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Stepwise

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Stepwisestep,wise

    Stepwise

墓取NORTHEAST.MATH.J

    ;21(1)(2oo5),117-126

    ;StepwiseMethodBasedonConfidenceBound

    ;andInformationIncorporationforIdentifying

    ;theMaximumTolerableDose

    ;WANGXue-li(王雪丽)

    ;(DepartmentoyMathematics,DaqingPetroleumInstitute,Daqing,Heilongfiang,163318; ;ResearchInstetuteoyMathematics.JilinUniversity.Changchun,13oo12, ;TAOJian(陶剑)andSHINing-zhong(史宁中)

    ;(DepartmentoyMathem0tics,NortheastNormalUniversity,Changchun,13ooe4) ;Abstract:TheprimarygoalofaphaseIclinicaltrialistofindthemaximumtoler- ;abledoseofatreatment.Inthispaper,weproposeanewstepwisemethodbasedon ;confidenceboundandinformationincorporationtodeterminethemaximumtolerable ;doseamonggivendoselevels.Ontheonehand,inordertoavoidsevereevenfatal ;toxicitytooccurandreducetheexperimentalsubjects,thenewmethodisexecuted ;fromthelowestdoselevel,andthengoesoninastepwisefashion.Ontheotherhand, ;inordertoimprovetheaccuracyoftherecommendation.thefinalrecommendationof ;themaximumtolerabledoseisaccomplishedthroughtheinformationincorporation ;ofanadditionalexperimentalcohortatthesamedoseleve1.Furthermore,empirical ;simulationresultsshowthatthenewmethodhassomerealadvantagesincomparison ;withthemodifiedcontinualreassessmentmethod.

    ;Keywords:confidencebound,continualreassessmentmethod,informationincoro ;poration,maximumtolerabledose,phaseIclinicaltrials,stepwisemethod ;2000MRsubjectclassification:62P10

    ;CLCnumber:0212

    ;DocumentCOde:A

    ;ArticleID:1000-1778(2005)0101l7_l0

    ;1Introduction

    ;Theclinicaltestingofanexperimentaldrugisnormallydoneinthreephases.AphaseI ;clinicaltrialisprimarilyconcernedwithassessingthedrug’Stoxicity,andthemainobjective

    ;ofaphaseIclinicaltrialistoestimatethemaximumtolerabledose(MTD)ofthedrug.By ;now,severaldesignshavebeenproposedforthispurpose.

    ;‘~eceiveddate:Oct.24,2003.

    ;Foundationitem:TheNNSF(10201006,10001008)ofChina.

    ;

    ;118NoRTHEAST.MATH.

    ;JVoL.21

    ;ThetraditionalmethodtofindtheMTDisthesocalledupanddownmethod.

    ;Thedose

    ;levelisescalatedtothenexthigherlevelwhendonotshowthedoselimitingtoxicity(DLT). ;IfoneofthreepatientsshowstheDLT,thenthreemorepatientsareaddedtothesamedose ;leve1.IfmorethanonepatientshowstheDLTamongoriginalthreepatientsormorethan ;twoamongsixpatients,thenthetrialisstoppedandtheresultingdoseisdeclareda,sthe ;MTD.

    ;Thismethodhasquitesomedrawbacks.Forexample,itcannotaccountforindication ;orpatientpopulationspecificDLTrates,anditcreatesproblemsifoneofthethreepatients ;isnotevaluableandhastobereplacedorifmorethanthreepatientshavebeentreatedat ;thesamedoseleve1.AlsotheprecisionoftheestimatoroftheMTDisfairlyimpreciseand ;themethoddoesnotescalatethedosequicklyenoughifthestartingdosehasbeenselected ;toolowandriskstounderexposetoomanypatients.

    ;Thecontinualreassessmentmethod(CfU)introducedbyO’Quigleyeta1.IJisaBayesian

    ;procedurethattakesapriordistributionofaparameterofaoneparameterdose-response ;modelasastartingpoint.Thisdistributionis’updated’accordingtoBayestheoremas

    ;soonasnewinformationabouttheoccurrenceofDLTsbecomesavailable.Themethodcan ;accountfordifierentnumbersofpatientsperdoseandcanbetargetedtoanypreselected ;DLTrateandcanevencopewiththesituationwheninvestigatorsfeelthattheyhaveto ;overruletheproposalbytheCRMforthenextdosetobeused.

    ;TheCRMissuperiortotheup--and--downdesignintermsofmoreaccuraterecommenda- ;tionofthetargetlevel,moreefficientuseofaccruedinformation,and,inparticular,asthe ;trialcontinues,thepropertyofconvergencetothetargetlevel(see[1-3).However,some

    ;seriousobjectionshavebeenraisedtotheCRM:(i)theCRMmethodtendstotreatmore ;patientsabovetheMTDthanthetraditionalmethod(see[4);and(ii)thecRMcallsinto

    ;playanexplicitdose-toxicitycurvethatisartificialandcanaddcomplicationsfortheuser ;(see[5).TherehavebeenmanymodificationsoftheoriginalCRMthatattempttoaddress ;theseproblems(see[6,7).

    ;Inthispaper,profitingfromsomeadvantagesoftheexistingmethods,weproposeanew ;stepwisemethodbasedonconfidenceintervalandinformationincorporation.Ontheone ;hand,inordertoreducethepossibilityofincreasedtoxicityandavoidexcessiveexperimen

    ;tationathigherdoselevels,assuggestedbyGoodmaneta1.【】andFaries[4,

    ;ourmethodis

    ;executedfromthelowestdoselevel,andthengoesoninastepwisefashion.Furthermore, ;skippingadoseshouldnotbeallowedduringdoseescalationforethicalreasons.Onthe ;otherhand,inordertoimprovetheprecisionoftherecommendation,thefinalrecommen

    ;dationoftheMTDisdeterminedthroughincorporatinganadditionalexperimentalcohort ;atthesamedoseleve1.

    ;Thepaperisorganizedasfollows.InSection2,thenewstepwisemethodbasedon ;confidenceboundandinformationincorporationandamodifiedCRMaredescribed.Inorder ;toinvestigatetheperformanceofthenewmethod,someempiricalsimulationcomparisons ;aregiveninSection3.Finally,inSection4,someconclusionsaredrawn.Anasymmetrical ;confidenceintervaloftheparameterinbinomialdistributionisconsideredinAppendix, ;

    ;NO.1WANG.L.eta1.METHODFORIDENTIFYINGMAXIMUMTOLERABLEDOSE119

    ;whichisessentialtothenewproposedprocedure.

;2Designs

    ;Inthissection,wefirstproposethenewstepwisemethod,andthenreformulateamodified ;CRMwiththepurposeofcomparingtheperformanceofthetwomethods.

    ;2.1Stepwisemethodbasedonconfidenceboundandinformationincorporation ;FollowingcommonpracticeinphaseIclinicaltrials,weassumethatkfixedlevelsaxeto ;betested.Letthesebedenotedbydl<d2<???<dk.Associatedwiththeselevelsaxe ;nondecreasingprobabilitiesoftoxicity,0plp2…Pk.Ourobjectistodeterminethe

    ;MTDamongthegivendoses,thatis,theMTDshouldbethehighestdosewithtoxicityrisk ;notexceedingthetolerable(target)toxicity,p(T).

    ;However,thedefinitionoftheMTDhastwopopularapproaches.Thefirstapproach ;attemptstodeterminethedose,amongallthedosesunderconsideration,withtoxicityrisk ;closesttothetolerable(target)toxicity,p(T)(see1115]and91).Someresearchersfind

    ;thisapproachinadequateinaddressingtheethicalquestionofprotectingthepatientsfrom ;severetoxicity.ThisleadstothesecondapproachbyLeungandWang[71andBabband ;eta1.(see10,11)forwhichtheMTDshouldbethehighestdosewithtoxicityrisknot ;exceedingp(.

    ;AlthoughbothdefinitionsoftheMTDarereasonable,weusethesecond

    ;definitionfromapracticalandethicalviewpoint.

    ;Inclinicaltrials,weassumethattheresponseresultisbinary(toxic/nontoxic).Small

    ;cohortsofmsubjectsaxesequentiallyassignedtovariousdoselevelsofadrugoneatatime, ;startingwiththelowestdose.Fhrthermore,skippingadoseshouldnotbeallowedduring ;doseescalationforethicalreasons.Thatis,thedoseforthenextcohortisthenextdoseup ;orthesamedose.

    ;Letdenotetheobservednumberoftoxicresponsesatdose.LetYi(1)denotethe ;observednumberoftoxicresponsesofonecohortatdoseleveldi.Let(2)denotethe ;observednumberoftoxicresponsesoftwocohortsatdoseleveldi.Let(1,denotethe ;frequencyoftoxicresponsesoftheonecohortatlevel(i.e_,)=),and)the

    ;frequencyoftoxicresponsesofthetwocontinuouscohortstreatedatthesameleveldi(i.e., ;(21,

    ;Pi(2)J’

    ;ForgivenoL>0andtargettoxicity,supposethat

    ;P{p(T)Pi(1)):1,

    ;andthisisequivalentto

    ;P{p(T)X{(1)p(T)+}=1.

    ;Thus,byusinganasymmetricalconfidenceintervaloftheparameterinbinomialdistribution ;givenintheAppendix,wecanobtainthevaluesofa,ndjbyletting

    ;p(T

    ;

    ;12ONORTHEAST.MATH.JVoL.21

    ;P+=

    ;respectively.Similarly,notethat

    ;P{p(T)一黜2))=1Q

    ;isequivalentto

    ;{p’T一碟;)p(T)+))=1.,

;an

    ;

    ;d

    ;,

    ;thusby

    ;,

    ;usingtheconfidenceboundsgivenintheAppendix,wecanobtainthevalues

    ;;and)_

    ;Afterthefirstcohortofpatientsisassignedtothelowestdosedl,wecanobtainthetoxic

    ;responseresults,andthenthesubsequentcohortsareassignedaccordingtothefollowing

    ;universalsteps:

    ;steP(i):Underdoseleveldi,cohortsofsubjectsareallocatedaccordingtotwosubsteps:

    ;Substep(i,1)Assignacohortatthedoseleveld.

    ;IfIp(Tt(1)I(j+6(u.)J,w,2,thengotoSubstep,2);

    ;otherwisegotoStep(i+1).

    ;Substep(i,2)Addanothercohortatthecurrentdoseleveldi. ;If}p(TA(2)l(+5(uJ,,)w2,then:

    ;?Iffii(2)>p(),thenrecommend盔一1astheMTDandstop;

    ;?otherwiserecommenddiastheMTDandstop,

    ;otherwisegotoStep(i+1).

    ;Remark2?1Intheabovemethod,thevaluesanddependonOlli(1)andOt2i(1),

    ;respectively,where

    ;0~1i(1)+a2i(x)-

    ;So,thechoicesofOtli(1)andOt2i(1)arefairlyimportant.Fromtheconfidenceboundsgiven

    ;intheAppendix,weknowthat

    ;PiP~(1)p)1Q1{(1)

    ;and

    ;P{15(1)pX{)1OL2i(1).

    ;Inordertoguaranteethesafetyofthismethodp(U ;.

    ;jshouldberelativelysmall;equivalently,

    ;Ot2i(1)shouldbechosentoberelativelylargeorot1i(1)toberelativelysmallwhenOtisgiven.

    ;So,wechoose

    ;.

    ;)(

    ;入一

    ;=1,L

    ;cl

    ;n

    ;

    ;d

    ;n

    ;a

    ;

;NO.1WANGX.L.eta1.METHODFORIDENTIFYINGMAXIMUMTOLERABLEDOSE121

    ;where(1)ISdefinedasfollows:if(1)=0,thenwetake

    ;:,

    ;otherwise,wetake

    ;,

    ;min(/~t(1),p(T))

    ;()=

    ;Obviously,wehave

    ;al.(1)2i(1).

    ;For(2),wesimilarlychoose

    ;and

    ;mln(2),p())

    ;C2)a

    ;Q2l(2)Qali(2)?.

    ;Remark2.2Theabovestepwisemethodisessentiallyasequentialprocedure.Themedical ;experiments(especiallytoxicitytrials)executedinastepwisefashionhavemanyadvantages. ;Anobviousadvantageisthatstepwisemethodssatisfyethicalimperative,thatistosay,they ;needfewerpatientstobetreatedandavoidtheoccurrenceofsevereevenfataltoxicity.So, ;stepwiseproceduresareeasytobeacceptedandusedinmedicalexperiments(see[12-141). ;Remark2.3InordertoimprovetherecommendationprecisionoftheMTD,thefinalre(:- ;ommendationoftheMTDisbasedontheincorporationofanadditionalcohortofpatients. ;Forthisreason,wecalltheaboveprocedureasthestepwisemethodbasedonconfidence ;boundandinformationincorporation(SMC).

    ;Remark2.4Theabovestepwisemethodis

    ;implementationsincethedecisionruledoesnot

    ;isnoassumedmode1.

    ;particularlyattractiveforitssimplicityin

    ;involvecomplicatedcalculationsandthere

    ;Remark2.5Basedontheinherentcharacteristicofthestepwisedesignitselfandthe ;subsequentsimulationresults,itisnotdifficulttofindthatthestepwisemethodisfairly ;conservative.Inordertotoavoidoverconservativeresults,weutilizesymmetricconfidence ;intervals.

    ;2.2A121odifiedCRM

    ;BecausethereareseveralproblemswiththepracticalimplementationoftheoriginalCRM, ;bythegreaterduration,aswellasbyconcerningwithexcessiveexperimentationathigh ;dosagelevels,andwithmorefrequentandseveretoxicity,someresearchers(seel4and

    ;81)haveproposedmanymodifications.Thesemodificationsaddressallofthemostserious ;criticismsoftheCRM,reducingthedurationofthetrial,reducingtoxicityincidence,and ;loweringtoxicityseverity.whichmaketheCRMacceptabletoclinicalinvestigators.Inthis ;section,followingsomeoftheirsuggestions,wereformulateamodifiedCRMwhichwillbe ;usedinsimulationcomparison.Themodificationsaremainlyasfollows: ;

    ;122NORTHEAST.

    ;MATH.JV0L.21

    ;?Initiationofexperimentation:Experimentationalwaysstartsatthelowestdoseleve1. ;?Doseescalation:Skippingofadoselevelisnotallowed.

    ;??u7n6er0su6jects2,er2evel:3subjectsatatimeareassignedtoadoseleve1.thatis ;tosay,thecohortsizemisfixedtobe3.

    ;?T

    ;,

    ;rialtermination:Thetrialwillstopatoneofthreefixedsamplesizes12,15,andn1

    ;TheCRMstartswithaone-parametermodel,givenby

    ;Pi:(,a)(i=1,2,…,),

    ;wherea?(0,+..)issomeparameterandismonotonic,andthenusesBayestheoremwith ;accruingdatatoupdateapriordistributionona,givenby9(0).Definejasthehistory

    ;ofthenumberoftoxicitiesandpatientsthathavebeenobservedfromeachdoselevel,up ;toandincludingthe(J1)thcohort.Letf(a,)beanonnegativefunctionsummarizing

    ;theavailableinformationsuchthat

    ;J=1,2,?,n,

    ;f(a,71”1)=9(0).

    ;UsingBayes’theorem,wecanevaluate

    ;f(a,j+1),J=1,2,…,n

    ;sequentiallyvia

    ;f(a,J+1).(f(a,)?(d,0)?J?(1(d,0)),,

    ;ifdoseisused,where.

    ;jdenotesthebinomialrandomvariabledefinedinSection2.1.

    ;Priormeanresponseprobabilityforjthcohortandithdoseis

    ;,J=/(,0)?,(0,~j)da.

    ;ApproximatingmeanresponseprobabilitywiththeUseofpriormeaD.ofparametera, ;.,:fo’Joa?f(a,rcj)da

    ;1S

    ;,

    ;j=(,D)).

    ;Asforthedoseselectionforthejthcohort.weselectthedoseleveltosatisfytheminimiz. ;ingcriteri0n

    ;i一极1I,j—p’TIinstcad.f?minJ—p’TIbecause.fthereducti.nin

    ;thenumberofinfiniteintegrationtobeperformed,thusreducingcomputationalresource. ;Goodmaneta1.Isjalsomentionedthatthecalculationoftheexpectedaproducesessentially ;thesameresultascalculatingtheexpectedvaluesoftheprobabilitiesthemselves,withSUb- ;stantiallylesscomputation,andthesamerulehasbeenadoptedinallsubsequentsimulation ;studiesbytheotherauthorswithoutcriticalexamination.Theaboveprocesscontinuesun

    ;tilapredeterminedfixedsamplesizen.andthentheMTDisthedoselevelthatsatisfies ;t

    ;rain

    ;1

    ;‰+1p’T)1.

    ;1

    ;l1

;0

    ;d

    ;,

    ;0

    ;,

    ;f

    ;?

    ;n

    ;e

    ;h

    ;C

    ;d

    ;n

    ;a

    ;

    ;NO?1WANGX.L.eta1.METHODFORIDENTIFYINGMAXIMUMTOLERABLEDOSE123

    ;3SimulationComparison

    ;Inthesimulationstudy,itisnecessarytoimposeboundaryconditions,asonecannotlorer ;thedoselevelatdlorraisethedoselevelatdk.Forourpurposewesimplyassignthe ;subjecttodlordk,respectively,ifsuchconditionsoccur.Inpractice,ifeitherofthese ;eventsoccursthedosespaceisusuallyre-evaluated.

    ;Herearesomedetailsforthesimulationstudy:

    ;?Thecohortsizemisfixedtobe3.

    ;?Thetargettoxicityprobabilityp(is0.2.

    ;?Theconfidencelevel1Qis0.95inthesimulationfortheSMC.

    ;?Thenumberofreplicationsforeachsimulationistakentobei000.

    ;ForthistrialweselectQ=005.whichisacompromisebetweenthetherapeuticaimof ;thetrialandtheneedtoavoidtreatmentattributabletoxicity.seeEichhornandZacks[151. ;Furthermore,weusethemodifiedCRMwithdose-toxicitymodel

    ;(,a):(1+e-2d1),

    ;andtaketheexponentialdistributionwithparameteroneasthepriorfora.Assuggestedby ;O’Quigleyetal-I,sixdoselevelsaretakentobedl=-i.47,d2=1.1,d3=0.69,d4=

    ;-

    ;0.42,ds=0,d6=0.42.

    ;IntheproposedSMC,thedecisionruleisbasedontheconfidenceboundoftheparameter ;Pineachstep.TheconfidenceboundsforaparameterPcorrespondingtoaconfidencelevel ;1QaregiveninAppendix.Forethicalreasons.weutilizethelowerconfidenceboundwith ;level1Q/2,andtheupperconfidenceboundwithlevella2i(1)(or1a2i(2))toobtain

    ;(.rand(.r6(L)Irespectively.However,itisw.rthy.fp.intingoutthatthe ;calculationseemstobesimplerwhenthedecisionruleisbasedonthefrequencyestimates ;ofPdirectly.

    ;Weexecutealargenumberofsimulationstudies.Forbrevity,wereportonlyasmallpart ;ofsimulationresultsthatspanafairlybroadrangeofpossibledose-toxicityrelationships. ;TableIgivessimulationresultsbasedonthetwomethods.Thesimulationresultscontain

    ;themostimportantmeasuresofsuccess:(i)finalrecommendationpercentages;(ii)average ;numberofcohorts;and(iii)averagenumberofpatients.Furthermore,thepercentages ;observedatthetrueMTDareinboldforeachscenario.

    ;ScenarioJ.Alldoselevelshaveunacceptabletoxicityprobabilities.Thetargetisdose ;d1.ThesimulationresultsshowthatboththestepwisemethodandthemodifiedCRMare ;accurateforrecommendationMTD.However,thestepwisemethodneedsfewerpatients, ;andprotectsefficientlythesubjectsfromsufferingseveretoxicity. ;Scenarios2and3.Thetoxicityprobabilitiesaresteadilyincreasingandthetargetsare ;dosesd3andd4,respectively.Thesimulationresultsshowthatthestepwisemethodhasmore ;chancestorecommendationthetrueMTD,butthemodifiedCRMtendstorecommendation ;asuboptimaldose,whichishigherthanthetrueMTD.Inotherwords,thestepwisemethod ;

    ;124NoRTHEAST.MATH.JVoL.21

    ;ismorepreferablethanthemodifiedCRMasfarastheprecisionoftherecommendation ;andthesafetyareconcerted.

    ;Scenari08and5.Thetoxicityprobabilitiesforthefirstthreeorfirstfourdoselevelsare ;relativelylower,butdosed5andhasaclearshifttoanunacceptabletoxicityprobabi

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