StartoftheImplementationofa NewHPLCMethod
1.1COLLECTIONOFINFORMATIONANDPLANNINGFORA NEWMETHOD
Forgoodresults,thecollectionofasmuchinformationaspossiblebeforestartingthe implementationofanewHPLCmethodofanalysisisveryimportant.Itisalsoimportant toassesshowreliablethevariousitemsofthecollectedinformationare.Theinformation collectedbeforestartingaselectionforananalyticalmethodshouldcoveranumberofaspectsthatincludethefollowing:(1)thepurposeofanalysis,(2)generalinformationabout thesamples,(3)thesample’sconstituents,(4)therequiredqualityoftheresults,(5)instrumentationavailability,expertiseinthelaboratory,andfunding,(6)informationaboutvarious methodsofanalysis,and(7)newdevelopmentsininstrumentation.
Thecollectionofinformationregardingthemethodofanalysisshouldcontinueevenafter themethoddevelopmenthasstarted.Anyadditional findingsobtainedafterthedevelopmenthasstartedandthefeedbackfromtheinitialtrialsofaspecificmethodarevaluable datafortheimprovementand finalselectionofanadequatemethod.Forthisreason,the collectionofinformationshouldbeconsideredacontinuousprocessthatmayevenhave aniterativepath,theresultsobtainedfromthe firstsetofrunsleadtotheneedformore “starting ” informationandsoonuntilthemethodiswellestablished.
ThePurposeofAnalysis
Thepurposeofanalysisshouldbethe firsttypeofinformationobtainedbeforestartingthe implementationofanewanalyticalmethod.Sourcesofinformationaboutthepurposecanbe verydiverse,themostcommonbeingthedirectrequestfromacustomer(includingselfimposedrequests).Anumberofitemscanbelistedasimportanttoknowregardingthepurposeofachemicalanalysis.Thisinformationshoulddescribewhytheanalysisisperformed, whatkindofresultsareexpectedfromtheanalysis,andpossiblytheutilityoftheanalysis.A widerangeofrequestscanbemadeforananalysis,andsamplesmustbeanalyzedfor numerousreasons.Inindustrialenvironmentssuchpurposesmayincludeof ficialorlegal requirements,assessingthequalityofrawmaterials,processcontrolortroubleshooting,
assuringthequalityof finishedproducts,research,reverseengineering,anddevelopment purposes.Samplesarefrequentlyanalyzedforhealth-relatedpurposes(e.g.,medicalanalyses,analysisofpharmaceuticals,analysisofmetabolites),forevaluatingenvironmentalissues,forforensicpurposes,forexploratoryreasons,andforfundamentalresearch. Dependingontheanalysisreason,speci ficdecisionsaremadeabouttheanalyticalprocess. Thepurposeofanalysisshoulddescribeiftheanalysisisrelatedtoamaterial,aprocess, orboth.Itisimportanttoknowifthewholesamplemustbeanalyzed(allconstituentsare inthiscaseanalytes)oronlyaspecificpartofthesample.
Importantinformationmustbecollectedindicatingtherequiredtypeofanalysisregarding qualitative,quantitative,semiquantitative,orbothqualitativeandquantitative.Itshouldbe indicatedifaspecialanalysisistobeperformedsuchasseparationofenantiomersor regardingstructuralelucidation.Thepurposeshouldspecifyiftheanalysishasaspeci fic target,ifitisonlyexploratory,orthegoalofperformingtheanalysisisvague.Also,itshould beknownifthereisaplantoperformtheanalysisonarepeatedbasisinthefutureoronlyat onetime.Thenumberofsamplestobeanalyzedatonepointorinanextendedperiodoftime shouldbeknown.Therapiditywithwhichtheresultsmustbedeliveredshouldalsobe knownbeforestartingthedevelopmentorimplementationofanewmethod.
Generalinformationmustspecifywhetheraspeci ficprotocolmustbefollowedduringthe analysisorthatnoregulationsareimposed.Someanalysesarerequiredtobenondestructive, andincertaincasestheanalysisisdoneinconjunctionwithpreparativepurposes,whichalso shouldbeknown.Thereareimportantissuesrelatedtoanalysesassociatedwithpreparatory purposessincesomeofthemmodifythenatureofthesample.Ifthebiologicalactivityofthe samplesmustbepreserved,thisaspectshouldalsobeknown.
Thelevelofaccuracyandprecisionoftheanalysisshouldbedescribed,indicatingifthe analysisisgearedtowardmajorconstituentsofthesample,minorconstituents,ortraces. Also,informationshouldbecollectedregardingthenoveltyoftheplannedanalysisorifit waspreviouslyperformed.Knowledgeregardingotheranalysesalreadyperformedonthe samplesisalwaysimportant,andoccasionallyitisusefultohaveinformationaboutanalyses thatareplannedtobeperformedonthesamplesinthefuture.
Thegeneralinformationregardingthepurposeofanalysisallowsearlyplanningofthe typeofanalysistobeperformedonthesamples.Incasethatverylittleinformationisavailableaboutthesamples,preliminaryanalysesshouldsometimesbeperformed.Thispreliminaryanalysiscanbequalitativeorsemiquantitative.Ifonlyqualitativeanalysisisrequired, high-performanceliquidchromatography(HPLC)isnotusuallythebestchoiceasthe methodofanalysis.Wheninformationexiststhatthesamplecontainsuniquecompounds oritisamixtureofveryfewcomponents,nonchromatographictechniquesmaybeconsideredforobtainingqualitativeinformation.Veryusefularethespectroscopicprocedures suchasultraviolet(UV),infrared(IR),ornuclearmagneticresonance(NMR).Whenthesampleshavemorecomponents,chromatographictechniquesaretypicallynecessaryforaninitial separationofsamplecomponents. “Scantype” chromatographictechniquesassociatedwith detectionbymassspectrometrysuchasgaschromatography massspectrometry(GC MS) areprobablythebesttoolsforcompoundidentification,providedthesamplesareamenable forthistypeofanalysis(containvolatileanalytes).Insomecases,whenthesampleshavea limitednumberofcomponentswhichareknowntobepotentiallypresent,HPLCorthinlayerchromatography(TLC)canbeusefulforidentifyingsamplecomposition.Mass
spectrometryhyphenatedwithliquidchromatography(LC)suchasLC-MSorLC-MS/MS mayprovidequalitativeinformationonunknownsamples,althoughtheidenti ficationofunknowncompoundsbyLC-MSorLC-MS/MStechniqueshaslimitations.Forquantitative analysis,chromatographyistypicallypreferred,includingGCorLC(HPLC).Also,anarray oftechniquesmustbenecessaryfortheanalysisofcertainsamples.Forlargenumbersof samplestobeanalyzed,automationmustbeconsidered.
GeneralInformationAbouttheSamples
Theinformationaboutthematerial(s)tobeanalyzed(indicatedas sample)shouldcover chemicalaspectsofthesample,physicalpropertiesinformation,aswellasanyothercontingentinformation.Twopartsofasamplearetobeconsideredinachemicalanalysis,the analytes andthe matrix.The analytes arethemolecularorionicspeciesofinterestinthesample andthe matrix indicatestherestofthesamplecomponents.Similartothecaseofinformation aboutthepurposeofanalysis,asmuchinformationaspossibleaboutthesamplesisdesirable, althoughinmanypracticalcasesthisinformationremainsincomplete.Acriticalpieceofinformationisrelatedtothesamples’ complexity.Thistypeofinformationisanearlycriterion relatedtothechoiceoftheneedforahyphenatedtechnique(separation þ measurement)for theanalysis.Informationaboutthesamplingprocessisalsoveryuseful,inparticularindicatingthesamples’ homogeneity,theageofthesamples,andpotentialofcontamination.If nonhomogeneoussamplesneedtobeanalyzeditshouldbeknowniftheanalysismustbe performedinaspecificway(e.g.,afterhomogenization,orselectingpartsofthesample). Also,itmustbeknownifthewholesampleshouldbeanalyzed,oronlyaspecificpart(surface,solublecomponent,selectedpoints,etc.).Informationaboutthesamples’ physicalstate shouldbecollected,indicatingifthesamplesaregas,liquid,solid,solution,semisolid,or mixedphases.Also,knowledgeshouldbecollectedwhetherthesamplesareinorganic, organic,composite,ofbiologicalorigin,environmental,orofaspecialsource.Otherdata aboutthesamplesshouldindicatetheamountavailable(largequantity,smallquantity, readilyavailable,uniqueness,etc.)andalsothevalueofthesamples.Insomecasesthesamplesmustbereturnedtotheproviderafterasmallamounthasbeenusedforanalysisandthis shouldalsobeknown.Otherusefulinformationincludesthesamples’ thermalstabilityand perishability,aswellasanysafetyconcerns.Theinformationaboutthesamplesshouldalso indicateifthesamplesareofanewtypeorifsimilarsampleswerepreviouslyanalyzed.
Thegeneralinformationaboutthesamplesiscloselyrelatedtothesamplingprocessifthe sampleswerenotalreadycollected.Itshouldguidetheprecautionsregardingsamplingto avoid,forexample,contamination,amountofnecessarysample,therequirementsofsamples transportation,andthesafetymeasuresrelatedtosampling.Nonhomogeneoussamplestypicallyneedcarefulsamplingfollowedbytheseparationofphases.Samplepreparationis frequentlyneeded.Besidesthesamplingprocess,thegeneralinformationonthesamples hasinputregardingtheinitialsampleprocessing,suchasphysicalprocessing(grinding,drying,homogenization),andalsosamplesstorageandpreservation.Inorganicsamplesaretypicallyprocesseddifferentlythanorganicorcomplexsamples.Sometimes,dissolutionof inorganicsamplesisanimportantsamplepreparationstep.Thermallylabilesamplesmay needspecialstorageandhandling,andHPLCtypetechniquesarefrequentlyusedforthis
typeofanalysis.Perishabilitymaybemodifiedusingpreservatives,andtheadditionofsuch compoundsmustbeknown.Theanalysisofcomplexsamplesmayalsorequiresample preparation.Samplecomplexityalsodeterminestheuseofaseparationtechniquehyphenatedwiththemeasurement.Themostcommonsuchtechniqueischromatography.Forthis reason,chromatographyisthetechniquepreferredinmanyorganic,biological,andenvironmentalanalyses.Gasesandthecompoundsthatarethermallystableattheirboilingpointare preferablyanalyzedbyGC,andsamplepreparationmaynotbenecessary.Manyorganic compoundscanbeanalyzedbyHPLC,thistechniquebeingnecessaryformoleculesthat cannotbevolatilized,butcanalsobeusedasachoiceforvolatileandsemivolatilemolecules. Appropriatemeasuresmustbetakenifhazardconcernsarepresent.
TheSampleConstituents
Theinformationaboutsampleconstituentscanbeplacedintotwocategories:(1)analytical and(2)generalproperties.Analyticalinformationincludesdataaboutthechemicalnatureof theconstituents,thestrengthofbondingoftheanalytestothematrix,rangeofconcentrations oftheanalytes,etc.Generalpropertiesincludephysicochemicaldatasuchasvolatilityofanalytesandmatrix,solubilityinvarioussolvents,acid basecharacter,hydrophilic hydrophobic properties,reactivity,etc.Theknowledgeaboutthechemicalnatureoftheanalytesisessential. Itispreferabletoknowthelistofcompounds(singleormultiplecomponentanalysis)thatmust beanalyzedoratleasttheclassoftheanalytes(inorganic,organic,functionalgroupsinorganic compounds,ioniccharacter,etc.).Whentheanalytesofinterestareknown,itisusefultoknowif theanalytesarevolatile,nonvolatile,ionic,polymeric,weakinteractioncompounds,ora mixtureofspeciesofthesamecompound.Ifthisinformationismissing,itmustbeknownat leastiftheanalytesaresmallmoleculesorpolymeric.Inthecaseofsmallanalytemolecules, dataregardingvolatility,solubility,andreactivityareveryuseful.Formacromolecules,ageneralcharacterizationistypicallynecessary.Otherdataregardingtheanalytesarehelpful,such asinformationontheestimatedlevelofanalytesinthesamples(ultra-lowtrace,trace,minor constituent,mediumlevels,ormajorconstituent).Thelevelofanalytesmaybetotallyunknown andthenpreliminaryanalysesmaybenecessary.Fromthegeneralinformationaboutthe samplesitmaybepossibletoestimatetherangeofconcentrationfortheanalytesinthesamples. Theiterativeaspectofcollectinginformationaboutthesamplesallowsfortheimprovementof theinformationaboutthesamplecomposition,andastheanalyticalprocessstartstobeimplemented,additionalinformation,aswellasadditionalquestions,mayemerge.
Similarinformationtothatabouttheanalytesshouldbeexaminedforthematrix.Itis importanttoknowifthematrixishomogeneousornot,ifitisinorganic,organic,hasanionic character,ifitispolymeric,orifitisofbiologicalorigin.Dataregardingthevolatilityand stabilityofthesamplematrixarealsoveryuseful.Theinteractionbetweenthematrixand theanalytesmayplayanimportantroleintheanalysis.Thematrixmayformstablemolecularassociationswiththeanalytes(e.g.,salts,clathratesincludinghost guestcomplexes, inclusioncompounds),ormaygenerateartifactanalytesinspeci ficconditions(e.g.,generationoffreeaminoacidsfromproteins).
Thisinformationisrelatedtoplanningofdifferentoperationsintheanalysisandcango backtoinstructionsaboutsampling(ifnotrestrictedtoasamplealreadyprovided).It
includesthechoiceofpreliminarysampleprocessing(physicalprocessing),andtheselection ofasamplepreparationprocess,suchassamplecleanupandanalyteconcentration.The informationaboutthechemicalnatureoftheanalytesandthematrixarethemaindata neededinplanningaspecifictypeofanalysis.Inorganicanalytescanbeanalyzedusingchromatographicmethodssuchasionchromatography(IC),buttechniquessuchasinductively coupledplasma(ICP)oratomicabsorption(AA)aremorefrequentlyapplied.Formost organicandcomplexsampleschromatographyisthepreferredtechnique.Sampleswith smallvolatilemoleculesaretypicallyanalyzedusingGC(withorwithoutsamplepreparation).SinceGCcanbeeasilycoupledwithMSasadetector,andbecauseoftheexistence oflargelibrariesofMSspectra,GC MSisthemostpowerfultechniqueforcompoundidentification,andisalsoveryusefulforquantitation.HPLCisthemostcommonanalytical techniqueforthequantitationofawiderangeoforganicmolecules.TheutilityofHPLC forqualitativeanalysisissomewhatlimitedevenwhenitiscoupledwithMSdetection. However,forquantitationpurposes,HPLCisthemostappropriateandcommonlyutilized analyticaltechniquefororganicmolecules.PolymerscanbeanalyzedbynonchromatographicmethodssuchasIR,ortheycanbeseparatedandanalyzedusingsizeexclusion chromatography(gelpermeation[GPC]orgel filtrationchromatography[GFC]).Sometimes thepolymersarealsoanalyzedafterpreliminaryhydrolysisorpyrolysisthatreducesthe initialmacromoleculestosmallermolecules.
Aspartofthe “analyticalinformation” theknowledgeabouttherangeoftheanalytescontainedinthesamplesallowsfortheselectionoftheappropriatesensitivityforthemethodof analysis.Sincespecificsensitivitiesarecharacteristicfordifferentanalyticaltechniquestheinformationabouttherangeoftheanalytelevelsmaybenecessaryforselectingthetypeof analyticaltechniqueandanalyticalinstrumentation,andnotonlyforadjustingmethodsensitivity.Dataaboutthelevelofanalytesmayalsoguidetheneedforsamplepreparationsuch asanalyteconcentrationprocedures.
TheRequiredQualityoftheResults
Theresultsofthechemicalanalysisaretypicallyexpectedtoanswerspecificquestions relatedtothepurposeofanalysis.Inordertohaveappropriateanswers,theresultsmust satisfyspecificrequirementsregardingaccuracy,precision,andthenumberofsamplescharacterized.Also,itisimportanttoknowiftheresultsarepartofalargerstudy,iftheycanbe comparedwiththeresultsfromotherlaboratoriesorwithresultsonstandardmaterials.
Thetypeofnecessaryresultshasimplicationsonvariousaspectsofplanninganew methodofanalysis.Onestartswithsampling.Aspecificallydesignedsamplingprotocol maybenecessarywhenspecificstatisticalanalysisofthedataisrequired.Also,thelevel ofprecisionoftherequireddatainfluencestheplanningforspecificsensitivityoftheanalyticalmethod.Theuseoftheresultsinspeci ficcollaborativestudiesmayalsoimposeagiven protocolforthemethodofanalysis.
InstrumentationAvailability,ExpertiseintheLaboratory,andFunding
Theselectionofananalyticalprocedureisfrequentlydeterminedbyvariousexternal factorsbesidesthepurposeofanalysisandtypeofsamples.Amongsuchfactorsarethe
instrumentavailability,theexpertiseinthelaboratory,thetimeavailablefordevelopingor implementingaspecificanalyticalprocedure,andalsofunding.Inadditiontoinstruments, informationshouldbecollectedaboutthematerialsavailable,indicatingforexamplethe needforpreliminarypurchasesofchromatographiccolumns,reagents,solvents,and standards.Relatedtothesafetyconcerns,appropriatelaboratoryconditionsmustalsobe assuredfortheanalysisofcertainsamples.Informationaboutalltheseaspectsisofprimary importancesuchthattheanalysiscanbeperformed.
Theinformationabouttheresourcesavailableforperformingaspecificanalysisina selectedmanneriscriticalforsuccessfulresults.Missingthenecessaryinstrument,for example,isanobviousbarrierforplanningthedevelopmentofanewmethodanalysis thatwouldusethatinstrument.Potentialpurchaseofnewinstrumentationmustbeconsideredinspeci ficcases.Theexpertiseavailableinthelaboratorytoperformtheanalysisisalso animportantfactor,includinginformationabouttheneedfortraining.Theperformanceof someanalysesisaverysimpletask,whileotheranalysesrequireconsiderableresources. Amongtheexternalfactorsthatcouldalsobeconsideredaretheeffectsofusedchemicals fromthelaboratoryontheenvironment.Forthisreasontheutilizationofenvironmentfriendlysolventsandreagentsisrecommended,aswellasofanalyticalproceduresthat minimizetheconsumptionofdangerousunavoidablechemicals.Forexample,organic solventsthatcouldbeconsidered “ green ” [1] aredesirableinchromatographicanalysis becausetheiruseavoidstoxicityandenvironmentalproblems [2,3]
InformationAboutVariousMethodsofAnalysis
Alargebodyofinformationisavailableregardingmethodsofanalysis.Thisinformationis availableintheformofarticlespublishedinpeer-reviewedjournals,books,companyapplicationnotesanduserguides,variousformsofInternetinformation,aswellasnumerous coursesatuniversitiesandprivateorganizations(allfurtherindicatedas “literature ”).Before startingthedevelopmentorimplementationofanewmethodofanalysisacriticalstepisthe reviewingoftheavailableinformationrelatedtotheanalysisofinterest.Foralmostanyanalyte,therearemethodsofanalysisthataredescribedintheliterature.Thealready-reported methodsmaynotbeintendedforthespecifictypeofsampleonwhichthenewmethod shouldbeappliedormaynotsatisfycertainrequirementsforthenewmethod.Itisalso possiblethatthemethoddescribedintheliteraturecannotbeimplemented,forexample, duetothelackofinstrumentation.However,thisliteratureinformationalwaysoffersavaluablestartingpointinanewmethoddevelopment.Whenamethodisavailableandcanbe utilized,itisfrequentlybene ficialtoimplementthemethodanddirectlyevaluateits adequacyforthenewpurpose.Incertaincases,wherenomethodisavailableintheliterature, proceduresfortheanalysisofsimilarcompoundsasthoseplannedfornewanalysisare helpful.
Thechoiceofamethodofanalysisrequiresknowledgeaboutalltheaspectspreviously described.Whenamethodisalreadyreportedforthesametypeofsample,orforsimilar samples,itistypicallyadvisabletoimplementthatmethodasis,evenifmodificationsand adjustmentsaretobemadeafterward.Inmostsituations,multiplechoicesforananalytical methodareavailable,evenwhentheanalysishasbeenpreviouslypracticedonthesametype ofanalytesandsamples.Itiscommonthatmorethanonechoicecangivegoodresults,when
mostoftherequiredconditionsforanappropriateanalysisarefulfilled.Themethodof analysiswithallitsaspectsincludingthenecessarysampleprocessing,thetypeofanalysis, anddataprocessingmaybemoreorlessconvenientdependingonparticularchoices. Differentanalyticaltechniquesforthesameanalytesandthesametypeofsamplecanoffer advantagesandmayhavedisadvantages.Also,somemayofferexcellentqualitiesalthough notnecessaryneeded.Forexample,theuseofanLC-MS/MSmethodversusanHPLC methodwithUVdetectionmayofferhighersensitivityandpositiveidentificationoftheanalytes,althoughthismaynotbenecessaryforaspeci ficanalysis.Arbitrarychoicescanbe madeforsomeanalyses,andthequalityoftheresultmayormaynotbeaffected.Although anoptimizationoftheanalyticalmethodissometimespossible,abalancemustalwaysbe madebetweentheeffortofimprovingthemethodandusingitasis,iftheresultsarestill satisfactory.Thechoiceofaspeci ficanalyticalmethodcanbecritical,butitisalsopossible thatanumberofalternativesleadtothesamedesiredanalyticalgoalandonechoiceor anotherisnotcritical.Theselectionofamethodofanalysismaybedeterminedbymanyreasons,forexampletheinstrumentavailability,personneltraining,limitedfunding,andmay bethesubjectofvariousconstraintsthatdonotallowtheselectionofthebestpossiblealternative.Nevertheless,thegoalofthisbookistodescribethebestchoicesforselectingan HPLCmethodofanalysis.
ForHPLCanalysis,awiderangeofchoicescanbemadewiththeendresultofsuccessfully performingaspeci ficanalysis.ThisbookdiscussesvariousaspectsofHPLCanalysisthat mustbetakenintoconsiderationfortheselection,development,andimplementationofa successfulmethodusingthistechnique.
NewDevelopmentsinInstrumentation
Theimprovementsofanalyticalinstrumentation,ofaccessories,ofconsumables,andthe developmentofnewtechnologies,andnewmaterialsisacontinuousandveryactiveprocess. Allofthesenewdevelopmentsmayoffernewpossibilitiesfortheanalysiseitherthrougha varietyofimprovementsinsensitivityofanalyticalinstruments,mediaforbetterseparations, simplificationsinthenecessaryoperationsforspecificanalyses,costreductionofanalyses, speedingtheanalyticalprocess,enhancingreliabilityoftheanalyses,etc.Avarietyofsources areavailableforprovidinginformationregardingallthesenewdevelopmentssuchasscientific meetingsandconferences,literatureprovidedbymanufacturers,variousInternetsites,etc.
Knowledgeaboutallthenewdevelopmentsrelatedtotheanalyticalcapabilitiesisvery importantforadoptingthebestpathforaspecificanalyticaltask.However,theimplementation inthelaboratoryofsuchnewdevelopmentsisnotalwayspossible.Thecostofnewinstruments maybeprohibitive,thetrainingofthepersonnelmaybelacking,andthenewdevelopmentsdo notalwaysrepresentbetterwaystosolveaspecificanalyticalproblem.Nevertheless,itisalways usefultobeinformedaboutnovelalternativesforachemicalanalysis.
1.2OVERVIEWOFANANALYTICALTECHNIQUE
Chemicalanalysiscanbequalitative,semiquantitative,quantitative,oraddressingstructuralproblems,canbedesignedfortheanalysisofmaterialsorforprocesscontrol,analysis
FIGURE1.2.1 Simplifiedschematicsforthepathofachemicalanalysis.
ofgases,liquids,solids,ormixedphases,andcanbedesignedfororganicmaterials, inorganicones,ormixedorganic inorganic,etc.Itcanbeperformedusingawidevariety ofprocedures.Beforethedevelopmentofmoderninstrumentsforanalysis,wetchemistry waswidelyusedforqualitativeanalysis,andgravimetricandvolumetricmethodswere usedforquantitativeanalysis.Althoughtheseolderprocedurescanstillbeutilized,thisis donemuchlessfrequently,andothermethodsmuchmoresensitiveandselectivebasedon variousinstrumentsarenowpracticed.Amoresystematicpathforthechemicalanalysisis alsousuallyimplemented.Anoverallpathforamodernchemicalanalysiscanbeschematicallydescribedbythediagramin Fig.1.2.1.Theschemeforthepathofachemicalanalysis canbemuchmorecomplicatedthanthesimplifiedschemeindicatedin Fig.1.2.1.However, theschemedescribesthebasicoperationsinamodernchemicalanalysis.
Thedecisionsregardingvariousselectionsforthewholeanalysisarehighlyinterrelatedin thesequence:samplecollection / samplepreparation / coreanalyticalprocedure / data analysis.Dependingonthesample,aspecificsamplepreparationmustbeselectedwiththe goalofaspecificcoreanalyticalprocedure.Atthesametime,thegoalofusingaspecificcore analyticaloperationiscriticalfortheselectionofthesamplepreparationstep.Aniterative selectionprocessofsamplepreparation 4 coreanalyticalprocedureisfrequentlynecessary forthebestresults.Regardingsamplecollection,thisisnotalwaysoptionalfortheanalyst, butinsomeinstancessamplecollectioncanbeadjustedtomatchthesamplepreparation andcoreanalyticalprocedure.
SampleCollection
Samplecollection(sampling)istheoperationofobtainingthe rawsample anddeliveringitin anappropriateformtotheanalyticalprocess.Samplingisaveryimportantprocessandalarge bodyofinformationpresentsthissubjectindetail(see,e.g., [4 7]).Theimplicationsofsample collectionontheselectionofthemethodofanalysisandinparticularontheselectionofHPLC techniquearesignificant.Forexample,thecollectionmayincludeaspecificmatrix,ormaybe associatedwithsometypeofconcentrationoftheanalyteswhichinfluencethefurtherselection ofthemethodofanalysis.However,thesubjectofsamplecollectionisbeyondthepurposeof thischapter.Startingwithagivensample,variousdecisionsmustbemaderegardingtheanalysis,andsuchdecisionswillbefurtherdiscussedonlyrelatedtoHPLCanalysis.
SamplePreparationStep
Therawsampleisfrequentlysubjectedtoasamplepreparationstep.Samplepreparation hastheroleofmodifyingtherawsampletomakeitmoreamenableforthecoreanalytical operation.Theoperationofsamplepreparationcanrangefromaminorsteptoverycomplex processingofthe rawsample (sampleasobtainedfromsampling).Amongthepotential
operationsinsamplepreparationaretheroutinemanipulationsofthesamplesuchashomogenization,reductionofthesamplesize,drying,weighing,volumemeasuring,andsample dissolution.Alsotheseoperationsmayincludecleanupprocedureswhenthematrixofthe sampleissimplified,fractionation,concentrationoftheanalytes,andchemicalmodifications. Variousseparationoperationscanbeappliedtotherawsamplesuchasmechanical (e.g., filtration),phasetransformationofthemixtureofcomponents(e.g.,distillation,dissolution,crystallization),extractionwithsolvents,separationsusingsolidphaseextraction, migrationina fieldforseparation(gravitational,magnetic,electric),etc.Theinitialrawsamplecanalsobesubjectedtovariouschemicalmodificationsinordertomakeitmore amenableforthecoreanalysis.Chemicalmodificationmayincludereactionswithoneor morereagents(derivatizations),chemicaldegradations(appliedforexampletopolymers), andpyrolysis(thermaldegradation).Alltheseoperationswilldelivera processedsample thatismoreamenableforfurtheroperationsofseparationanddetectionthatcomposethe coreanalyticaloperation.Samplepreparationisalwaysperformedconsideringtheplanned coreanalyticalprocessesthatfollowintheanalysis(furthercomponentseparationanddetection).Sincesamplepreparationisperformedwiththegoalofrenderingtherawsamplemore amenableforthecoreanalyticalprocess,thecharacteristicsofthecoreanalyticaloperation arealwaystakenintoconsiderationduringsamplepreparation.
Besidesthepreparationoftherawsampleinordertodeliveraprocessedsample,other operationsmustbeconsideredinthisstageoftheanalysis.Theseinclude,forexample,the preparationofstandardsforthegenerationofcalibrationsorforobtainingpreliminaryinformationonthelevelsofmaterialsthatarefurtherusedintheanalysis.Mostanalytical methodsincludeasamplepreparationstepsuchthatsamplepreparationisanimportant partofthewholeanalysis.Forthisreason,thesubjectofsamplepreparationisextensively presentedintheliteratureanddedicatedbooks,andothermaterialspresentasastandalone subjectthemethodologiesforsamplepreparation(see,e.g., [8 11]).Samplepreparation cannotbedisconnectedfromtheselectionofthecoreanalyticalprocedure,anddepending ontheextentandnatureofsamplepreparation,anadequatecoreanalyticalmethodis selected.However,thisbookpresentsthespeci ficsfortheselectionofanHPLCanalysis whenagivensamplemustbeanalyzedafterpreliminaryprocessing.Forthisreason,the subjectofsamplepreparationisnotfurtherdiscussed,althoughwithinacompletestrategy forachemicalanalysissamplepreparationisanintegralpartoftheanalyticalprocess.
Samplepreparationisfrequentlyatime-andmanpower-consumingstep,althoughit renderstothecoreanalyticalmethodacleanerandpossiblymoreconcentratedprocessed sample.Forthisreason,thetendencytousesimplerandshortersamplepreparationtechniquesiscommon.However,thesesimplertechniquesmayresultinaprocessedsample havingalesssimplifiedmatrixandwithamodestornoincreaseintheanalyteconcentration. Asaresult,thecoreanalyticaltechnique(e.g.,HPLC)hasamoredifficulttaskinproviding accurateresults,andthedevelopmentofanewandbettercoreanalyticaltechniqueisacommontask.
TheCoreAnalyticalOperation
Thecoreanalyticaloperationreceivingtheprocessedsamplehastheroleofdeliveringthe analyticalresultsthatarefurtherevaluatedinthedataanalysisstepforprovidingthe
Core analytical operation
Detection and measurement
Data analysis Processed sample
FIGURE1.2.2 Schematicdescriptionsofthe flowofoperationsina nonhyphenated coreanalyticaloperation.
necessaryinformationaboutthesample.Thecoreanalyticaloperationisusuallyperformed usinginstrumentaltechniquesthatmayinvolvespectroscopic,electrochemical,radiochemical,thermal,orotherinstrumentalprocedures.Whenappliedonpurecompounds,or whenacompoundinamixturehasuniquephysicochemicalproperties,theanalysiscanbe performedwithouthavingaseparationstepaddedtothecoreanalyticaloperation(sample preparationshouldbeconsidereddifferentfromacoreanalyticalprocedure).Suchtechniquescanbeindicatedas nonhyphenated (separationfordifferentionscouldbeimbedded inthetechniqueitself). Fig.1.2.2 showstheschematicdescriptionsofthe flowofoperations inanonhyphenatedanalyticaloperation.
Becauseofthenecessitytoanalyzemoreandmorecomplexsamples,manymodernanalyticaltechniquesthatareabletoaddresssuchanalyseshaveaseparationstepimbeddedinthe coreanalyticaloperation.Thesetechniquescanbeindicatedas hyphenated.Samplepreparation mayuseseparationtechniquesforsampleprocessing(see,e.g., [8,9]),buttheyarenotonline andarenotpartofthecoreanalyticalstep.Theprocessedsamplemayhavealesscomplex matrixthantherawsampleormayhaveahigherconcentrationofanalytes,butitiscommon thatfurtherseparationsareneededforanalyticalpurposes.Whensuchaseparatorystepis necessaryanditisimbedded(online)inthecoreanalyticalprocedure,thetechniquecanbe indicatedas hyphenated Fig.1.2.3 showstheschematicdescriptionsofthe flowofoperations inananalyticaltechniquethatcontainsthecoreanalyticaloperationsinaseparationstep. Theseparationcanbeperformedbyseveralprocedures,butthemostcommonischromatography. Chromatography designatesseveralsimilartechniquesthatallowtheseparationof differentmolecularspeciesfromamixture.Forthechromatographicseparation,thesample isplacedina fluidcalled themobilephase,whichcarriesthesamplethroughamaterialcalled the stationaryphase.Theseparationinchromatographyisachievedbecausethestationary phaseretainsstrongerorweakerdifferentmolecularspeciesfromthe flowingmobilephase andreleasesthemseparatelyintimebackintothemobilephase(differentmolecularspecies willhavedifferent retentiontimes).Theprocessofmovingtheanalytesthroughthestationary phaseisknownas elution.Thedetectionandmeasurementinanalyticalchromatography isusuallybasedonacertainphysicochemicalpropertyoftheseparatedmolecule (UV-absorption,refractiveindex, fluorescence,molecularmassandfragmentationina
Core analytical operation
Separation of sample components
Detection and measurement
Data analysis Processed sample
FIGURE1.2.3 Schematicdescriptionsofthe flowofoperationsinananalyticaltechniquethatcontainsinthecore analyticaloperationsaseparationstep(hyphenated technique).
massspectrometer),apropertywhichisdifferentfromthatofthemobilephase.Foranalyte detection,manyinstrumentaltechniquesdevelopedindependentofachromatographic separationstepwereadaptedtobeusedinanalyticalchromatography.Asaresult,numerous chromatographicdetectorsarebasedonUV-absorption,refractiveindex, fluorescence, molecularmass,andfragmentationinamassspectrometer,etc.
Besideschromatography,otherseparationtechniquescanbeconnectedonlinewithan instrumentaltechniquefordetection.Amongtheseelectro-separationsandmembraneseparationscanbementioned.Also,morecomplicatedpatternsofoperationscanbeencountered, suchascombinationsofhyphenatedtechniques(e.g.,inbidimensionalseparations).
Importantdecisionsaremaderegardingtheselectionofthetypeofcoreanalyticalprocedure.Dependingonseveralfactors,suchaspropertiesoftheanalytes,propertiesofmatrix, requestedturnaroundtimeoftheanalysis,availableequipment,etc.,atechniqueincluding aseparationstep(ahyphenatedtechnique)isorisnotselected.Thesubjectisfurtherdiscussedinthisbook.
DataEvaluation
Dataevaluation,alsoapartofachemicalanalysis,isdiscussedinmanypublishedmaterials.Dataevaluation(oranalysis)typicallyinvolvesstatisticalevaluations(withnumerous chemometricstools),butincludesmoregeneraloperationssuchasinspecting,cleaning,transforming,andmodelingofthedatawiththegoalofdiscoveringusefulinformation.Dataanalysisisamuchwider fieldthanappliedtochemicalanalysis,butevenrestrictedtothissubject, dataanalysiswillbeonlybrieflypresentedinthisbook(see Section1.3).Detailedinformation regardingstatisticaldataprocessingcanbefoundinmanypublications(see,e.g., [12 15].For detailsregardingdataanalysis,thededicatedliteratureforthissubjectisrecommended.
QualitativeandQuantitativeAnalysis
Theinformationtypicallygeneratedinachemicalanalysiscanberelatedtothe findingof thenatureoftheanalytes(qualitativeanalysis),leveloftheanalyte(quantitativeanalysis),or both.Qualitativeanalysiscanrefertovariousdegreesofknowledgebeginningwithabrute formula,indicationsofthecompoundclass,presenceofspecificfunctionalgroups,uptoa trueidentificationofthecompound(possiblyincludingstereo-isomericstructure,detailed structureofthemolecule,etc.).Quantitativeanalysisdescribeshowmuchanalyteispresent inaspeci ficsample,andthisinformationmayalsorangefromasemiquantitativeanalysisto averypreciseone.Thecapabilityofdifferentanalyticaltechniquestoprovidequalitative and/orquantitativeinformationaboutthesamplesisfurtherdiscussedinChapters2and3.
Forthequantitativeanalysis,theidealcaseiswhenthesignaloftheanalyticalinstrument dependslinearlyontheanalyteconcentrationinagivenvolume(oramount)ofsample.The concentrations C ofinterestcanbedeterminedfromthesignal y usingtherelation:
Thevalue b istheslopeforthedependenceandcanbeobtainedusingcalibrationswith standardsofknownconcentration.Thestandardsaretypicallyusedaspurecompounds andthecalibrationsaredoneindependentlyofthesample.Thecalibrationstandardscan
beconsideredexternalstandards.(An externalstandard isanalyzedinadifferentrunfromthe sample,whilean internalstandard isaddedandanalyzedtogetherwiththesample.)Inmany practicalapplicationsitispreferabletomakethecalibrationsbyaddingdifferentlevelsofthe calibrationcompoundtoablanksamplethatdoesnotcontaintheanalyte.Thisprocedure makestheanalysisofthesamplescontainingthecalibrationstandardsascloseaspossible totheanalysisofarealsampleandallowsthesubtractionoftheoverallinfluenceofthe matrixintheanalysis.Forcompoundsthathavesimilarstructures,thecalibrationcurve foronlyoneofthecompoundsisutilizedsometimes,anddifferentcompoundsarequantitatedbasedonthesamecalibration.However,thisisnotarecommendedpracticeand calibrationcurvesareusuallynecessaryforeachanalytethatmustbequantitated.
Inordertobettermimicthein fluenceofthematrixontheanalyteresponse,itiscommon touseintheanalysesaninternalstandardatconstantconcentrationthatwillgeneratea signal yi.s. foranystandardorsample.Thecalibrationsarethengeneratedusingtheformula:
Theuseof expression1.2.2 forcalibrationinsteadof expression1.2.1 isexpectingthatany influenceofthematrixonthevalueof y willalsoaffect yi.s. suchthatthematrixeffectswillbe compensated.
Somelinearcalibrationsdonotpassthroughtheorigin,andthecalculationoftheconcentrationfromtheresponseisdoneusingarelationoftheform:
Thistypeofdependencemayindicatesomeproblemswiththeparticularanalytical method,suchassampledecompositionorlossofsampleinthemeasurementprocess when a isnegative,orabackgroundinterferencewhen a ispositive.
Besideslineardependenciesbetweentheconcentrationoftheanalyteandthesignal y, othertypesofdependenciesarealsoencountered.Forexample,quadraticdependenceis rathercommonforsignalvs.concentrationdependence,mainlyforwiderangesofconcentrationsoftheanalyte.
Adifferentquantitationtechniquebesidestheexternalcalibrationisthatofstandardaddition.Thestandardadditionmethodcanbeusedtoanalyzeanunknownsamplewithoutthe useofacalibrationcurveobtainedinseparateruns.Thesampletobeanalyzedhasan unknownconcentration C0 ¼ q0/V0,where q0 istheunknownamountinthesampleand V0 istheknownvolumeofthesampletobeanalyzed.Itmustbeassumed,however,that thedependenceofthesignalontheconcentrationfollows expression1.2.1 (andnot expression1.2.3).Asetofknownamountsofanalyte{qi} i ¼ 1,2.n areaddedtothe unknownsample,leadingtotheconcentrations Ci ¼ (q0 þ qi)/(V0 þ Vi),where Vi isthevolumeoftheaddedsolutionwiththestandardand C0 ¼ q0/V0.Therelationbetweentheconcentration Ci andthesignal yi isinthiscasegivenbytherelation:
Thevaluesfor C0 and b (asunknownparameters)canbeobtainedfromtheadded amounts{qi}i ¼ 1.n andthesignalmeasurements{yi}i ¼ 0,1.n using,forexample,leastsquares fitting.Thestandardadditionmethodcanbeusedevenwithasingleaddedamount
C ¼ a þ by (1.2.3)
totheunknownsample.Ifonesingleaddition q1 ismadetothesample,twosignals y0 and y1 aregeneratedcorrespondingto C0 and C1.Thetwoequationsoftheform 1.2.4 areasfollows:
Thevaluefor qx canbeobtainedfrom expression1.2.5 andthevaluefor Cx ¼ C0 isgivenby theformula:
Whentheadditionofthestandarddoesnotdilutethesample(V1 ¼ 0), expression1.2.6 can bewrittenintheform:
Theaccuracyofstandardadditionissignifi cantlybetterwhenmorethanonestandard additionisperformed.Thecalculationofconc entrationusingstandardadditionusingonly oneadditionisinsomerespectsequivalenttousingacalibrationcurvewithonlytwo points.
Otherprocedurescanalsobeusedforquantitation(see,e.g., [8]).Oneoftheseprocedures usesaresponsefactorforquantitation.Thisresponsefactoriscalculatedastheratioofthe responses yA and yB oftwocompounds A and B atthesameconcentration, A beingthe analyteand B theinternalstandard.Theratioofthetwosignalsisusuallyobtainedasan averageofseveralmeasurementsgivingtheresponsefactor:
Ideally,thevaluefor fA remainsconstantforanintervalofvaluesforthepairsofconcentrationsofthestandardandtheanalyte.Theunknownconcentration CA,x ofcompound A is thenobtainedbymeasuringinthesamerunthesignalofthecompound A tobeanalyzed (atunknownconcentration)andthesignalofthestandard B (atknownconcentration CB) usingtheformula:
Typically,itisrecommendedthattheanalyte A andtheinternalstandard B arechemically similarorevenidenticalbutisotopicallylabeled.
Theselectionofaspecificquantitationproceduredependsonnumerousfactorssuchas propertiesofthematrix,concentrationoftheanalyteinthesample,andavailabilityofisotopicallylabeledstandards.Althoughideallyallproceduresshouldgeneratethesameresult, somedifferencesexistbetweendifferentprocedures.Thecalibrationwithpurestandards hasthedisadvantagethatthematrixmayinfluencetheinstrumentresponse,andforreal samplestheresponsetothesameconcentrationoftheanalytemaybedifferentthanthe responseforpuresolutions.Theuseofaninternalstandardanduseof expression1.2.2 for calibrationrequiresthattheinternalstandardresponsemustbeconstantineverysample.
FIGURE1.2.4 Diagramsuggestingtheiterativeprocessofestablishingamethodforanalysis.
Forthecaseofanalyzingmultipleanalytesandoneinternalstandard,anyerrorinthe internalstandardmeasurementwillaffectalltheothermeasurements,whichcanbeadisadvantage.Theuseofstandardadditiontakesintoaccounttheinfluenceofsamplematrix,and whenusedwithmultiplestandardadditionsprovidessufficientreliabilityoftheprocedure. However,conditionssuchasthelinearityofthedependenceandzerovaluefortheresponse intheabsenceoftheanalyte(dependenceoftheform 1.2.1)mustbesatisfiedforitsapplicability.Theuseofaresponsefactorforquantitationrequiresidenticalbehaviorofthe compoundusedasinternalstandard B andoftheanalyte A.
DecisionProcessinSelectingaMethodofAnalysis
Thepathforselectionofamethodofanalysisisaniterativeprocess.Eachpartoftheanalysishasitsowncharacteristicsandthesamplecharacteristics,decisionsonsample preparationandeachcomponentofthecoreanalyticalmethodmustbetakenintoconsiderationwithimplicationsfromonephasetoanother.Thisprocessisschematicallyillustratedin Fig.1.2.4.
Thecycleindicatedin Fig.1.2.4 canbefollowedonlyonce,butitispossibletoberepeated asdecisionsatonestepareinfluencedbythosefromprevioussteps.Inthedecisionprocess allthepreliminaryinformationshouldbeincluded,butalsoanyadditionalinformation gatheredalongthedecisionprocessandpossiblyresultsfrompreliminaryexperiments. Thetimefactorisalsotobetakenintoconsiderationinthedecisionprocess,theimplementationofananalyticalmethodusuallyhasadeadline.
1.3STATISTICALEVALUATIONOFDATAANDCRITERIAFOR METHODVALIDATION
Theresultsfrombothqualitativeandquantitativeanalysiscanbeprocessedusingstatisticalconcepts(see,e.g., [8]),althoughstatisticsismorefrequentlyappliedfortheevaluation ofthequantitativeresults.Accuracy(nearnesstothetruevalue)andprecision(repeatability oftheresults)arethemostcommonparametersgeneratedusingstatisticalevaluations. However,otheraspectsrelatedtoanalyticaldatacharacterizationhaveastatistical interpretation.
PrecisionandAccuracyinQuantitativeChemicalAnalysis
Inmostquantitativeanalyticaldeterminationsthemeasurementoftheamountorofthe concentrationofananalyteisperformedusingthedependenceofameasuredsignal y on theconcentration(orquantityoftheanalyte)
Theamountortheconcentration x canbecalculatedbasedonthedependencedescribed byrelation 1.3.1 usinganumberofproceduresthatgenerateacalibrationfunction:
Whenmorethanonemeasurementisperformed,theresultsfor x aretypicallyscattered aroundaspeci ficvalue,themeasurementsbeingalwaysaffectedbyerrors.Theerrorsofmeasurementsareclassi fiedassystematic(determinate)orrandom(see,e.g., [16]).Systematic errorsaregeneratedbyaspecificcause,andtheyaffecttheaccuracyoftheresults.Random errorsdonothaveanassignedcause,andtheyaffecttheprecisionofthemeasurement. Precision referstothereproducibilityofmeasurementwithinaset,indicatingthescatteror dispersionofasetofmeasurementsabouttheircentralvalue [12].Forexample,repeated determinationoftheamountorconcentrationofananalytegeneratesforeachmeasurement j avalue xj.Ifthenumberofmeasurementsis n,theywillgenerateasetofmeasurementsalso indicatedinstatisticsasa sample {x1, x2.xn}ofmeasurements(nottobeconfusedwith sample ¼ specimeninachemicalanalysis).Forthisset,an average (orthe mean) m,anda standarddeviation (SD) s canbeobtainedusingthefollowingformulas:
and:
Thestandarddeviation s showsthedistributionofmeasurementsaboutthemeanand characterizes precision.Asmallstandarddeviationindicatesgoodprecisionforthesetof measurements.Both m and s areexpressedinthesameunits.Arelativestandarddeviation %(RSD%)isfrequentlyusedinsteadofstandarddeviation s forthecharacterizationof precision,anditisexpressedbytheformula:
Insteadofstandarddeviation s,itisalsocommontouseforprecisioncharacterizationthe value s 2 called variance. Instatistics,anyobtainedsetofmeasurements{x1, x2.xn}representsa sample ofvalues takenbyarandomvariable x.This sample isapartofaninfinitesetofvaluesthatvariable x mayhave.Thisinfinitesetofvaluesisknownasa population.Similartotheaverage m
andstandarddeviation s forone sample,the population ischaracterizedbythemean m ofthe population andthestandarddeviation s ofthe population whicharegivenbytheformulas:
Sinceaninfinitenumberofmeasurementscannotbeperformed,thetruevaluesfor m and s arenotknown,andtheyareapproximatedwith m and s forlargenumberofmeasurements n Thetrue(ideal)valuefor m representstheidealresultforallthemeasurements,butdoesnot necessarilyrepresentthetruevalueofthemeasuredquantity.Thedifferencebetween m and thetruevalue(ifknown)ofaquantityiscalledthe bias.Thistruevaluecanbe,forexample, thelevelofananalyteinastandard(regardlessofthefactthatmakingofthestandardcan alsobeaffectedbyerrors).Anotherconventionforatruevalueistheassumptionthatitisthe one “generallyaccepted.” Accepting m0 asatruevalueofthemeasurementandtakingthe populationmean m z m,the bias isapproximatedbythefollowingformula:
Amethodisconsideredaccuratewhenthebiasisverysmall(orevenzero).
Forthesamequantity,morethanonesetofmeasurementscanbeperformed(e.g.,setsof measurementsofthesamesampleperformedindifferentlaboratories),generatinganumber of samples ofdata,eachonewithitsownmean m1, m2, . mk.Althoughexpectedtobecloseto eachother,thesemeansarenotnecessarilyidentical.Atotalmean m canbeobtainedfrom thesepartialmeans, m beingequalwiththeaverageofallmeasurements(additionisassociative).Thestandarddeviationofthemean m isknownastandarderrorofthemean sm anditis expressedbytheformula:
Onequestionregardingthedistributionofmeasurementsabouttheirmeanistheexpected frequencyofoccurrenceofanerrorasafunctionoftheerrormagnitude.Itisexpectedthat largerrandomerrorswillbelessfrequentthansmallerrors.Themostcommonlyutilized function,whichdescribeswelltherelativefrequencyofoccurrenceofrandomerrorsinlarge setsofmeasurements,isgivenbytheGaussformula:
Thisfrequencyfunction f(x)(knownasGaussiandensityfunction)showsthatthepointof maximumfrequencyisobtainedforthemean(when x ¼ m),thedistributionofpositiveand negativeerrorsissymmetrical,andasthemagnitudeofthedeviationfromthemean increases,anexponentialdecreaseinthefrequencytakesplace.Theerrorswiththerelative frequencyofoccurrencegivenbyrelation 1.3.10 haveaso-callednormaldistribution N(m,s).BesidesGaussiandensityfunctions,otherfrequencyfunctionsareknown(e.g., Studentfrequencyfunction).
For x withanormaldistributionN(m,s),thefollowingsubstitution:
leadstoavariable y withanormaldistributionN(0,1).
Thevalue(x m)isso-calledmean-centeredvalue,andbydivisionwith s, y isexpressed in s units(oritisstandardized).Mean-centeredstandardizedvariables y (standardizedvariates)arecommonlyusedinstatisticaldataprocessing.
Theareaunderthecurve f(x)for x < a,with f(x)givenbyformula 1.3.10,willgiveacumulativefrequencydistributionexpressedbytheformula:
Thecumulativefrequencydistributionisequaltotheprobability P for x tohaveavalue below a inanymeasurement.Theintegralof f(x)overthewholespacegives P ¼ 1.Thevalues offunction F(a)(for f(x)aGaussianfunction)areknownandtabulated(e.g., [14])orgivenin computerstatisticalpackages(e.g., [15]).
Themean mk ofasampleof n values{x1, x2.xn},isitselfavalueofarandomvariable m. Assumingthat x hasanormaldistribution N(m,s),therandomvariable m (mk isoneofthe possiblevaluesof m)takesacontinuousrangeofvalueswithanormaldistribution N(m,s/n1/2).
Amean-centeredstandardizedvariabledefinedbytheformula: willhaveanN(0,1)distribution.Withthehelpofvariable z itispossibletoevaluatehowclose thevaluesof m and mk areforacertainpopulation.Forthevariable z givenbyrelation 1.3.13, twovalues za/2 and z1 a/2 canbefoundsuchthattheprobabilityfor z ofbeingoutsidethe interval(-za/2,z1 a/2)isequalto a (areas a/2indicatedin Fig.1.3.1).Theinterval( za/2, z1 a/2)isindicatedasconfidenceinterval.For z beinginsidetheinterval( za/2,z1 a/2),or:
thevalueforprobabilitywillbe P ¼ 1 a.
Expression1.3.14 canindicatethelimitsfor mk withtheprobability P ¼ 1 a.Forthis purpose,itshouldbenotedthat za/2 ¼ z1 a/2,andrearranging expression1.3.14 theresult isasfollows:
Thevaluesfor z1 a/2 foraselectedprobability P andGaussiandistributionareavailablein theliterature(see,e.g., [14,17]).Suchvaluesaregivenfor P ¼ 1 a (two-sidednormaldistribution)orfor P ¼ (1 a)/2(one-sidednormaldistribution). Fig.1.3.1 showsthecurveN(0,1) onwhichareindicatedtwovalues za/2 and z1 a/2 suchthattheprobabilityfor z ofbeing outsidethecon fidenceinterval( za/2,z1 a/2)isequalto a (areaunderthecurve).Larger valuesfor a havecorrespondinglysmallervaluesfor P andsmallervaluesfor z1 a/2
SensitivityandLimitofDetection
Thesensitivityofaquantitativeanalyticalmethodcanbedefinedastheslopeofthecurve thatisobtainedwhentheresultofaseriesofmeasurementsisplottedagainsttheamount(or
FIGURE1.3.1 GaussiancurveN(0,1),showingtwovalues, za/2 and z1 a/2,suchthattheprobabilityfor z of beingoutsidetheinterval( za/2, z1 a/2)isequalto a (andareaunderthecurveis P ¼ 1 a showingprobability insidethesameinterval).
concentration)thatistobedetermined.Forthedependencedescribedby expression1.3.1,the sensitivityisdefinedasthe firstderivativeofthefunction FðxÞ or:
Itiscommonthatthedependencedescribedby expression1.3.16 isalinearfunction(see expression1.2.3)andhastheexpression:
Inthiscase,thesensitivity S isequaltotheconstant b.Sensitivitycanthereforebedeterminedfromthecalibrationcurveforthemethod.Fornonlineardependenciesthedefinition stillcanbeapplied,butthesensitivityisnotconstantforallconcentrations.Manymethods haveonlyarangeoflineardependence,withanupperregionandalowerregionthatare notlinear.Inthenonlinearregions,thesensitivityvaries,andforthisreasonsensitivityis notnecessarilyaconvenientwaytocharacterizeananalyticalmethod.Itiscommonthatconcentrationsbelowacertainvalueshowasensitivitydecrease.Thedeviationfromlinearityat lowerconcentrationsinchromatographicanalysisislikelyduetoacombinedeffectoftheloss ofasmallamountofsamplethatisadsorbedirreversiblyinthechromatographicsystem.
Theconcentrationofananalytebelowwhichthedetectionisnotpossibleistypicallyindicatedaslimitofdetection.Foraquantitativecharacterization,thedetectionlimitcanbediscussedintermsofsignalandtransformedintermsofamountorconcentrationusingthe calibrationfunction.Thesignal y forananalyticalmeasurement(seerelation 1.3.1)isinfact thedifferencebetweenthesignalforasample ys andthatofablank yb,bothaffectedbyerrors. Thesignalsfortheblank(orthenoise)areassumedtohaveamean mb andthesignalsfromthe sampletohaveamean ms.Thestandarddeviationsforboththeblankandthesamplecanbe consideredequaltothesamevalue sy ¼ s.Aspecificvalueofthesignalcanbeselected(arbitrarily)andconsideredasnotgeneratedbythenoise.Thisvalueisindicatedasdecisionlimit L [18] wherethesignalisnoticeableand:
Assumingthattheerrorsinthesignaloftheblankandsamplehaveanormaldistribution, theprobabilitytoobtainsignalsfromtheblankhigherthanthedecisionlimit L isgivenbythe expression:
In expression1.3.19,thefunction f isgivenbyformula 1.3.10 (Gaussiandensityfunction) thathas m ¼ mb (s,thestandarddeviationisthesameforboththeblankandthesample).The decisionlimit L intheintegralcanbeexpressedintermsofsignalsfortheblankusingthe expression:
Theprobabilitytoconsideranoiseasbeingsignalfromtheanalyteforavaluehigherthan L isgivenby P ¼ a.For k ¼ 2.33theresultingvaluefortheone-sidednormaldistribution gives a ¼ 0.01(or1%ifexpressedinpercent).Therefore,ifthesignalishigherthan mb þ 2.33 s,theprobabilityoffalsepositives(signalfromthebackgroundtobeconsidered analyte)is1%.
Asignal y with ms ¼ L has,however,theproblemofpossiblygeneratingfalsenegatives. Theprobabilityoffalsenegativesisgivenbytheexpression:
In expression1.3.21, f(ys)isaGaussianwith m ¼ ms.Thisprobabilityis P ¼ 0.5(or50%) becausethenormalcurveissymmetricalaroundthemean.Therefore,thepossibilityoffalse negativesisveryhigh,and50%ofthetruepositiveswillbeconsiderednoise.Thissituationis depictedin Fig.1.3.2.
Ahighersignal(highervaluesfor ms)andmaintaining L ¼ ms willcontinuetodiminishthe chancesforfalsepositivessincethearea a willbecomesmallerandsmallerasthesecond Gaussianmovestohighervaluesof y.However,maintaining L ¼ ms doesnotmodifythe probabilityoffalsenegatives.Forachosenprobability P ¼ 0.01ofobtainingafalsenegative, thecorrespondingvalueofthesignalcanbecalculated.Thisvalueisnotedby D,shouldbe smallerthan ms,andhastheexpression:
Fortheprobability b ¼ 0.01,theresultingvaluefortheone-sidednormaldistributionis k’ ¼ 2.33.
ForL ¼ D,thesubtractionofrelation 1.3.22 fromrelation 1.3.20 leadstotheexpression:
Theoverlappingoffrequencyfunctionsforthetwoprobabilitiesisshownin Fig.1.3.3
Theresultoftheseconsiderationsisthatinordertohaveaprobabilityof1%forafalsenegativesignal,andaprobabilityof1%forafalse-positivesignalrequires k þ k’ ¼ 4.66, andinthiscaseitcanbewritteninthefollowingformula:
to
FIGURE1.3.2 Graphshowingonthehorizontalaxisthesignal y,thevalues mb and ms aswellasthedecisionlimit
L ¼ ms.Theverticalaxisgivesthefrequencies f(y)ofoccurrenceforthevalueofameasurement,where f isgivenby relation 1.3.10 anddescribesanormaldistribution.
FIGURE1.3.3 Graphshowingonthehorizontalaxisthesignal y,thevalues mb and ms aswellasthedecisionlimit
D ¼ ms k’ s Theverticalaxisgivesthefrequencies f(y)ofoccurrenceforthevalueofameasurementwhere f isgiven byrelation 1.3.10 anddescribesanormaldistribution.
Ifthesignalisconsidered S ¼ ms mb andthenoiseistakenas N ¼ s,formula 1.3.24 is equivalentwiththeformula: S=N z4:66(1.3.25)
Forthedetectiontobepossibleinanalyticalpractice,avalueof S=N z3thatisslightlysmaller than4.66wasconsideredtobesufficient(withlowerprobabilitythan1%forfalsenegatives,and higherthan1%togivefalsepositives).Thevalue S=N ¼ 3hasbeenadoptedinanalyticalpracticeforthedefinitionof detectionlimitorlimitofdetection (LOD) [19 21].Theestimation of S=N ¼ 3inasetofchromatographicmeasurementscanbedonedirectlyontheplotof chromatograms(typicallyusingthecapabilitiesofthedata-processingsoftwareofthechromatographicinstrument).Theconcentration(oramount)generating S=N ¼ 3istakenasLOD.
Althoughthetheorysupportstheselectionofthevalue S=N z3forLOD,inchromatographicpracticethisdefinitionisnotveryinformative.Thenoiseistypicallymeasuredas thevariationindetectorresponseatthebaseline(differencebetweentheminimumandthe maximumofthebaselinesignal),andthesignal S forapeakismeasuredasthedifference
fromthemaximumofthepeakandtheaveragevalueofthebaselineforarangearoundthe peak.Thesimplestwaytocheckthebaselinenoise N istomeasurethevariationofthedetectorresponseforachromatogramwithoutmakinganinjection [22].Itisalsocommonto measurethevariationofthedetectorresponseinanypartofthechromatogramwhereno extraneouspeaksarepresent.Theplacewherethenoiseismeasuredstillremainsarbitrary, inspiteofsomerecommendationsthatthenoiseshouldcoveraspeci ficrangeoftime,wider thanthechromatographicpeakoftheanalyte.Inadditiontothat,thenoisecanbedifferent fromonechromatogramtoanother.Thesignalandnoisemeasurementsareexemplifiedin Fig.1.3.4 thatshowsthemeasurementoftwosignalsandtwoselectionsforthenoiseleading totwodifferentvalues N 1 and N 2 .
AbetterprocedureforLODevaluationisbasedonrepeatedmeasurementsandtheuseof theapproximations ms mb z m and s ¼ s (s isalsoindicatedasSD).Thechoice S=N ¼ 3is equivalentwiththeexpression:
Expression1.3.26 indicatesthatfordetectiontobepossible,aconcentration(orlevel)at leastequalto3s forasetofmeasurementsatlowconcentrationisnecessary.Basedon expression1.3.26 itresultsthatforasetofmeasurementsatlowanalyteconcentrationthedefinition ofLODshouldbebasedonthefollowingexpression [23,24]:
LOD ¼ 3s (1.3.27)
TheconventionoftakingLODequalwith3increasestheprobability P ofobtainingafalse positiveanddecreasestheprobability P ofobtainingafalsenegative(in Fig.1.3.3 asmaller coef ficientfor s movesthepoint D towardlowervalues).Toaddressthisproblem,anadditionalparameterindicatedas limitofquantitation (LOQ)hasbeenintroduced.Thedefinitionof LOQisthefollowing:
LOQ ¼ 10s (1.3.28)
LOQindicatestheminimumamount(orconcentration)thatcanbequantitatedbya speci ficmethod.Thechoiceof10s forLOQdecreasestheprobability P forobtainingafalse positivebutincreasestheprobability P ofobtainingafalsenegative.Topreventtheproblem
Signal 1 (S1)
Noise 1 (N1)
Noise 2 (N2)
Signal 2 (S2)
Average baseline
Ret. time
FIGURE1.3.4 Exampleofmeasurementoftwosignalsandtwoselectionsforthenoiseleadingtotwodifferent values, N 1 and N 2 .
ofgeneratingfalsenegatives,themeasurementswithresultsbelow10s arenotlabeledas “analyteabsent” but “analytebelowLOQ.”
PracticalLODandLOQ(PLODandPLOQ)
TheestablishingofLODandLOQforananalyticalmethodisveryimportant.Some variationintheprocedurefortheestablishingofLODandLOQisdescribedwitheach analyticalmethod.Althoughitiscommontomeasure s (orSD)forasetoflowstandards, thereisaknownfactthatitispossiblefortheanalysisofstandardstogeneratecleanerchromatogramsthantheanalysisofthesameanalytesinamatrix.Evenwhensamplepreparation offersproceduresformatrixcleanup,inmanyinstancesonlyapartialcleanupisperformed, andbesidestheanalytessomeadditionalmatrixcompoundsmayremainintheprocessed sample(residualmatrix).Theresidualmatrixcanbetoleratedwhenitdoesnotinterfere significantlywiththeanalytemeasurement,butthisdoesnotnecessarilyimplythatat verylowleveloftheanalytetheresidualmatrixhasnoeffectonsensitivity.Forthisreason, incertainanalyses,althoughthemeasurementisaccuratefortheanalytesaboveacertain level,thedeterminationsclosetotheLOQ(LOQdeterminedwithstandards)isdifficultor evennotpossible.Forthisreason,forsomeproceduresinsteadofreportingLODandLOQ obtainedwithstandards,(practical)PLODandPLOQarereported.Inthesecases,themeasurementofstandarddeviation s orthesignaltonoise S=N aremeasuredforalowlevelof analyteinthesamplewithresidualmatrixcomponents(possiblyafterpartialcleanup).Thisis notalwayspossiblesincesampleswithlowlevelsofanalytemaynotbeavailable.Theconcentrationoftheanalyteinthesamplecanbemeasuredasusual,usingstandardsorstandard additionprocedures.ThevaluesofPLODandPLOQaremoreusefulforpracticalapplication ofamethodwhentheyaredifferentfromLODandLOQ.
LinearityoftheInstrumentalResponseandLeastSquareRegression
Lineardependenceofananalyticalsignalandtheconcentration(ortheamount)ofananalyteisthecommonprocedureforquantitativeanalysis.Thedependencetypicallyhastheform ofaregressionlinewiththeequationgivenbythefollowingformula(see expression1.2.3):
Thesameformisvalidfor y,ananalyticalsignaland x,theanalyteconcentration,orfor y, theanalyteconcentration,dependinglinearlyonasignal x (e.g.,peakareainachromatogram). Inpractice,foranumberofvaluesfor x,{x1, x2.xn}anumberofresponses(measurements){y1, y2.yn}areobtained.Thevaluesof a and b mustbeobtainedsuchthattominimize(inabsolute value)forall i thedifferencesbetweenexperimentaldata yi andcalculateddata a þ bxi:
Thevalues ri arecalledresiduals,andfortheminimizationofall ri (i ¼ 1,2,.n),the followingexpressionmustbeminimal:
Sincetheprocedureinvolvestheminimizationoftheerrorsinthecalculatedvaluesfor y,it isknownaslinearminimummean-square(leastsquare)error fitting.Forexpressingfurther resultsrelatedtotheminimizationofexpression E(a,b),severalnotationsmustbeintroduced. Theaverageof xi valueswillbenotedwith mx,andtheaverageof yi valueswillbenotedwith my.Alsoseveralpartialsumsarenotedasfollows:
Theminimumisachievedfor E(a,b)whenitspartialderivativewithrespectto a and b isset tozero.Thisisexpressedbytheformulas:
Expression1.3.33 leadstoapairofequationsin a and b.Thesolutionofthesetwoequationsleadstothefollowingresultsfortheminimizationof E(a,b):
Thesameminimizationgeneratesresultsforthestandarddeviationsfortheresiduals sr, fortheslope sb,andfortheintercept sa
Anadditionalparametercalled coefficientofdetermination, R2,isutilizedforthecharacterizationoftheregressionline.Theformulafor R2 isthefollowing:
Perfect fitofexperimentaldata{y1, y2 yn}andcalculateddata{a þ bx1, a þ bx2, a þ bxn} isindicatedby R2 ¼ 1when Pn i ¼ 1 r2 i ¼ 0.Forlargerresiduals,thevalueof R2 decreasesand canhavevaluesbetween1and0.Thecalculationsfor a and b fortheregressionlineaswellas of sr, sb, sa,and R2 areavailableinExcelcomputerpackages(see,e.g., [25]).
Theminimummean-squareerror fittingcanbeappliednotonlytolineardependences,but alsotoothertypesofdependencesuchasthequadraticone.Thevaluefor R2 isidenticalfor
