Stepwise
墓取NORTHEAST.MATH.J
;21(1)(2oo5),117-126
;StepwiseMethodBasedonConfidenceBound
;andInformationIncorporationforIdentifying
;theMaximumTolerableDose牛
;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)01—01l7_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
;ThetraditionalmethodtofindtheMTDistheso—calledup—and—downmethod.
;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)(see【111【5]and【91).Someresearchersfind
;thisapproachinadequateinaddressingtheethicalquestionofprotectingthepatientsfrom ;severetoxicity.ThisleadstothesecondapproachbyLeungandWang[71andBabband ;eta1.(see【10】,【11】)forwhichtheMTDshouldbethehighestdosewithtoxicityrisknot ;exceedingp(.
;AlthoughbothdefinitionsoftheMTDarereasonable,weusethesecond
;definitionfromapracticalandethicalviewpoint.
;Inclinicaltrials,weassumethattheresponseresultisbinary(toxic/non—toxic).Small
;cohortsofmsubjectsaxesequentiallyassignedtovariousdoselevelsofadrugoneatatime, ;startingwiththelowestdose.Fhrthermore,skippingadoseshouldnotbeallowedduring ;doseescalationforethicalreasons.Thatis,thedoseforthenextcohortisthenextdoseup ;orthesamedose.
;Letdenotetheobservednumberoftoxicresponsesatdose.LetYi(1)denotethe ;observednumberoftoxicresponsesofonecohortatdoseleveldi.Let(2)denotethe ;observednumberoftoxicresponsesoftwocohortsatdoseleveldi.Let(1,denotethe ;frequencyoftoxicresponsesoftheonecohortatlevel巩(i.e_,)=二),and)the
;frequencyoftoxicresponsesofthetwocontinuouscohortstreatedatthesameleveldi(i.e., ;(21,
;Pi(2)J’
;ForgivenoL>0andtargettoxicity,supposethat
;P{一p(T)一Pi(1)):1一,
;andthisisequivalentto
;P{p(T)一X{(1)p(T)+}=1一.
;Thus,byusinganasymmetricalconfidenceintervaloftheparameterinbinomialdistribution ;givenintheAppendix,wecanobtainthevaluesofa,nd蹦jbyletting
;p(T瑞
;
;12ONORTHEAST.MATH.JVoL.21
;P+=
;respectively.Similarly,notethat
;P{一p(T)一黜2))=1一Q
;isequivalentto
;尸{p’T一碟;辄)p(T)+榷))=1一.,
;an
;…
;d
;,
;thusby
;,
;usingtheconfidenceboundsgivenintheAppendix,wecanobtainthevalues
;;and龆)_
;Afterthefirstcohortofpatientsisassignedtothelowestdosedl,wecanobtainthetoxic
;responseresults,andthenthesubsequentcohortsareassignedaccordingtothefollowing
;universalsteps:
;steP(i):Underdoseleveldi,cohortsofsubjectsareallocatedaccordingtotwosubsteps:
;Substep(i,1)Assignacohortatthedoseleveld.
;IfIp(T一t(1)I(矗j+6(u.)J,w,2,thengotoSubstep,2);
;otherwisegotoStep(i+1).
;Substep(i,2)Addanothercohortatthecurrentdoseleveldi. ;If}p(T一A(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)1一Q1{(1)
;and
;P{15(1)pX{)1一OL2i(1).
;Inordertoguaranteethesafetyofthismethodp(U ;.
;jshouldberelativelysmall;equivalently,
;Ot2i(1)shouldbechosentoberelativelylargeorot1i(1)toberelativelysmallwhenOtisgiven.
;So,wechoose
;.
;)(
;入一
;=1,L
;cl
;n
;瓤
;d
;n
;a
;
;NO.1WANGX.L.eta1.METHODFORIDENTIFYINGMAXIMUMTOLERABLEDOSE121
;where(1)ISdefinedasfollows:if(1)=0,thenwetake
;:,
;otherwise,wetake
;,
;min(/~t(1),p(T))
;()了=厂’
;Obviously,wehave
;al.(1)2i(1).
;For(2),wesimilarlychoose
;and
;mln(2),p())
;伽C2)a
;Q2l(2)Q—ali(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.Inordertotoavoidover—conservativeresults,weutilizesymmetricconfidence ;intervals.
;2.2A121odifiedCRM
;BecausethereareseveralproblemswiththepracticalimplementationoftheoriginalCRM, ;bythegreaterduration,aswellasbyconcerningwithexcessiveexperimentationathigh ;dosagelevels,andwithmorefrequentandseveretoxicity,someresearchers(seel4】and
;『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:Thetrialwillstopatoneofthreefixedsamplesizes一12,15,andn1
;TheCRMstartswithaone-parametermodel,givenby
;Pi:(以,a)(i=1,2,…,后),
;wherea?(0,+..)issomeparameterandismonotonic,andthenusesBayestheoremwith ;accruingdatatoupdateapriordistributionona,givenby9(0).Define不jasthehistory
;ofthenumberoftoxicitiesandpatientsthathavebeenobservedfromeachdoselevel,up ;toandincludingthe(J！1)thcohort.Letf(a,)beanon—negativefunctionsummarizing
;theavailableinformationsuchthat
;J=1,2,?一,n,
;f(a,71”1)=9(0).
;UsingBayes’theorem,wecanevaluate
;f(a,不j+1),J=1,2,…,n
;sequentiallyvia
;f(a,不J+1).(f(a,)?(d,0)?J?(1一(d,0))一,,
;ifdoseisused,where.
;jdenotesthebinomialrandomvariabledefinedinSection2.1.
;Priormeanresponseprobabilityforjthcohortandithdoseis
;,J=/(,0)?,(0,~j)da.
;ApproximatingmeanresponseprobabilitywiththeUseofpriormeaD.ofparametera, ;.山,:fo’Joa?f(a,rcj)da
;1S
;,
;j=(,D)).
;Asforthedoseselectionforthejthcohort.weselectthedoseleveltosatisfytheminimiz. ;ingcriteri0n
;i一极1I,j—p’TIinstcad.f?min川J—p’TIbecause.fthereducti.nin
;thenumberofinfiniteintegrationtobeperformed,thusreducingcomputationalresource. ;Goodmaneta1.Isjalsomentionedthatthecalculationoftheexpectedaproducesessentially ;thesameresultascalculatingtheexpectedvaluesoftheprobabilitiesthemselves,withSUb- ;stantiallylesscomputation,andthesamerulehasbeenadoptedinallsubsequentsimulation ;studiesbytheotherauthorswithoutcriticalexamination.Theaboveprocesscontinuesun—
;tilapredeterminedfixedsamplesizen.andthentheMTDisthedoselevelthatsatisfies ;t
;rain
;件1
;‰+1一p’T)1.
;1上
;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.
;?Theconfidencelevel1一Qis0.95inthesimulationfortheSMC.
;?Thenumberofreplicationsforeachsimulationistakentobei000.
;ForthistrialweselectQ=005.whichisacompromisebetweenthetherapeuticaimof ;thetrialandtheneedtoavoidtreatmentattributabletoxicity.seeEichhornandZacks[151. ;Furthermore,weusethemodifiedCRMwithdose-toxicitymodel
;(吨,a):(1+e-2d1)一,
;andtaketheexponentialdistributionwithparameteroneasthepriorfora.Assuggestedby ;O’Quigleyetal-I,sixdoselevelsaretakentobedl=-i.47,d2=一1.1,d3=一0.69,d4=
;-
;0.42,ds=0,d6=0.42.
;IntheproposedSMC,thedecisionruleisbasedontheconfidenceboundoftheparameter ;Pineachstep.TheconfidenceboundsforaparameterPcorrespondingtoaconfidencelevel ;1一QaregiveninAppendix.Forethicalreasons.weutilizethelowerconfidenceboundwith ;level1一Q/2,andtheupperconfidenceboundwithlevell—a2i(1)(or1一a2i(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.
;Scenari08彳and5.Thetoxicityprobabilitiesforthefirstthreeorfirstfourdoselevelsare ;relativelylower,butdosed5and魂hasaclearshifttoanunacceptabletoxicityprobabi