Rapid optimization of gradient liquid chromatographic (LC) separations often utilizes analyte retention modelling to predict retention times as function of eluent composition. However, due to the dwell volume and technical imperfections, the actual gradient may deviate from the set gradient in a fashion unique to the employed instrument.
Trang 1journalhomepage:www.elsevier.com/locate/chroma
Tijmen S Bosa,c,∗, Leon E Niezenb,c, Mimi J den Uijlb,c, Stef R.A Molenaarb,c, Sascha Leged,
Peter J Schoenmakersb,c, Govert W Somsena,c, Bob W.J Pirokb,c
a Division of Bioanalytical Chemistry, Amsterdam Institute for Molecular and Life Sciences, Vrije Universiteit Amsterdam, De Boelelaan 1085, 1081 HV
Amsterdam, The Netherlands
b Van ’t Hoff Institute for Molecular Science (HIMS), University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
c Centre for Analytical Sciences Amsterdam (CASA), The Netherlands
d Agilent Technologies, R&D and Marketing GmbH, Hewlett-Packard-Strasse 8, 76337 Waldbronn, Germany
a r t i c l e i n f o
Article history:
Received 28 July 2020
Revised 15 October 2020
Accepted 9 November 2020
Available online 13 November 2020
Keywords:
optimization
multi-step gradients
gradient deformation
retention modelling
response functions
a b s t r a c t
Rapid optimizationofgradient liquidchromatographic(LC) separationsoftenutilizesanalyteretention modellingtopredictretentiontimesasfunctionofeluentcomposition.However,duetothedwell vol-ume and technicalimperfections, the actual gradient may deviate fromthe set gradient in afashion uniquetotheemployedinstrument.ThismakesaccurateretentionmodellingforgradientLCchallenging,
inparticularwhenveryfastseparationsarepursued.Althoughgradientdeformationhasbeenaddressed
inmethod-transfersituations,itisrarelytakenintoaccountwhenreportinganalyteretentionparameters obtainedfromgradient LCdata,hamperingthecomparisonofdatafromvarioussources.Inthisstudy,
aresponse-function-basedalgorithmwasdevelopedtodetermineanalyteretentionparameterscorrected forgeometry-induceddeformations byspecificLC instruments.Outofanumber ofmathematical dis-tributionsinvestigatedas response-functions,theso-called“stablefunction” wasfoundtodescribethe formedgradientmostaccurately.Thefourparametersdescribingthemodelresemblethestatistical mo-mentsofthedistributionandarerelatedtochromatographicparameters,suchasdwellvolumeandflow rate.Theinstrument-specificresponsefunctioncanthenbeusedtopredicttheactualshapeofanyother gradientprogrammedonthatinstrument.Toincorporate thepredictedgradient inthe retention mod-ellingoftheanalytes,themodelwasextendedtofacilitateanunlimitednumberoflineargradientsteps
tosolvetheequationsnumerically.Thesignificanceandimpactofdistinctgradientdeformationforfast gradientswasdemonstratedusingthreedifferentLCinstruments.Asaproofofprinciple,thealgorithm and retentionparameters obtainedonaspecific instrumentwereused topredict theretention times
ondifferentinstruments.The relativeerrorinthepredicted retentiontimeswentdownfroman aver-ageof9.8%and12.2%onthetwootherinstrumentswhenusingonlyadwell-volumecorrectionto2.1% and6.5%,respectively,whenusingtheproposedalgorithm.Thecorrectedretentionparametersareless dependentongeometry-inducedinstrumenteffects
© 2020TheAuthors.PublishedbyElsevierB.V ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)
1 Introduction
The majorityofmethods inliquidchromatography(LC) utilize
gradient elution,wherethefractionofstrongsolvent(e.g.the
or-ganicmodifierinreversed-phaseLC)ϕisgraduallyincreased
Ana-∗ Corresponding author: Tijmen S Bos Division of Bioanalytical Chemistry, Am-
sterdam Institute for Molecular and Life Sciences, Vrije Universiteit Amsterdam,
De Boelelaan 1085, 1081 HV Amsterdam, The Netherlands Telephone number:
+ 31640951663
E-mail address: t.s.bos@vu.nl (T.S Bos)
lyteretentiondependsonthemobile-phasecompositionand,thus,
ontheappliedgradientwhentheanalytemovesthroughthe col-umn.Consequently,modelsthat describetheretentionofanalytes whenusingagradientmustaccuratelyaccountforthetrueshape
oftheprogrammedgradient.Toautomateandacceleratethe devel-opmentofeffectivegradient-elutionmethods,computer-aided op-timizationtools,suchasChromSword[1],PEWS[2],Drylab[3]and MOREPEAKS(formerlyPIOTR)[4],employscanningexperimentsto establishtherequiredretentionparametersforeachanalyte [5,6] The gradient delayarising from thedwell volume (V D) of the LC system[7,8]is generallytakeninto accountduringretention
pre-https://doi.org/10.1016/j.chroma.2020.461714
0021-9673/© 2020 The Authors Published by Elsevier B.V This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )
Trang 2it isgenerallyassumedthat apartfromthisdelaytheactual
gra-dientdeliveredtothecolumnisidenticaltotheprogrammed
gra-dient.However,otherinstrument-relatedfactors,suchaserrorsin
temperatureandflowrate,willalsoinfluencetheseparation[9,10]
Gritti et al. have extensivelyinvestigated gradientdeformation in
reversed-phase LCand theeffects thereof onthe separation [11–
13].They were able to improveretention predictionfor fast
gra-dients ona singleinstrumentbytakingthe adsorption isotherms
ofindividual analytesintoaccount [11].Inthe samestudyitwas
shownthatforless-retainedcompoundstheresolutionwould
col-lapsewhenfastgradientsareappliedandtheauthorsproposedto
modifythegradienttopreventthisbehaviour
Gradientdeformationscanbecaused,forexample,byflow
im-perfectionscausedbyamixerorbyregulardispersioninthe
con-nection tubing Modest gradient deformation may be of limited
concern when theretention parameters obtainedusinga specific
instrument are exclusively used for optimization of gradients on
thesameinstrument However,becausedeformation ofthe
gradi-ent is dependent on themobile-phase delivery assembly, the
in-stalled injection devices and the (pre-column) connectors of the
instrument, the obtainedretention parameters cannot be usedto
accuratelypredictanalyteretentiononotherLCsystems.Acorrect
comparisonof(reported)retentionparametersacquiredonvarious
LC gradient instruments is only possible afteraccounting forthe
differencesintheactualgradientshapes[14]
Geometry-induceddeviationsfromprogrammedeluent
compo-sitionsarerelativelymostprominentinveryfastgradients,suchas
thoseencounteredinultra-high-performanceLC(UHPLC)orinthe
second dimensionofcomprehensivetwo-dimensionalliquid
chro-matography (LC× LC).Quarry et al. showedalreadyinthe1980s
thattheactualshapeoffastgradientprogramsinparticularcanbe
significantly deformed[15] Deformations can be induced by the
specific (mixing) properties and interactions of the two solvents
formingthegradient,aswellasbythegeometricalfeaturesofthe
LCinstrument.Retention-scanningexperimentscaninprinciplebe
conductedusingisocraticelution.However,whenapplyingthe
re-tention parameters thus obtained for predictinggradient
separa-tions,correctionforgradientdeformationisstillrequired[15,16]
The deformationofalineargradient dependsontheflow rate
(F)andtheslopeofthegradient,whichisthechangeinthe
vol-umefractionofmodifier( ϕ )dividedbytheduration ofthe
lin-ear segment of the gradient (t G) [17] The most-accurate
experi-mental methodto reveal the truegradient profile isthrough
de-tection ofachromophoricagentdissolved inoneofthe
gradient-forming solvents[15].Anotherapproachis throughinterpretation
ofisocraticallyacquiredretentionparameters[18],butthisrequires
a large number of runs [19] In silico accounting for the
gradi-ent deformation arising fromthe LC system wouldbe an
attrac-tive next step,as it can potentially be automated and requiresa
minimal numberof measurements Ideally, itwould improvethe
accuracyofpredictedoptimalgradientseparations
Inthispaper,wepresentanovelcomputationalstrategyto
es-tablish the effects ofgradient deformationscausedby the
geom-etry of the instrument, yielding geometry-independent retention
parameters from a limited number of gradient experiments We
demonstrate that theinfluence ofgradient deformation is
poten-tially significant and that it is worthwhile to correctfor this As
input data, ouralgorithm employs a measured gradient delay in
a water-watersystem Forouralgorithm,multiple response
func-tions were tested to determine the most accurate and best
in-terpretable model To incorporate the gradient deformation into
retention modelling, new models were derived that support any
numberofgradientsteps.Theusedexperimentalsetupexclusively
provides informationon thegeometry-inducedgradient
deforma-tion,butexcludesanyeffectsofthesolventandmixtures,suchas
viscosity,densityandmiscibilityeffects.Solventadsorption[20,21] andsolvatochromiceffectswerenotstudied
2 Theory
In this paper we employ the log-linear (“linear solvent strength”,LSS)modelforretentionprediction,butother retention modelsmaybeusedaswell
2.1 Retention time in LSS gradient elution with linear gradient
Inthelog-linearmodel(Eqn.1),k0representstheextrapolated retentionfactoratϕ=0andS representsthemagnitudeofchange
inlnkwithincreasingeluentstrength(ϕ)
This model is often referred to as the linear solvent strength (LSS)modelincombinationwithlineargradients[23]
Intheeventthatananalyteelutesbeforeaprogrammed gradi-ent,theretentiontime(t R ,before)isgivenby
wherekinitistheanalyteretentionfactoratthestartofthe gradi-entandt0 depictsthecolumndeadtime.ByincorporatingtheLSS modelintothegradientequationitfollowsthat
1
B
ϕfinal
∫
ϕinit
dϕ
k ϕ +t R−τ− t G
k final =t 0−t init+t D
k init
(3)
whereB (dϕ/d t) is theslope of a gradientrunning fromϕinit to
ϕfinal, tinit is the initial isocratic time, t D the dwell time, k ϕ the retentionfactoratacertainfractionofstrongsolventandτ=t D+
tinit+t0.Schoenmakerset al.derivedequationstopredictretention timesduringalineargradient[5]
t R ,gradient= 1
SB ln
1+SB · kinit
t 0−t init+t D
k init
aswellasretentiontimesintheeventthattheanalyteelutesafter thegradient,withretentionfactoratthefinalconditionskfinal,and gradienttimet G
t R,after= k final
t 0−t init+t D
k init
BS
1−k final
k init
2.2 Describing the shape of the geometry-corrected gradient
Characterizing the shape of the geometry-corrected gradient (GCG) startsby finding a modeland relatedparameters that ac-curatelydescribeshowtheprogrammedgradientisinfluencedby the instrument Thisresponse of thesystemcan be expressed in the form of a distribution or a so-called response function The whole gradient experiences the same geometrical effects as ex-pressedthroughthisresponsefunction.Summingall priorsignals resultingfromthe response function atanypoint intime results
intheGCG.Examplesofaprogrammedgradient,the correspond-ingresponsefunctionandtheGCGaredepictedinFig.1
The response function can be expressed using a mathemati-cal distribution, the properties of which can be described using itsstatisticalmoments [24].Agraphicaloverviewofthemoments [25] and their parametrized symbols, which are used inthis pa-perasinstrumentparameters,isshowninFig.2.The correspond-ingequationscanbefoundinSupplementaryMaterialsectionS-1 The zeroth moment (A) isthe area Inour case, thismoment is adjustedtobeidenticaltothecompositionatacertaintimepoint The first moment (μ ) is normalized for the area and gives the centreofgravity ofthedistribution(mean), whichisequaltothe dwelltime ofthesetup.Thevariance(σ2)orscaleisthe central-izedsecondmoment(i.e.correctedtothefirstmoment)andwhich
Trang 3Fig 1 Schematic illustrating the conversion of a programmed linear gradient to the
GCG using response curves at different time points Blue line: programmed gradi-
ent Red lines: Response function Magenta: GCG shape
Fig 2 The common properties of a distribution with the corresponding moment,
visualisation thereof, traditional symbols and symbols ( A , μ , σ , S, K) of the param-
eterized moments ( ∼, δ , γ , β , α )
iscorrelatedtothewidthofthedistribution,whichcapturessome
of the flow profile of the instrument The skewness (i.e
magni-tudeoftailing/fronting)isthestandardizedandcentredthird
mo-ment (i.e.corrected tothevariance andthefirst moment) andis
instrumentdependentandcorrelatedtothekurtosis(i.e.degreeof
flattening).Thekurtosis(K)isthestandardizedandcentredfourth
moment(i.e.correctedtothevarianceandthefirstmoment).The
skewness andthekurtosis together describethe degreeoftailing
andtheshapeofthedistribution.Theresponsecurvethuscan
de-scribe the deviation from the programmedgradient arising from
anypossiblesource,suchasthedwellvolume,flowand
imperfec-tions therein (e.g. flow turbulence causedby the mixer or sharp
bendsintubing)
3 Experimental
3.1 Instrumental
Experimentswere carriedout onthreeAgilentLCinstruments
(Agilent Technologies, Waldbronn, Germany) Instrument 1 was
an Agilent 1290 Infinity II series equipped with a binary pump
(G7120A) equipped with a 35-μL JetWeaver mixer, an
autosam-pler(G7129B),acolumnoven(G7116B)andadiode-arraydetector
(DAD,G7117B).Instrument2wasanAgilent1100seriesequipped
withaquaternarypump(G1311A),anautosampler(G1313A),a
col-umn oven (G1316A) and a DAD (G1365B) Instrument 3 was an
Agilent 1290 Infinity II series equippedwith a quaternary pump
(G7104A)equippedwitha35-μL JetWeavermixer,an autosampler
(G7167B),acolumnoven(G7116B)andaDAD(G7114B)
For all measurements involving chromatography, the same XBridge BEHShield RP18column (50mm × 4.6mm i.d.,2.5-μm particles;Waters,Milford,MA,USA)wasused
3.2 Chemicals
The eluent was prepared using deionised water (resistivity 18.2 M cm; Arium 611UV, Sartorius, Germany) Acetone and acetonitrile (ACN) of HPLC grade were obtained from Biosolve (Valkenswaard,TheNetherlands) Emodin(EMOD),sudan I(SUD), phenol(PHEN),anthracene(ANT),toluene(TOL)andthioureawere obtainedfromMerck– Sigma-Aldrich(Darmstadt,Germany)
3.3 Analytical procedures 3.3.1 Sample preparation
All test-compound solutions were prepared in ACN and con-tained100mg•L−1ofthioureaast0marker.Theapproximate con-centrationsofthe solutionswere:EMOD,250mg•L−1;TOL, 1500
mg•L−1;SUD,PHENandANT,each500mg•L−1
3.3.2 Chromatographic method
Measurementsoftheactualgradientshapewereperformedon allLCinstrumentswithoutacolumnatflowratesof0.25,0.5and 0.75mL•min−1.SolventAwaswaterandsolventBwaswater con-taining0.1vol% acetone.Aninitialisocratic time(100%A) of0.25 minwasused.Thegradientranfrom0to100%Bin0.5,1.0or1.5 min.All gradientmeasurements were performedintriplicate.UV detectionwas performedat 210 nm with a bandwidth of 4nm; the referencewavelength was 360nm witha bandwidth of 100
nm anda slitsize of 4nm The samplingrateforInstruments 1 and3was160 Hz,whileforInstrument 2itwas20 Hz.Column ovensweresetto30°C
Retention-timemeasurementsofthetestcompoundswere per-formed on all instruments withthe LC columninstalled using a flowrateof0.5mL•min−1 withaninitialisocratictime(100% sol-vent A) of 0.25 min The gradient ran from 0 to 100% B in 0.5, 1.0or1.5min,followedbya10-minisocratichold.SolventAwas ACN-water (5:95, v/v) and solvent B wasACN-water (95:5, v/v) Betweenmeasurements,10minofequilibrationtimewasallowed Forallanalytemeasurements,thesametwobottlesofsolvent mix-tures(A (5:95, v/v)andB (95:5, v/v))were used whichwere ul-trasonicated before use.The injectionvolume was 5μL All test-compoundsolutionscontainedat0 markerandweremeasured in-dividually.Measurementswere repeated4timesandthus5 mea-surements per solution Detection was at 254 nm with a band-widthof4nm;thereferencewavelengthwassetto360nmwith
abandwidthof100nmandaslitsizeof4nm.Thesamplingrate was160 Hzforinstruments 1and 3and20Hz forinstrument2 Columnovensweresetto30°C
3.3.3 Data treatment
AllalgorithmswerewritteninMATLAB2019bupdate3 (Math-works, Natick, MA, USA) Measurements of the actual gradient shape were first normalizedbetween 0 and1, after which three identical measurements were averaged to minimize noise The recordedgradient measurements weretruncatedto aperiod of6 minutes for establishing the response-function parameters to re-duce the computation time Retention times and t0 values were averagedbeforefittingtheretentionmodel
4 Results and Discussion
Our strategy encompasses three steps to correct the mea-sured retention parameters for the actual gradient Firstly, the instrument-specificresponsefunctionthatdeterminestheshapeof
3
Trang 4Table 1
Obtained sum-of-squared errors (SSE) values for the regression experiments determined on Instrument 1 with flow rates of 0.25, 0.5 and 0.75 mL •min −1 and gradients times of 0.5, 1 and 1.5 min Color scale from red through yellow (50%) to green representing high to low SSE values
theactualgradientisdetermined.Secondly,thisresponsefunction
isusedtopredictthecorrectedshape(GCG)ofagradientof
inter-est TheGCG canthen beused tomoreaccurately determinethe
LSS model parameters describing analyte retention Finally, these
latterretentionparameters areusedto predicttheretentionona
differentinstrument,usingitsspecificresponsefunction
4.1 Gradient-profile description
4.1.1 Selecting the optimum response function
Generally,twomethodsexisttodescribethegradient
deforma-tion One relieson a direct fit of the gradient curves The other
method, describes how every timepoint of the initial gradient
passesthroughthedetector.Thefirstapproachshouldallowa
de-scriptionofthestart(quickbend),middle(linear),andendofthe
gradient (slowbend).Thismay,forexample,be achievedwithan
alternative-skew exponential power distribution [26], which was
slightly alteredforthispurpose,resultingin anaccurate
descrip-tion of the profile However, this first approach works only for
linear gradients and no chromatographically meaningful
correla-tions betweentheparameters describingdifferentgradientscould
be found,resultinginlargeerrorswhenpredictingnewgradients
Therefore,inthispaperthesecondapproachisfollowed.Whichis
notlimitedtolineargradients
Multiple mathematical distribution functions were tested for
their suitabilitytodescribetheresponsefunctiontoconstructthe
GCG by applying the function to all time points The
gradient-profilemeasurementsonInstrument1wereusedfortheinitial
ex-ploration ofthe differentresponsefunctions The sumofsquared
errors(SSE)ofthefitteddistributionsarereportedinTable1and
Fig 3 The four-parameter stable function is referred to as
“4p-Stable”.Totesttheinfluenceoftheasymmetryandtailingfactorin
thestablefunction,twostablefunctionswithoneofthese
param-etersfixedtoits“Gaussian” stateweretested.Thesewerereferred
toas“3p-Stable”.BesidesthestableandGaussian functions,other
distributions that can express asymmetrical tailing were tested
Thesespecific distributionsweretestedbecauseoftheirabilityto
describesignificanttailing[22],whichisrequiredtocovertheslow
roundingattheendoftheactualgradient.Equationsofthetested
distributions canbefound inSupplementary MaterialsectionS-2
Twoexamplesper fitteddistributioncan befoundin
Supplemen-tary Material section S-3.The four-parameterstablefunction was
foundtoyieldthesmallestSSEandisreferredtoasthestable
dis-tributionintherestofthepaper
The selected stable distribution contains four parameters
(δ,γ,β,α ) which respectively resemble the four statistical
mo-ments(μ,σ, S , K),althoughtheyaredefineddifferently[22],as
in-dicatedinFig.2
Fig 3 Boxplot of the SSE values per type of distribution used for the response
function describing the observed gradient determined on Instrument 1 with flow rates of 0.25, 0.5 and 0.75 mL •min −1 and gradients times of 0.5, 1 and 1.5 min
The response functions tested all are distribution functions, basedonthesameunderlyingmathematics,buttheirunique prop-erties can be described through their characteristic function ex-pressedasϑX(u).Thiswayofdescribingthestablefunctionisthe mostconcisewaytocoverallpossibleinterpretationsofthestable function[22].TheϑX(u)isdefined,whileuisintherealdomain, as
ϑX(u )= E e iuX
−∞e iux dF(x )= ∞∫
−∞e iux f (x )dx (6)
whereuisthexdomainuptotheupperlimitofx,iisthe imag-inary unit and e is Euler’s number E(x) is the expected value,
F(x)isthecumulative distributionfunctionand f(x)isthe prob-abilitydensityfunction X states thevariables intheequation In caseofthestabledistributiontheseequal( δ,γ,β,α ).The tailing parameter( α )isrestrictedbetween0<( α )≤ 2andthe asymme-tryparameter( β )isrestrictedbetween-1≤β≤ 1.Whenαequals
2,thedistributionisGaussian.Whenβ ispositivethedistribution
istailingandwhenβ isnegativethedistributionis fronting.δ is themeanparameterandγ thescaleparameter Bothare positive realnumbersfortheapplicationasresponsefunction
TheϑX(u)ofthestabledistributionisdefinedasEqn.8,where
αdoesnotequal1andX equals( δ,γ,β,α ).Inthisequation,the signlogicisdefinedasfollows:
sign (u )=
−1 u < 0
1 u > 0
(7)
Trang 5Table 2
Response function parameters obtained by fitting gradient-profile measurements with the four-parameter stable function for various settings on Instrument 1
Flow rate
(mL •min −1 ) t G (min)
Mean parameter
( δ ) Mean parameter ( δ )∗ Flow rate
Scale parameter
( γ ) Scale parameter ( γ )∗ Flow rate
Asymmetry parameter ( β ) Tailing parameter ( α )
Table 3
Response function parameters obtained by fitting the gradient profiles obtained on three different instruments with the four-parameter stable
function Parameters were obtained by fitting a series of gradient profiles (see Table 2 ) simultaneously The variance and mean parameters
were normalized by multiplying by the flow rate
Instrument Mean parameter ( δ )∗ Flow rate Scale parameter ( γ )∗ Flow rate Asymmetry parameter ( β ) Tailing parameter ( α )
Fig 4 Representation of the response functions of Instruments 1 (A), 2 (B), and 3 (C) for a flow rate of 0.5 ml/min AU indicates arbitrary units
ϑ (u |X )=E e iuX
= e −γ α|u|α(1+i β ∗sign ( u )·tanπα
2 ·(( γ | u|)1 −α−1)+i δ u) (8)
Application of the Fourier-inversion theorem to the ϑX(u)
yields thefollowing equation forthe probability densityfunction
ofthestabledistribution,wheret isthetime
f X(t| ϕ )= ϕ
2π
∞
∫
The above results(Table1,Fig 3)show that the stable
distri-butionfunctiondescribesthedatabest.Anadditionaladvantageis
thatitsparametersresemblethestatisticalmoments,allowingthe
usertoexplain differencesbetweeninstrumentsinaless-abstract
waythanwithsomeoftheotherdistributions
4.1.2 Determining instrumental parameters
Fittingtheselectedresponsefunctiontoeachexperimental
gra-dient profile obtained with Instrument 1 yielded scale ( γ ) and
mean ( δ ) parameters that appeared inversely correlated to the
flow rate Table2 provides the best-fit response-function
param-eters for each setting Moreover, the asymmetry factor (β) was
found tobe alwayspositive anda tailingfactor(α) lowerthan 2,
indicating tailing of the distribution function This was expected
since any additional instrument component of the flow-delivery
systemmayinduceflowimperfectionswhichisexpectedtoresult
ina gradientdelay,andnot anacceleration.Thiswouldprimarily
be observable inthemean parameter(δ ),butifthisdelayisnot
uniformitleadstotailing.Flowimperfectionscanoccurdueto ir-regularflowinthemixerandotherpartsoftheLCinstrument In-teractionbetweenthesolventandtheLCinstrumentisexpectedto
beminimalinawell-designedandwell-maintainedsystem.Inthe presentexperiments(mixingwaterwithwatercontainingacetone) onlyinteractionofacetonewithsystemcomponentswouldbe re-flectedinthegradientprofile,whichwasassumedtobeminimal Parametersforeachinstrumentwereobtainedbyfittingallthe gradient-profile measurements simultaneously with the response curve (i.e the stablefunction), so asto incorporate theeffect of theflow ratewhichintroduces anadditionalerrorbetween mea-surements.Incaseof anrandom errortheresponse function can
bedefinedinlessdetailandthusresultsinamoreGaussianshape
oftheresponsefunction.Theobtainedparametersforthe individ-ual responsecurves are shownin Table3 InFig 4the resulting response functions are plotted for a flow rate of 0.5 mL•min−1 ForInstrument2theresponse functionis aGaussian distribution sincethetailingparameter( α )is2.Thiswascausedbythe down-ward drift in the detector response, which the regression model attemptedto compensateasseeninSupplementary Material sec-tion S-4 While this effect wascaused by the detector and thus hasno influenceof theactual gradient, themeasured data is in-fluenced However, this resulted in a Gaussian response function whichmeans thatthe deformationinthe formofbendingatthe endislesswelldescribedandthusmakingthecorrectionless ac-curate
5
Trang 6Fig 5 Measured (blue line) and modelled gradient profile (GCG; red dashed line)
for Instrument 1 using a gradient time of 1 min and a flow rate of 0.5 mL •min −1
AU indicates arbitrary units SSE between the measured and the modelled was
5.6 •10 −4
These responsecurveswere usedto describe theGCG.An
ex-ample is shown in Fig 5, where the measured gradient of
In-strument 1 (F = 0.5 mL•min−1, t G = 1 min) is plotted (blue
curve) with the reproduced gradient (GCG) overlayed (dashed
redcurve)
4.2 Retention modelling using n gradients
In order toimprove retentionmodelling, themethod to
com-pute retention times hadto be adapted to accommodate forthe
established GCG shapes The equations for the retention time
Eqns.4and(5)werederivedbyincorporatingtheretentionmodel
(Eqn.1) inthe generalgradient equation (Eqn.3) forthe caseof
alineargradient.TheGCGsasshowninFig.5wereapproximated
numericallyasaseriesofshortlinearsegments.Tofacilitatethis,
an adjustedgradient equationwasderived tohandleanynumber
(n)ofgradientstepswithdurationt n,slopeB n andinitial
compo-sition ϕn (withϕn+ 1 =ϕn + t n B n).The retentionfactoratϕn is
denoted by k n The derived formulas are shown in Eqn.s 10 and
11(See SupplementaryMaterial section S-5fora detailed
deriva-tion)
ϕ n+B n T
∫
ϕ n
dϕ
k ϕ =B n
t 0−t 1+t D
k 1 − t 2
k 2 − −t n
k n − 1
B 1
ϕ2
∫
ϕ1
dϕ
k ϕ − 1
B 2
ϕ3
∫
ϕ2
dϕ
k ϕ − − 1
B n−1
ϕ n
∫
ϕ n−1
dϕ
k ϕ
(10)
t R, a f ter n gradients= k n+1
t 0−t1 +t D
k1 −t2
k2− −t n
k n− 1
B1
ϕ2
ϕ1
d ϕ
k ϕ − 1
B2
ϕ3
ϕ2
d ϕ
k ϕ − − 1
B n
ϕ n+1
ϕ n
d ϕ
k ϕ
+
t 0+t D+t 1+t 2+ +t n−1+t n+t G,1+t G,2+ +t G,n−1+t G,n
(11)
At this stage, retention models can be included, as shown
for the LSS model in Eqns 12 and 13 The derivation of the
Eqns 12 and13 canbe found inSupplementary Material section
S-5
t R, d uring n th grad ient= 1
B n S
1+B n S k n
t 0−t 1+t D
k 1 −t 2
k 2 − −t n
k n+ 1
B 1S
1
k 1− 1
k 2
B 2S
1
k 2− 1
k 3
+ + 1
B n−1S
1
k n−1− 1
k n
+t 0+t D+t 1+t 2+ +t n−1+t n+t G ,1+t G ,2+ +t G ,n−1
(12)
t R, a f ter n gradients=k n+1
t 0−t 1+t D
k 1 −t 2
k 2− − t n
k n+ 1
B 1S
1
k 1 − 1
k 2
B 2S
1
k 2 − 1
k 3
+ + 1
B n S
1
k n − 1
k n+1
+
t 0+t D+t 1+t 2+ +t n−1+t n+t G ,1+t G ,2+ +t G ,n−1+t G ,n
(13)
Eqn.s 12 and 13 can be used to numerically approximate the retentiontime forelution underGCG conditions However, since,
t DisalreadyincludedintheGCGitshouldbesettozero
4.3 Computation of retention parameters 4.3.1 Without correction for gradient deformation
Forcomparison,retention parameters forthe LSS modelwere first established assuming a perfect linear gradient, only consid-ering the dwell time of the instruments (i.e. not correcting for instrument-induceddeformation,usingEqns.4and5).Inessence, Eqn.s 12 and13 were applied to the theoretical gradient profile
to eliminate theerror dueto themodel, buttheseequations re-duce to Eqn.s 4 and 5 when there is no change inslope during thegradient.The dwelltimewastakenasthetime difference be-tween themidpoint ofthe programmedgradient andthat ofthe measured gradient The results are shown in Table 4 The mea-suredretentiontimesforeachtestcompoundandthe accompany-ingt0 valuescanbe foundinSupplementary MaterialsectionS-6 Whiletheretentionparametersobtainedforaspecificinstrument allowcomputationofretentiontimesofthetestcompoundsonthe sameinstrumentwithgoodaccuracy, asexpected, itisclear that theLSSparametersestablishedusingdifferentinstrumentsdeviate dramatically This is particularly evident for Instrument 2 These resultsconfirmthatonlyadjustingforthedwelltimedoesnot suf-fice to obtain correct retention parameters, and clearly illustrate theadverseeffectsofthegradientdeformation
4.3.2 Incorporating correction for geometry-induced gradient deformation
The LSS parameters of the test compounds were also estab-lishedusingEqns 13and14,i.e.correcting forgradient deforma-tionemployingtheestablished GCGs foreachLC instrument The GCGs werecalculatedusing theparametersfromTable 3and ap-proximatedby 100 linear-gradient steps The resultsare listed in Table 5 Gradient correction yields higher lnk0 and S values as comparedtothoseobtainedby correctingonlyforthedwelltime (see Section4.3.1,Table4).In allcasesbutANT onInstrument1, themodelshowedamuchbetterfittotheretentiontimes,as in-dicatedbytheSSEvalues.Theobtainedretentionparameters(lnk0
andS)still varysignificantly.Althoughgeometry-inducedgradient deformation have been eliminated still, solvent-related deforma-tions remain,which maybe significant, duetothe differencesin pumptechnologybetweentheLCinstruments
Inordertotestwhethertheretentionparametersobtained us-ingthe GCGcorrectionyield moreconsistentretentionprediction
Trang 7Table 4
LSS parameters ( Eqn 1 ) determined for the test compounds on each LC instrument using uncorrected gradient parameters and Eqns 4 and 5 Colour scale from red through yellow (50% of Table 4 and 5 ) to green representing high to low SSE values of the predicted versus the experimental retention times respectively
Table 5
LSS parameters Eqn 1 ) as determined for the test compounds on each LC instrument using corrected gradient parameters and
Eqns 12 and (13) Colour scale from red through yellow (50% of Table 4 and 5 ) to green representing high to low SSE values of the
predicted versus the experimental retention times respectively
amongtheLCsystems,theerrorinpredictedretentiontimeswas
calculated Theretentionparameters determined onInstrument1
were usedtopredictretentiontimesonInstruments 2and3,
us-inguncorrectedgradientdata(dwellvolumeonly)orincorporating
gradientcorrection(usingGCGs).Theobtainedresultsare
summa-rized inFig 6(see Supplementary Material section S-7 fora
de-tailedoverviewofeachcompoundandinstrumentcombination)
Fig 6 indicates that after GCG correction for the influence of
theinstrumentationtherelativeerrordecreasesinallsituationsin
comparison withcorrectingforthe dwellvolume only Especially
the retentiontime predictionforinstrument2 isapproaching
ac-ceptableerrors.Itwasnotexpectedthatthispredictionwould
per-formbestsincetheresponsefunctionwasinfluencedbythe
down-ward driftresultedina Gaussianresponse functionwhichshould
resultinalessaccuratedefinedGCG
The error formanycompounds isstill in the highsingle
dig-its Thismaybe due– atleastinpart– to solvent-related
defor-mations The fact that we aim to correlatea binarysystem with
two quaternarysystemsmaycontributetodifferentdeformations
Quaternary pumpstend togive rise tocomposition errorsdueto
volumecontractionduringproportioningatthemulti-channel
gra-dient valve, while the same effect causes flow errors for binary
pumps
Additionally,themeasurementsofthegradientprofilesmaybe
improved In thisresearch, acetonewas used,which is a volatile
compound.Non-constantlossesintheonlinedegasserofthe
flow-deliverymodulemaycauseerrors.Thismaybeverifiedbyrunning
abackwardgradient(from100%to0%B).Inthatcasethe
acetone-containingsolventwillspendlesstimeinthedegasserwhen
chan-nelBhasnoflow.Anon-volatileUV-absorbinganalyte,whichdoes
not adsorbto thedegassermembraneorother surfaces,may
im-prove theaccuracyofthe measuredgradientprofiles andthe
de-rivedinstrumentparameters.Anotherlimitationofourapproachis theassumption thatthe responseoftheUV detectorislinearfor acetoneandthatsolvatochromiceffectsdonotoccur.However,the effectofthelatterisnotexpectedinthewater-waterandacetone systemusedinthisstudy.Finally,theresultsobtainedfor Instru-ment2showedthat detectordriftcanaffecttheobtained instru-mental parameters (See example in Supplementary Material sec-tionS-4).Additionalstudiesonalloftheabovepointsmayfurther refinethecorrections
Althoughnotallinstrumentinfluencescouldbecorrectedfor,a significantreductioninpredictionerrorswasachieved,whichmay improve retention modelling andmethod transfer andmay con-tributetodetermininginstrument-independentretention parame-ters
5 Concluding remarks and outlook
Wehavedevelopedanalgorithmtocorrectretentionmodelling forgradientdeformationinducedbyinstrumentgeometry.Several mathematicaldistributions were evaluated fortheir abilityto de-scribe the response function associated with the gradient defor-mation The four-parameter stable distribution was found to be mostsuitable for thispurpose Using this response function, the geometry-correctedgradient(GCG)shapeforwater-basedsystems couldbe accuratelydescribed Boththevariance andthemeanof the response function proved inversely proportional to the flow rate
For retention prediction the deformed (i.e. non-linear) gradi-entprofilewasapproximatedbyahundredsmalllinearsegments Equations were derived to compute retention times for a com-pound eluting during and after such complex multi-step gradi-ents This allowed correcting for the GCG shape and resulted in
7
Trang 8Fig 6 Relative errors (%) in the predicted retention times of the test compounds on Instruments 2 (top) and 3 (bottom) obtained when using retention parameters deter-
mined for the test compounds on Instrument 1 at different flow rates Relative errors obtained using uncorrected gradients and Eqns 4 and 5 (left) and using GCG correction and Eqns 12 and 13 (right)
morecomparableretentionparametersbetweeninstruments.Most
importantly, we found that correcting the retention parameters
geometry-induceddeformationssignificantlyimprovedthe
predic-tion of retentiontimeson other instruments The average
reduc-tion of the prediction error depended on the instrument Using
dataobtainedononeinstrument(Instrument1)andthenewly
de-velopedalgorithmimprovedtheaveragerelativeerrorinretention
time from 9.8%and 12.2% down to2.1% and6.5 % fortwo other
instruments(Instrument2and3,respectively)incomparisonwith
aconventionalapproach(onlycorrectingforthedwellvolume)
However, whilepredictionaccuracycouldbeimproved,alarge
spread remained between retention parameters for various
an-alytes obtained using different instruments Thus, such
parame-ters should not be interpreted as the true retention parameters
In our proof-of-principle study we corrected for gradient
defor-mation measured withwaterandwatercontaininga tracer
(ace-tone)asthegradient-formingsolvents.Thisallowedcorrectionfor
geometry-induceddeformationofthegradient.Whiletheresponse
function describedthiswater-watersystemadequately,additional
effects due to viscosityand densitydifferences andpossible
vol-umecontractionorexpansionareexpectedwhenmixingdifferent
solvents Taking thesesolvent effects on the gradient shape into
accountmaypotentiallyimprovetheaccuracyoftheretention
pa-rametersandthusfurtherreducetheeffectoftheinstrumentation
on the obtained retention parameters However, this correction
may be more complex, asmore variables relatedto the solvents
used, additives,temperature, pressure, etc. may need to be
con-sidered Whenusingdifferentsolventssolvatochromiceffectsmay
alsooccur,whichmayaffectthemeasuredgradientprofile
There-fore,methodstoaccountforchangesintheabsorptioncoefficient
may also need to be explored Furthermore, it should be noted thatourmeasurements wereexclusivelyconductedusingfast gra-dients The resultsshow that the extentof deformation depends
onthe employed flowrate Lowerflow ratesmaybe expectedto yieldimprovedpredictionaccuracies.Moreover, thepresentstudy waslimited to the log-linear (“linear-solvent-strength”) retention model Further improvements may be obtained by investigating othermodels
Nevertheless, correction using the current algorithm yielded significantlyimprovedpredictionaccuraciesacrossdifferent instru-ments
Credit author statement Tijmen S Bos:Conceptualization,Methodology,Writing Orig-inalDraft,Visualization,SoftwareFormalanalysisLeon E Niezen:
Investigation, Writing Review & Editing Mimi J den Uijl: In-vestigation, Writing Review& Editing Stef R.A Molenaar: For-mal analysis, Methodology, Software, Writing Review & Edit-ingSascha Lege:Conceptualization,Resources,Writing Review& Editing Peter J Schoenmakers: Supervision, Writing Review& EditingGovert W Somsen:Supervision,Writing Review& Edit-ing.Bob W.J Pirok:Conceptualization,Supervision,Funding acqui-sition,Projectadministration,Writing Review&Editing
Declaration of competing interest
Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper
Trang 9TB, LN, SM, PS, GS and BP acknowledge the UNMATCHED
project, which is supported by BASF, DSM andNouryon, and
re-ceives funding from the Dutch Research Council (NWO) in the
framework of the Innovation Fund for Chemistry and from the
Ministry of Economic Affairs in the framework of the
“PPS-toeslagregeling” BP acknowledges the Agilent UR grant #4354
MU acknowledgethe TooCOLDproject,whichis partof theTTW
OpenTechnologyProgrammawithprojectnumber15506whichis
(partly)financedbytheDutchResearchCouncil(NWO)
This workwasperformedin thecontext oftheChemometrics
andAdvancedSeparationsTeam (CAST)withintheCentre for
An-alyticalSciences Amsterdam(CASA).Thevaluablecontributionsof
theCASTmembersaregratefullyacknowledged
Supplementary materials
Supplementary material associated with this article can be
found,intheonlineversion,atdoi:10.1016/j.chroma.2020.461714
References
[1] E.F Hewitt, P Lukulay, S Galushko, Implementation of a rapid and auto-
mated high performance liquid chromatography method development strat-
egy for pharmaceutical drug candidates, J Chromatogr A 1107 (2006) 79–87
https://doi.org/10.1016/j.chroma.2005.12.042
[2] E Tyteca, A Périat, S Rudaz, G Desmet, D Guillarme, Retention modeling
and method development in hydrophilic interaction chromatography, J Chro-
matogr A 1337 (2014) 116–127 https://doi.org/10.1016/j.chroma.2014.02.032
[3] J.W Dolan, D.C Lommen, L.R Snyder, Drylab® computer simulation for high-
performance liquid chromatographic method development II, Gradient Elution,
J Chromatogr A 485 (1989) 91–112 https://doi.org/10.1016/S0021-9673(01)
89134-2
[4] B.W.J Pirok, S Pous-Torres, C Ortiz-Bolsico, G Vivó-Truyols, P.J Schoenmak-
ers, Program for the interpretive optimization of two-dimensional resolution,
J Chromatogr A 1450 (2016) 29–37 https://doi.org/10.1016/j.chroma.2016.04
061
[5] P.J Schoenmakers, H.A.H Billiet, R Tijssen, L De Galan, Gradient selection in
reversed-phase liquid chromatography, J Chromatogr A 149 (1978) 519–537
https://doi.org/10.1016/S0 021-9673(0 0)810 08-0
[6] B.W.J Pirok, A.F.G Gargano, P.J Schoenmakers, Optimizing separations in on-
line comprehensive two-dimensional liquid chromatography, J Sep Sci 41
(2018) 68–98 https://doi.org/10.10 02/jssc.20170 0863
[7] D Abate-Pella, D.M Freund, Y Ma, Y Simón-Manso, J Hollender, C.D Broeck-
ling, D.V Huhman, O.V Krokhin, D.R Stoll, A.D Hegeman, T Kind, O Fiehn,
E.L Schymanski, J.E Prenni, L.W Sumner, P.G Boswell, Retention projection
enables accurate calculation of liquid chromatographic retention times across
labs and methods, J Chromatogr A 1412 (2015) 43–51 https://doi.org/10.1016/
j.chroma.2015.07.108
[8] P.G Boswell, J.R Schellenberg, P.W Carr, J.D Cohen, A.D Hegeman, A study on
retention “projection” as a supplementary means for compound identification
by liquid chromatography–mass spectrometry capable of predicting retention
with different gradients, flow rates, and instruments, J Chromatogr A 1218
(2011) 6732–6741 https://doi.org/10.1016/j.chroma.2011.07.105
[9] A Beyaz, W Fan, P.W Carr, A.P Schellinger, Instrument parameters controlling retention precision in gradient elution reversed-phase liquid chromatography,
J Chromatogr A 1371 (2014) 90–105 https://doi.org/10.1016/j.chroma.2014.09
085 [10] I.A.H Ahmad, F Hrovat, A Soliven, A Clarke, P Boswell, T Tarara, A Blasko, A
14 Parameter Study of UHPLC’s for Method Development Transfer and Trou- bleshooting, Chromatographia 80 (2017) 1143–1159 https://doi.org/10.1007/ s10337- 017- 3337- 8
[11] F Gritti, G Guiochon, Calculated and experimental chromatograms for dis- torted gradients and non-linear solvation strength retention models, J Chro- matogr A 1356 (2014) 96–104 https://doi.org/10.1016/j.chroma.2014.06.030 [12] F Gritti, G Guiochon, Separations by gradient elution: Why are steep gradi- ent profiles distorted and what is their impact on resolution in reversed-phase liquid chromatography, J Chromatogr A 1344 (2014) 66–75 https://doi.org/10 1016/j.chroma.2014.04.010
[13] F Gritti, G Guiochon, The distortion of gradient profiles in reversed-phase liq- uid chromatography, J Chromatogr A 1340 (2014) 50–58 https://doi.org/10 1016/j.chroma.2014.03.004
[14] A.P Schellinger, P.W Carr, A practical approach to transferring linear gradient elution methods, J Chromatogr A 1077 (2005) 110–119 https://doi.org/10.1016/ j.chroma.2005.04.088
[15] M.A Quarry, R.L Grob, L.R Snyder, Measurement and use of retention data from high-performance gradient elution, J Chromatogr A 285 (1984) 1–18 https://doi.org/10.1016/S0021-9673(01)87732-3
[16] G Vivó-Truyols, J.R Torres-Lapasió, M.C Garcı´a-Alvarez-Coque, Error analysis and performance of different retention models in the transference of data from/to isocratic/gradient elution, J Chromatogr A 1018 (2003) 169–181 https: //doi.org/10.1016/j.chroma.2003.08.044
[17] N Wang, P.G Boswell, Accurate prediction of retention in hydrophilic interac- tion chromatography by back calculation of high pressure liquid chromatogra- phy gradient profiles, J Chromatogr A 1520 (2017) 75–82 https://doi.org/10 1016/j.chroma.2017.08.050
[18] M.H Magee, J.C Manulik, B.B Barnes, D Abate-Pella, J.T Hewitt, P.G Boswell,
“Measure Your Gradient”: A new way to measure gradients in high perfor- mance liquid chromatography by mass spectrometric or absorbance detection,
J Chromatogr A 1369 (2014) 73–82 https://doi.org/10.1016/j.chroma.2014.09
084 [19] P.G Boswell, J.R Schellenberg, P.W Carr, J.D Cohen, A.D Hegeman, Easy and accurate high-performance liquid chromatography retention prediction with different gradients, flow rates, and instruments by back-calculation of gradi- ent and flow rate profiles, J Chromatogr A 1218 (2011) 6742–6749 https: //doi.org/10.1016/j.chroma.2011.07.070
[20] A Velayudhan, M.R Ladisch, Effect of modulator sorption in gradient elution chromatography: gradient deformation, Chem Eng Sci 47 (1992) 233–239 https://doi.org/10.1016/0 0 09- 2509(92)80217- Z
[21] W Pi ˛atkowski, R Kramarz, I Poplewska, D Antos, Deformation of gradient shape as a result of preferential adsorption of solvents in mixed mobile phases,
J Chromatogr A 1127 (2006) 187–199 https://doi.org/10.1016/j.chroma.2006 06.018
[22] J Nolan , Stable Distribution: Models for Heavy-Tailed data, Birkhauser, 2019 [23] L.R Snyder, J.W Dolan, J.R Gant, Gradient elution in high-performance liq- uid chromatography, J Chromatogr A 165 (1979) 3–30 https://doi.org/10.1016/ S0 021-9673(0 0)85726-X
[24] D.W Morton, C.L Young, Analysis of Peak Profiles Using Statistical Moments, J Chromatogr Sci 33 (1995) 514–524 https://doi.org/10.1093/chromsci/33.9.514 [25] T.S Bos, W.C Knol, S.R.A Molenaar, L.E Niezen, P.J Schoenmakers, G.W Som- sen, B.W.J Pirok, Recent applications of chemometrics in one- and two- dimensional chromatography, J Sep Sci (2020) jssc.2020 0 0 011 https://doi.org/ 10.10 02/jssc.2020 0 0 011
[26] A.D Hutson, An alternative skew exponential power distribution formulation, Commun Stat - Theory Methods 48 (2019) 3005–3024 https://doi.org/10 1080/03610926.2018.1473600
9
... when there is no change inslope during thegradient .The dwelltimewastakenasthetime difference be-tween themidpoint ofthe programmedgradient andthat ofthe measured gradient The results are shown in. ..Eqns.4and(5)werederivedbyincorporatingtheretentionmodel
(Eqn.1) inthe generalgradient equation (Eqn.3) forthe caseof
alineargradient.TheGCGsasshowninFig.5wereapproximated
numericallyasaseriesofshortlinearsegments.Tofacilitatethis,... In- teractionbetweenthesolventandtheLCinstrumentisexpectedto
beminimalinawell-designedandwell-maintainedsystem.Inthe presentexperiments(mixingwaterwithwatercontainingacetone) onlyinteractionofacetonewithsystemcomponentswouldbe