity
Key Concepts
Mathematical Definition
●
A functionf(x)is linear if it satisfies the
following two properties:
●
Additivity:f(x+y)=f(x)+f(y)
●
Homogeneity:f(ax)=af(x), whereais a
constant.
●
Thesepropertiesensurethatthe
relationshipbetweenvariablesis
proportionalandadditive.

39. Linearity in Statistics
●
In statistics, linearity often refers to the
relationship between two variables in a
dataset.
●
A linear relationship can be assessed
using a scatterplot and quantified using
the correlation coefficient (r), which
measures the strength and direction of
the linear relationship.
Linearity
Graphical Representation
●
A linear relationship is represented by a
straight line on a graph.
●
The slope (m) determines the steepness
and direction of the line:
○
Positive slope:yincreases
asxincreases.
○
Negative slope:ydecreases
asxincreases.
○
Zero slope:yremains constant
asxchanges.

40. Linearity
Assumptions of Linearity
●
In regression analysis, linearity is a key
assumption for linear models.
●
It assumes that the relationship between
the independent variable(s) and the
dependent variable is linear.

41. Residual Analysis
●
In regression analysis, plot the
residuals (errors) against the predicted
values.●
If the residuals are randomly scattered
around zero, the relationship is likely
linear.
Hypothesis Testing
●
Use statistical tests (e.g., F-test) to
determine if a linear model fits the data
better than a nonlinear model.
Linearity
Testing for Linearity
Visual Inspection
Plot the data on a scatterplot and check if
the points roughly form a straight line.
Correlation Coefficient (r)
●
Calculaterto measure the strength and
direction of the linear relationship.
●
rranges from -1 to 1:
○
r=1: Perfect positive linear relationship.
○
r=−1: Perfect negative linear
relationship.
○
r=0: No linear relationship.

42. ●
Linearityensuresthattherelationship
betweentheresponseand
concentrationisproportional,whichis
essentialforaccuratequantification.●
Regulatoryguidelines(e.g.,ICHQ2)
requirelinearitytobedemonstrated
duringmethodvalidation.
●
Example:InHigh-PerformanceLiquid
Chromatography(HPLC),alinear
calibrationcurveisusedtodetermine
theconcentrationofanactive
pharmaceuticalingredient(API).
Significance of Linearity
Analytical Method Validation
●
Analyticalmethodsareusedtomeasure
theconcentrationofdrugs,impurities,
andothersubstancesinpharmaceutical
products.
●
Acalibrationcurveisaplotofthe
instrumentresponse(e.g.,absorbance,
peakarea)versustheconcentrationof
theanalyte.

43. Bioavailability and Bioequivalence Studies
●
Thesestudiescomparetherateand
extentofdrugabsorptionfromdifferent
formulations.
●
Therelationshipbetweendrug
concentrationinthebloodandtimeis
oftenlinearincertainphases(e.g.,
eliminationphase).Thislinearity
simplifiesthecalculationof
pharmacokineticparameterslikehalf-life,
clearance,andvolumeofdistribution.
●
Example:Alinearpharmacokineticprofile
ensuresthatdoublingthedoseresultsin
aproportionalincreaseindrugexposure.
Significance of Linearity
Dose-Response Relationships●
Understandingtherelationshipbetween
drugdoseanditseffectiscriticalfor
determiningthetherapeuticwindow.
●
Inmanycases,thedose-response
relationshipislinearwithinaspecificrange.
●
Thislinearityhelpsinpredictingtheeffect
ofdifferentdosesandoptimizingdrug
efficacyandsafety.
●
Example:Alineardose-responsecurvecan
helpdeterminetheminimumeffectivedose
andthemaximumtolerateddose.

44. QualityControlandManufacturing
●
Qualitycontrolensuresthat
pharmaceuticalproductsmeetpredefined
standards.
●
Linearityiscriticalfordevelopingassays
thatmeasurethepotencyofdrugsand
theconcentrationofimpurities.
●
Alinearrelationshipbetweentheassay
responseandanalyteconcentration
ensuresaccurateandreproducible
results.
●
Example:InUV-Visspectroscopy,linearity
isusedtoquantifytheconcentrationofan
APIinatabletorsolution.
Significance of Linearity
StabilityTesting
●
Stabilitytestingensuresthatadrugproduct
maintainsitsquality,efficacy,andsafety
overtime.
●
Thedegradationofadrugovertime(e.g.,
duetotemperature,humidity)isoften
linear,allowingforthepredictionofshelf
life.
●
Linearityindegradationkineticssimplifies
themodelingofstabilitydata.
●
Example:Alineardegradationcurvecanbe
usedtoestimatetheexpirationdateofa
drugproduct.

45. PharmacokineticModeling
●
Pharmacokineticmodelsdescribehowa
drugmovesthroughthebody.
●
Linearpharmacokineticsassumesthat
parameterslikeclearanceandvolumeof
distributionareconstant,simplifyingthe
modelingprocess.
●
Thisassumptionholdstrueformany
drugsattherapeuticdoses.
●
Example:Alinearpharmacokineticmodel
canpredictdrugconcentrationsinthe
bodyovertime,aidingindosingregimen
design.
Significance of Linearity
RegulatoryCompliance
●
Regulatoryagencies(e.g.,FDA,EMA)
requireevidenceoflinearityinanalytical
methodsandstudies.
●
Linearityisoneofthekeyparameters
assessedduringmethodvalidation,asper
guidelineslikeICHQ2(R1).
●
Demonstratinglinearityensuresthatthe
methodissuitableforitsintendedpurpose
andmeetsregulatorystandards.
●
Example:Avalidatedlinearmethodis
requiredforthereleasetestingofdrug
products.

46. DissolutionTesting
●
Dissolutiontestingmeasurestherateat
whichadrugdissolvesinamedium.
●
Alinearrelationshipbetweendissolution
rateandtimeisoftenobserved,
simplifyingtheanalysisofdrugrelease.
●
Thislinearityhelpsincomparingdifferent
formulationsandensuringbatch-to-batch
consistency.
●
Example:Alineardissolutionprofilecan
beusedtooptimizetheformulationofa
tablet.
Significance of Linearity
ImpurityProfiling
●
Impuritiesindrugproductsmustbe
identifiedandquantifiedtoensuresafety.
●
Linearityisessentialfordeveloping
methodstoquantifyimpuritiesatlow
concentrations.
●
Alinearrelationshipensuresaccurate
measurementofimpurities,evenattrace
levels.
●
Example:Ingaschromatography(GC),
linearityisusedtoquantifyresidual
solventsinadrugproduct.

47. Thesignificanceoflinearityinthe
pharmaceuticalfieldliesinitsabilityto:
●
Ensureaccurateandreliableanalytical
methods.
●
Simplifypharmacokineticand
pharmacodynamicmodeling.
●
Supportregulatorycomplianceandquality
control.
●
Optimizedrugdevelopmentand
manufacturingprocesses.
●
Reducecostsandimproveefficiency.
SignificanceofLinearity
CostandTimeEfficiency
●
Purpose:Linearitysimplifiesdataanalysis,
reducingthetimeandcostof
pharmaceuticalresearchanddevelopment.
●
Linearmodelsarecomputationallyefficient
andrequirefewerresourcescomparedto
nonlinearmodels.
●
Thisefficiencyisparticularlyimportantin
high-throughputscreeningandlarge-scale
manufacturing.
●
Example:Alinearcalibrationcurvecanbe
quicklygeneratedandvalidated,speeding
uptheanalyticalprocess.

48. ●
Inthisformula,σisthestandarddeviation,
x1isthedatapoint,µisthemean,andNis
thetotalnumberofdatapoints.
Standard Deviation
●
Astandarddeviation(orσ)isameasureof
howdispersedthedataisinrelationtothe
mean.
●
StandardDeviationisastatisticaltermused
tomeasuretheamountofvariabilityor
dispersionaroundanaverage.
●
Lowstandarddeviationmeansdataare
clusteredaroundthemean,andhigh
standarddeviationindicatesdataaremore
spreadout.
●
Astandarddeviationclosetozeroindicates
thatdatapointsareclosetothemean,
whereasahighorlowstandarddeviation
indicatesdatapointsarerespectivelyabove
orbelowthemean.

49. AnalyticalMethodValidation
●
Validateanalyticalmethodstoensurethey
areaccurate,precise,andreliable.
●
Standarddeviationisusedtoassessthe
precisionofananalyticalmethod.
●
Precisionismeasuredbyanalyzing
replicatesamplesandcalculatingthe
standarddeviationoftheresults.
●
Lowstandarddeviationindicateshigh
precision,whichiscriticalforreliable
measurements.
●
Example:Inassayvalidation,thestandard
deviationofrepeatedmeasurementsofa
drug'sconcentrationisusedtoconfirm
methodprecision.
SignificanceofStandard Deviation
QualityControlandManufacturing
●
Ensurethatpharmaceuticalproductsare
consistentlyproducedtomeetpredefined
qualitystandards.
●
Standarddeviationisusedtomonitorthe
variabilityincriticalqualityattributes(e.g.,
drugpotency,tabletweight,dissolutionrate).
●
Lowstandarddeviationindicatesconsistent
productquality,whilehighstandarddeviation
maysignalprocessvariabilityordefects.
●
Example:Intabletmanufacturing,the
standarddeviationoftabletweightsis
monitoredtoensureuniformity.

50. Batch-to-BatchConsistency
●
Ensurethatdifferentbatchesofadrug
productareconsistentinquality.
●
Standarddeviationisusedtocomparethe
variabilityofkeyparameters(e.g.,drug
content,dissolutionrate)acrossbatches.
●
Lowstandarddeviationacrossbatches
indicatesconsistentmanufacturing
processes.
●
Example:ThestandarddeviationofAPI
(activepharmaceuticalingredient)content
inmultiplebatchesismonitoredtoensure
consistency.
SignificanceofStandard Deviation
StabilityTesting
●
Ensurethatdrugproductsmaintaintheir
quality,efficacy,andsafetyovertime.
●
Standarddeviationisusedtoanalyze
variabilityinstabilitydata(e.g.,drugpotency
overtime).
●
Ithelpsidentifytrendsoroutliersthatmay
indicatedegradationorinstability.
●
Example:Thestandarddeviationofpotency
measurementsduringacceleratedstability
studieshelpspredictthedrug'sshelflife.

51. ClinicalTrials
●
Evaluatethesafetyandefficacyofnew
drugs.
●
Standarddeviationisusedtoanalyze
variabilityinclinicaltrialdata(e.g.,patient
responses,adverseevents).
●
Ithelpsdeterminethestatistical
significanceofresultsandthereliabilityof
conclusions.
●
Example:Thestandarddeviationofblood
pressurereductioninahypertensiondrug
trialhelpsassessthedrug'sefficacy.
SignificanceofStandard Deviation
PharmacokineticandPharmacodynamicStudies
●
Understandhowadrugbehavesinthebody
(absorption,distribution,metabolism,
excretion)anditseffects.
●
Standarddeviationisusedtoquantify
variabilityinpharmacokineticparameters
(e.g.,half-life,clearance)and
pharmacodynamicresponses(e.g.,efficacy,
sideeffects).
●
Ithelpsassessthereliabilityofstudyresults
andthepredictabilityofdrugbehavior.
●
Example:Thestandarddeviationofplasma
concentrationmeasurementsinaclinicaltrial
helpsevaluatethedrug'spharmacokinetic
profile.

52. RegulatoryCompliance
●
Meetregulatoryrequirementsfordrug
approvalandqualityassurance.
●
Regulatoryagencies(e.g.,FDA,EMA)
requirethereportingofstandarddeviation
inanalyticalandmanufacturingdata.
●
Itisakeyparameterindemonstrating
processcontrolandproductconsistency.
●
Example:Standarddeviationisincludedin
regulatorysubmissionstosupportthe
validationofanalyticalmethods.
SignificanceofStandard Deviation
ProcessOptimization
●
Improvemanufacturingprocessesto
enhanceefficiencyandproductquality.
●
Standarddeviationisusedtoidentifysources
ofvariabilityinmanufacturingprocesses.
●
Reducingstandarddeviationthroughprocess
optimizationleadstomoreconsistentand
higher-qualityproducts.
●
Example:Inagranulationprocess,the
standarddeviationofparticlesizedistribution
ismonitoredtooptimizetheprocess.

53. StatisticalProcessControl(SPC)
●
Purpose:Monitorandcontrolmanufacturing
processestoensurequality.
●
Standarddeviationisakeycomponentof
controlchartsusedinSPC.
●
Ithelpsdetectprocessvariabilityand
deviationsfromthetarget,enablingtimely
correctiveactions.
●
Example:Acontrolchartfortablethardness
usesstandarddeviationtoidentifyout-of-
specificationbatches.
SignificanceofStandard Deviation
RiskAssessment
●
Purpose:Identifyandmitigaterisksindrug
developmentandmanufacturing.
●
Standarddeviationisusedtoquantify
variabilityinriskfactors(e.g.,rawmaterial
quality,environmentalconditions).
●
Highstandarddeviationmayindicateahigher
riskofproductfailureorinconsistency.
●
Example:Thestandarddeviationofimpurity
levelsinrawmaterialsisassessedto
evaluatetheriskofcontamination.

54. ●
Ifthereisadifferencebetweenthe
observedandtheexpectedfrequencies
thenthevalueofChisquarewouldbemore
than0.
●
ThelargertheChi-squarethegreaterthe
probabilityofarealdivergenceof
experimentallyobservedfromexpected
results.
●
Thisstatisticaltestfollowsaspecific
distributionknownaschisquare
distribution.
Chi square test
●
Achi-squaretestisastatisticaltestusedto
compareobservedresultswithexpected
results.
●
Thepurposeofthistestistodetermineifa
differencebetweenobserveddataand
expecteddataisduetochance,orifitisdue
toarelationshipbetweenthevariablesyou
arestudying.
●
Thus,Chi-squareisameasureofactual
divergenceoftheobservedandexpected
frequencies.
●
Ifthereisnodifferencebetweenexpected
andobservedfrequenciesthevalueofChi-
squareis0.

55. UsesofChi-SquareTest
●
Althoughtestisconductedintermsof
frequenciesitcanbebestviewed
conceptuallyasatestaboutproportions.
●
Χ
2
testisusedintestinghypothesisandis
notusefulforestimation.
●
Chi-squaretestcanbeappliedtocomplex
contingencytablewithseveralclasses.
●
Chi-squaretesthasaveryusefulproperty
i.e.,‘theadditiveproperty’.Ifanumberof
samplestudiesareconductedinthesame
field,theresultscanbepooledtogether.
Thismeansthatχ2valuescanbeadded.
Chi square test
Theapplicationsofχ2
●
Testingthedivergenceofobservedresults
fromexpectedresultswhenourexpectations
arebasedonthehypothesisofequal
probability.
●
Chi-squaretestwhenexpectationsarebased
onnormaldistribution.
●
Chi-squaretestwhenourexpectationsare
basedonpredeterminedresults.
●
CorrectionfordiscontinuityorYates’
correctionincalculatingχ
2
●
Chi-squaretestofindependencein
contingencytables.

56. AdverseEventAnalysis
●
Monitorandanalyzethesafetyprofileof
drugs.
●
TheChi-squaretestisusedtodetermineif
thereisasignificantdifferenceinthe
frequencyofadverseeventsbetween
differentpatientgroupsordrug
formulations.
●
Example:TheChi-squaretestcancompare
theincidenceofnauseainpatientstaking
DrugAversusDrugB.
Significance of Chi square test
ClinicalTrials
●
Evaluatetheefficacyandsafetyofnew
drugs.
●
TheChi-squaretestisusedtoanalyze
categoricaldata,suchasthenumberof
patientsrespondingtoatreatmentversusa
placebo.
●
Ithelpsdetermineifthereisasignificant
associationbetweenthetreatmentandthe
outcome(e.g.,success/failure,adverse
events).
●
Example:Inaclinicaltrialforanewdrug,the
Chi-squaretestcancomparetheproportion
ofpatientsexperiencingsideeffectsinthe
treatmentgroupversusthecontrolgroup.

57. BioequivalenceStudies
●
Comparethebioavailabilityoftwodrug
formulationstoensuretheyareequivalent.
●
TheChi-squaretestcanbeusedtoanalyze
categoricaldata,suchastheproportionof
subjectsachievingacertainplasma
concentrationlevelfortwoformulations.
●
Example:TheChi-squaretestcancompare
theproportionofsubjectsreachingthe
therapeuticthresholdforagenericdrug
versusthebrand-namedrug.
Significance of Chi square test
QualityControl
●
Ensurethatpharmaceuticalproductsmeet
predefinedqualitystandards.
●
TheChi-squaretestisusedtocompare
observedfrequenciesofdefectsordeviations
inproductbatcheswithexpected
frequencies.
●
Ithelpsidentifywhethervariationsinquality
areduetorandomchanceorsystematic
issues.
●
Example:TheChi-squaretestcananalyzethe
distributionofdefectivetabletsindifferent
manufacturingbatches.

58. MarketResearchandDrugAdoption
●
Analyzetheadoptionandpreferenceof
drugsamonghealthcareprovidersor
patients.
●
TheChi-squaretestisusedtoanalyze
surveydataanddetermineifthereisa
significantassociationbetweencategorical
variables,suchasdrugpreferenceand
demographicfactors.
●
Example:TheChi-squaretestcananalyze
whetherthepreferenceforanewdrug
formulationvariesbygeographicregion.
Significance of Chi square test
PatientDemographicsandSubgroupAnalysis
●
Understandhowdifferentpatientgroups
respondtoadrug.
●
TheChi-squaretestisusedtoanalyze
associationsbetweencategoricalvariables,
suchasagegroups,gender,orethnicity,and
treatmentoutcomes.
●
Example:TheChi-squaretestcandetermine
ifthereisasignificantdifferenceindrug
efficacybetweenmaleandfemalepatients.

59. GenomicandPharmacogeneticStudies
●
Understandthegeneticfactorsinfluencing
drugresponse.
●
TheChi-squaretestisusedtoanalyze
associationsbetweengeneticmarkers(e.g.,
SNPs)anddrugefficacyortoxicity.
●
Example:TheChi-squaretestcandetermine
ifaspecificgeneticvariantisassociated
withanincreasedriskofadversedrug
reactions.
Significance of Chi square test
RegulatoryCompliance
●
Ensurethatpharmaceuticalproductsmeet
regulatoryrequirements.
●
TheChi-squaretestisusedtoanalyzedata
submittedtoregulatoryagencies,suchasthe
frequencyofadverseeventsorthe
distributionofproductdefects.
●
Example:TheChi-squaretestcanbeusedto
demonstratethattheobserveddistributionof
adverseeventsinaclinicaltrialisconsistent
withexpectedvalues.

60. ProcessValidation
●
Ensurethatmanufacturingprocesses
consistentlyproducehigh-qualityproducts.
●
TheChi-squaretestisusedtocompare
observedprocessoutcomes(e.g.,pass/fail
rates)withexpectedoutcomes.
●
Example:TheChi-squaretestcananalyze
thedistributionofdefectsina
manufacturingprocesstovalidateits
consistency.
Significance of Chi square test
Post-MarketingSurveillance
●
Purpose:Monitorthesafetyand
effectivenessofdrugsaftertheyare
approvedandmarketed.
●
TheChi-squaretestisusedtoanalyzedata
frompost-marketingstudies,suchasthe
frequencyofadverseeventsinreal-world
populations.
●
Example:TheChi-squaretestcancompare
theincidenceofsideeffectsindifferentage
groupsusingadruginreal-worldsettings.

61. ●
Thisisparametrictestusedtocompare
samplesformtwodifferentbatches.
●
Itisusuallyusedwithsmall(<30)samples
thatarenormallydistributed.
●
ttestdependsonthepropertiesofnormal
distributioncurves.
Student’s t test
●
TheStudent'st-testisastatisticalmethod
usedtocomparethemeansoftwogroupsto
determineiftheyaresignificantlydifferent
fromeachother.
●
Itisoneofthemostcommonlyused
hypothesistestsinstatisticsandis
particularlyusefulwhenworkingwithsmall
samplesizesorwhenthepopulation
standarddeviationisunknown.

62. Typesoft-tests:
●
Singlesamplettest:onlyonegrouptested
againstahypotheticalmean.
●
Independentsamplettest:Twogroups,two
means,norelationbetweengroups.
Exampletestdrugwithplacebo.
●
Pairedttest:Samplesofmatchedpairsof
similarunitsorgroupofunitstestedtwice.
Examplebeforeandaftertreatmenteffect.
Student’s t test
ConceptsoftheStudent'st-test
Purpose
●
Thet-testassesseswhetherthedifference
betweentwogroupmeansisstatistically
significantorduetorandomvariation.

63. Hypotheses:
●
NullHypothesis(H₀):Thereisnodifference
betweenthemeansofthetwogroups(μ₁=
μ₂).
●
AlternativeHypothesis(H₁):Thereisa
differencebetweenthemeansofthetwo
groups(μ₁≠μ₂foratwo-tailedtest,orμ₁>
μ₂orμ₁<μ₂foraone-tailedtest).
Student’s t test
Assumptions:
●
Normality:Thedataineachgroupshouldbe
approximatelynormallydistributed
(especiallyimportantforsmallsamplesizes).
●
Independence:Observationsineachgroup
shouldbeindependentofeachother.
●
Homogeneityofvariance(forindependentt-
test):Thevariancesofthetwogroupsshould
beequal(thisassumptioncanberelaxed
usingWelch'st-test).

64. Interpretation:
●
Comparethecalculatedt-valuetothe
criticalt-valuefromthet-distributiontable
(basedondfandα).
●
Alternatively,comparethep-valuetothe
significancelevel(α,usually0.05).
●
Ifp<α,rejectthenullhypothesis
(significantdifference).
●
Ifp≥α,failtorejectthenullhypothesis(no
significantdifference).
Student’s t test
TestStatistic(t-value):
●
Thet-valueiscalculatedas:
t=Differencebetweengroupmeans/Standard
errorofthedifference
●
Theformulavariesslightlydependingonthe
typeoft-test.
DegreesofFreedom(df):
The degrees of freedom depend on the type of
t-test:
●
Independent t test:df=n1 + n2 −2df = n1 +
n2 − 2
●
Paired t-test:df= n−1 (wherenis the
number of pairs).

65. ●
DeterminetheCriticalValueorp-value:Use
at-distributiontableorstatisticalsoftware
tofindthecriticalt-valueorp-value.
●
MakeaDecision:Comparethet-valuetothe
criticalvalueorthep-valuetoα.Rejector
failtorejectthenullhypothesis.
●
InterprettheResults:Explainthefindingsin
thecontextoftheresearchquestion.
Student’s t test
StepstoPerformat-test
●
StatetheHypotheses:Definethenulland
alternativehypotheses.
●
ChoosetheTypeoft-test:Decidewhetherto
useanindependent,paired,orone-samplet-
testbasedonthestudydesign.
●
CheckAssumptions :Verifynormality,
independence,andhomogeneityofvariance
(ifapplicable).
●
Calculatethet-statistic:Usetheappropriate
formulaforthechosent-test.

66. BioequivalenceStudies
●
Comparethebioavailabilityoftwodrug
formulations(e.g.,genericvs.brand-name
drugs).
●
Thet-testisusedtocomparethemean
pharmacokineticparameters(e.g.,AUC,
Cmax)betweentwoformulations.
●
Ithelpsdetermineiftheformulationsare
bioequivalent.
●
Example:Thet-testcancomparethemean
AUC(areaunderthecurve)ofageneric
drugtothatofthereferencedrug.
Significance of Student’s t test
ClinicalTrials
●
Evaluatetheefficacyandsafetyofnew
drugs.
●
Thet-testisusedtocomparethemean
responses(e.g.,reductioninbloodpressure,
cholesterollevels)betweentreatmentand
controlgroups.
●
Ithelpsdetermineiftheobserveddifferences
inoutcomesarestatisticallysignificantor
duetorandomvariation.
●
Example:Inaclinicaltrialforanew
antihypertensivedrug,thet-testcancompare
themeanreductioninbloodpressure
betweenthedruggroupandtheplacebo
group.

67. StabilityTesting
●
Ensurethatdrugproductsmaintaintheir
quality,efficacy,andsafetyovertime.
●
Thet-testisusedtocomparethemean
valuesofstabilityparameters(e.g.,drug
potency,impuritylevels)atdifferenttime
points.
●
Ithelpsdetermineifthereisasignificant
changeintheproductovertime.
●
Example:Thet-testcancomparethemean
potencyofadrugatthebeginningandend
ofastabilitystudy.
Significance of Student’s t test
QualityControl
●
Ensurethatpharmaceuticalproductsmeet
predefinedqualitystandards.
●
Thet-testisusedtocomparethemean
valuesofcriticalqualityattributes(e.g.,drug
potency,tabletweight)betweenbatchesor
manufacturingprocesses.
●
Ithelpsidentifysignificantdeviationsfrom
thetargetspecifications.
●
Example:Thet-testcancomparethemean
potencyofanactivepharmaceutical
ingredient(API)intwodifferentproduction
batches.

68. Dose-ResponseStudies
●
Determinetherelationshipbetweendrug
doseanditseffect.
●
Thet-testisusedtocomparethemean
responsesatdifferentdoselevels.
●
Ithelpsidentifytheminimumeffectivedose
andthemaximumtolerateddose.
●
Example:Thet-testcancomparethemean
reductionintumorsizeatdifferentdosesof
ananticancerdrug.
Significance of Student’s t test
PharmacokineticandPharmacodynamicStudies
●
Understandhowadrugbehavesinthebody
(absorption,distribution,metabolism,
excretion)anditseffects.
●
Thet-testisusedtocomparemean
pharmacokineticparameters(e.g.,half-life,
clearance)orpharmacodynamicresponses
(e.g.,efficacy,sideeffects)betweendifferent
groupsorconditions.
●
Example:Thet-testcancomparethemean
half-lifeofadruginhealthyvolunteersversus
patientswithrenalimpairment.

69. ProcessOptimization
●
Improvemanufacturingprocessesto
enhanceefficiencyandproductquality.
●
Thet-testisusedtocomparethemean
valuesofprocessparameters(e.g.,yield,
purity)beforeandafterprocesschanges.
●
Ithelpsdetermineifthechangeshavea
significantimpactontheprocess.
●
Example:Thet-testcancomparethemean
yieldofadrugbeforeandafteroptimizinga
synthesisstep.
Significance of Student’s t test
ComparativeStudies
●
Comparetheperformanceofdifferentdrugs,
formulations,ordeliverysystems.
●
Thet-testisusedtocomparethemean
outcomes(e.g.,efficacy,safety)oftwo
differenttreatments.
●
Example:Thet-testcancomparethemean
painreliefscoresoftwodifferentanalgesics.

70. Post-MarketingSurveillance
●
Purpose:Monitorthesafetyand
effectivenessofdrugsaftertheyare
approvedandmarketed.
●
Thet-testisusedtocomparethemean
outcomes(e.g.,adverseeventrates,
efficacy)inreal-worldpopulations.
●
Example:Thet-testcancomparethemean
incidenceofsideeffectsinpatientsusinga
drugindifferentagegroups.
Significance of Student’s t test
RegulatoryCompliance
●
Ensurethatpharmaceuticalproductsmeet
regulatoryrequirements.
●
Thet-testisusedtoanalyzedatasubmitted
toregulatoryagencies,suchasthe
comparisonofmeanvaluesinclinicalor
bioequivalencestudies.
●
Example:Thet-testcanbeusedto
demonstratethatthemeanbioavailabilityofa
genericdrugisnotsignificantlydifferentfrom
thatofthereferencedrug.

71. ●
Theanalysisofvarianceinvolves
determiningiftheobservedvaluesbelongto
thesamepopulation,regardlessofthe
group,orwhethertheobservationsinat
leastoneofthesegroupscomefroma
differentpopulation.
ANOVA
●
TheANOVAisusedtoidentifyandmeasure
sourcesofvariationwithinacollectionof
observations,hencethenameanalysisof
variance.
●
Analysisofvarianceisaparametric
statisticaltechniquethathasfoundextensive
applicationsinscientificresearch,mainly
becauseofitsflexibility.
●
Thismethodmaybeemployedtoanalyse
bothpairedandindependentdataandalsois
usedtosimultaneouslycomparelarge
numberofvariables.
●
Theone-wayANOVAisnothingmorethanan
expansionofthet-testtomorethantwo
groupsofsample.

72. ConceptsinANOVA
1.NullHypothesis(H₀)●
The null hypothesis in ANOVA states that
there are no differences among the
group means. In other words, all groups
have the same mean.
●
H0:μ1=μ2=μ3=⋯=μk
●
Whereμ1, μ2,…,μkare the means of the
different groups.
ANOVA
●
ANOVA,orAnalysisofVariance,isa
statisticalmethodusedtocomparethe
meansofthreeormoregroupsto
determineiftherearestatistically
significantdifferencesamongthem.
●
Itisanextensionofthet-test,whichis
usedforcomparingthemeansoftwo
groups.
●
ANOVAhelpstoidentifywhetherthe
observeddifferencesinsamplemeansare
duetogenuinedifferencesinthe
populationmeansorsimplyduetorandom
variation.

73. 3. F-statistic:
●
ANOVA uses the F-statistic to test the
null hypothesis. The F-statistic is the
ratio of the variance between the group
means to the variance within the
groups.
●
F=Variancebetweengroups /
Variancewithingroups
●
A higher F-value indicates a greater
degree of difference among the group
means.
ANOVA
2. Alternative Hypothesis (H₁)
●
The alternative hypothesis states that at
least one group mean is different from the
others.
●
H1 :At least oneμiis different.

74. 5. Assumptions of ANOVA
●
Independence: The observations in
each group are independent of each
other.●
Normality: The data in each group are
normally distributed.
●
Homogeneity of variances: The
variances among the groups are equal
(also known as homoscedasticity).
ANOVA
4. Partitioning Variance:
ANOVA partitions the total variance in the
data into two components:●
Between-group variance:Variance due
to the differences among the group
means.
●
Within-group variance:Variance due to
the differences within each group (also
called error variance).

75. 7. Post-hoc Tests
●
If the ANOVA results are significant
(i.e., the null hypothesis is rejected),
post-hoc tests are conducted to
determine which specific groups differ
from each other.
●
Common post-hoc tests include
Tukey's HSD, Bonferroni correction, and
Scheffé'smethod.
ANOVA
6. Types of ANOVA
●
One-way ANOVA: Used when there is only
one independent variable (factor) with three
or more levels (groups).
●
Two-way ANOVA: Used when there are two
independent variables, and it can also test
for interaction effects between the
variables.
●
Repeated Measures ANOVA: Used when
the same subjects are measured multiple
times under different conditions.
●
MANOVA (Multivariate ANOVA): Used when
there are multiple dependent variables.

76. ●
Compare the calculated F-statistic to the
critical F-value from the F-distribution
table, based on the degrees of freedom
and the chosen significance level (usually
0.05).
●
Make a Decision: If the calculated F-
statistic is greater than the critical F-value,
reject the null hypothesis. If not, fail to
reject the null hypothesis.
●
Conduct Post-hoc Tests (if necessary): If
the null hypothesis is rejected, perform
post-hoc tests to identify which groups
are significantly different.
ANOVA
Steps in Conducting ANOVA
●
State the Hypotheses:
●
Formulate the null and alternative
hypotheses.
●
Calculate the F-statistic:
●
Compute the between-group and within-
group variances.
●
Calculate the F-statistic using the ratio of
these variances.
●
Determine the Critical Value:

77. TwowayANOVA
●
Itisusedwhenthedataareclassifiedon
thebasisoftwofactors.
●
Itisstatisticaltestusedtodeterminethe
effectoftwonominalpredictorvariableson
acontinuousoutcomevariable.
●
Itanalysestheeffectoftheindependent
variablesontheexpectedoutcomealong
withtheirrelationshiptotheoutcomeitself.
●
Two-waydesignmayhaverepeated
measurementsofeachfactorormaynot
haverepeatedvalues.
ANOVA
OnewayANOVA
●
TisthesimplesttypeofANOVA,inwhich
onlyonesourceofvariation,orfactor,is
investigated.
●
Itisanextensiontothreeormoresamplesof
thettestproceduresforusewithtwo
independentsamples.
●
Inanotherwayttestforusewithtwo
independentsamplesisaspecialcaseofone
wayanalysisofvariance.

78. ANOVA
ApplicationsofANOVA
●
Similartottest.
●
Moreversatilethant-test.
●
ANOVAisthesynthesisofseveralideasand
isusedformultiplepurposes.
●
Thestatisticalanalysisdependsonthe
designanddiscussionofANOVAtherefore
includescommonstatisticaldesignsusedin
pharmaceuticalresearch.
●
Pharmacokineticandpharmacodynamicdata
canbeevaluated.

79. FormulationDevelopment
●
Developandoptimizedrugformulations.
●
ANOVAisusedtocomparethemean
performance(e.g.,dissolutionrate,stability)
ofdifferentformulations.
●
Ithelpsidentifythebestformulationbased
onkeyqualityattributes.
●
Example:ANOVAcancomparethemean
dissolutionratesofthreedifferenttablet
formulationstodeterminewhichone
performsbest.
Significance of ANOVA
ClinicalTrials
●
Evaluatetheefficacyandsafetyofnew
drugs.
●
ANOVAisusedtocomparethemean
responses(e.g.,reductioninbloodpressure,
cholesterollevels)acrossmultipletreatment
groupsordoses.
●
Ithelpsdetermineiftherearesignificant
differencesinoutcomesamongthegroups.
●
Example:Inaclinicaltrialforanew
antihypertensivedrug,ANOVAcancompare
themeanreductioninbloodpressureacross
threedifferentdoselevelsandaplacebo
group.

80. StabilityTesting
●
Ensurethatdrugproductsmaintaintheir
quality,efficacy,andsafetyovertime.
●
ANOVAisusedtocomparethemeanvalues
ofstabilityparameters(e.g.,drugpotency,
impuritylevels)atdifferenttimepointsor
storageconditions.
●
Ithelpsdetermineiftherearesignificant
changesintheproductovertimeorunder
differentconditions.
●
Example:ANOVAcancomparethemean
potencyofadrugstoredatthreedifferent
temperaturesoversixmonths.
Significance of ANOVA
QualityControl
●
Ensurethatpharmaceuticalproductsmeet
predefinedqualitystandards.
●
ANOVAisusedtocomparethemeanvalues
ofcriticalqualityattributes(e.g.,drug
potency,tabletweight)acrossmultiple
batchesormanufacturingprocesses.
●
Ithelpsidentifysignificantvariationsand
ensureconsistency.
●
Example:ANOVAcancomparethemean
potencyofanactivepharmaceutical
ingredient(API)acrossfivedifferent
productionbatches.

81. Dose-ResponseStudies
●
Determinetherelationshipbetweendrug
doseanditseffect.
●
ANOVAisusedtocomparethemean
responsesatdifferentdoselevels.
●
Ithelpsidentifytheminimumeffectivedose
andthemaximumtolerateddose.
●
Example:ANOVAcancomparethemean
reductionintumorsizeatfourdifferent
dosesofananticancerdrug.
Significance of ANOVA
PharmacokineticandPharmacodynamicStudies
●
Understandhowadrugbehavesinthebody
(absorption,distribution,metabolism,
excretion)anditseffects.
●
ANOVAisusedtocomparemean
pharmacokineticparameters(e.g.,half-life,
clearance)orpharmacodynamicresponses
(e.g.,efficacy,sideeffects)acrossmultiple
groupsorconditions.
●
Example:ANOVAcancomparethemeanhalf-
lifeofadruginhealthyvolunteers,patients
withmildrenalimpairment,andpatientswith
severerenalimpairment.

82. ProcessOptimization
●
Improvemanufacturingprocessesto
enhanceefficiencyandproductquality.
●
ANOVAisusedtocomparethemeanvalues
ofprocessparameters(e.g.,yield,purity)
acrossdifferentprocessconditionsor
settings.
●
Ithelpsdeterminetheoptimalprocess
conditions.
●
Example:ANOVAcancomparethemean
yieldofadrugproducedusingthree
differentsynthesismethods.
Significance of ANOVA
ComparativeStudies
●
Comparetheperformanceofdifferentdrugs,
formulations,ordeliverysystems.
●
ANOVAisusedtocomparethemean
outcomes(e.g.,efficacy,safety)ofmultiple
treatments.
●
Example:ANOVAcancomparethemeanpain
reliefscoresofthreedifferentanalgesics.

83. Post-MarketingSurveillance
●
Monitorthesafetyandeffectivenessof
drugsaftertheyareapprovedand
marketed.
●
ANOVAisusedtocomparethemean
outcomes(e.g.,adverseeventrates,
efficacy)inreal-worldpopulationsacross
differentgroupsorconditions.
●
Example:ANOVAcancomparethemean
incidenceofsideeffectsinpatientsusinga
drugindifferentagegroupsorgeographic
regions.
Significance of ANOVA
RegulatoryCompliance
●
Ensurethatpharmaceuticalproductsmeet
regulatoryrequirements.
●
ANOVAisusedtoanalyzedatasubmittedto
regulatoryagencies,suchasthecomparison
ofmeanvaluesinclinicalorbioequivalence
studies.
●
Example:ANOVAcanbeusedtodemonstrate
thatthemeanbioavailabilityofagenericdrug
isnotsignificantlydifferentfromthatofthe
referencedrugacrossmultiplestudies.

84. Thank you
Professor in Pharmaceutics,
Adarsh College of Pharmacy, Vita, Sangli
415311
[email protected]
[phone]