dfpk : An R package for Bayesian dose finding designs using Pharmacokinetics (PK) for phase I clinical trials

warwick.ac.uk/lib-publications

Original citation:

Toumazi, A., Comets, E., Alberti, C., Friede, T., Lentz, F., Stallard, Nigel, Zohar, S. and Ursino,

M. (2018) dfpk : An R-package for Bayesian dose-finding designs using Pharmacokinetics (PK)

for phase I clinical trials. Computer Methods and Programs in Biomedicine, 157. pp.

163-177. doi:

10.1016/j.cmpb.2018.01.023

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http://wrap.warwick.ac.uk/98032

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ContentslistsavailableatScienceDirect

Computer

Methods

and

Programs

in

Biomedicine

journalhomepage:www.elsevier.com/locate/cmpb

dfpk:

An

R-package

for

Bayesian

dose-finding

designs

using

pharmacokinetics

(PK)

for

phase

I

clinical

trials

A.

Toumazi

a

_,

_E.

_Comets

b,c

_,

_C.

_Alberti

d

_,

_T.

_Friede

e

_,

_F.

_Lentz

f

_,

_N.

_Stallard

g

_,

_S.

_Zohar

a,1

_,

M.

Ursino

a,1,∗

a_{INSERM,UMRS1138,Team22,CRC,UniversityParis5,UniversityParis6,Paris,France} b_{INSERM,CIC1414,UniversityRennes-1,Rennes,France}

c_{INSERM,IAMEUMR1137,UniversityParisDiderot,Paris,France}

d_{INSERM,UMR1123,HôpitalRobert-Debré,APHP,UniversityParis7,Paris,France} e_{DepartmentofMedicalStatistics,UniversityMedicalCenterGöttingen,Göttingen,Germany} f_{FederalInstituteforDrugsandMedicalDevices,Bonn,Germany}

g_{StatisticsandEpidemiology,DivisionofHealthSciences,WarwickMedicalSchool,TheUniversityofWarwick,UK}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received30June2017 Revised11January2018 Accepted24January2018

Keywords: Dose-finding

Maximumtolerateddose Pharmacokinetics PhaseIclinicaltrials Rpackage

a

b

s

t

r

a

c

t

Backgroundand objective: Dose-finding, aimingatfinding the maximumtolerateddose, and pharma-cokineticsstudiesarethefirstinhumanstudiesinthedevelopmentprocessofanewpharmacological treatment.Intheliterature,todateonlyfewattemptshavebeenmadetocombinepharmacokineticsand dose-findingandtoour knowledgenosoftwareimplementationis generallyavailable. Inprevious pa-pers,weproposedseveralBayesianadaptivepharmacokinetics-baseddose-findingdesignsinsmall pop-ulations.Theobjectiveofthisworkistoimplementthesedose-findingmethodsinan

R

package,called

dfpk

.

Methods: AllmethodsweredevelopedinasequentialBayesiansettingandBayesianparameter estima-tioniscarriedoutusingthe

rstan

package.Allavailablepharmacokineticsandtoxicitydataareusedto suggestthedoseofthenextcohortwithaconstraintregardingtheprobabilityoftoxicity.Stoppingrules arealsoconsideredforeachmethod.The

ggplot2

packageisusedtocreatesummaryplotsoftoxicities orconcentrationcurves.

Results: Forallimplementedmethods,

dfpk

providesafunction(

nextDose

)toestimatetheprobability ofefficacyandtosuggestthedosetogivetothenextcohort,andafunctiontoruntrialsimulationsto design atrial (

nsim

). The

sim.data

functiongeneratesateachdose thetoxicity valuerelatedto a pharmacokineticmeasureofexposure,theAUC,withanunderlyingpharmacokineticonecompartmental modelwithlinearabsorption.It isincludedasanexamplesincesimilar data-framescanbegenerated directlybytheuserandpassedto

nsim

.

Conclusion: Thedevelopeduser-friendlyRpackage

dfpk

,availableontheCRANrepository,supportsthe designofinnovativedose-findingstudiesusingPKinformation.

© 2018TheAuthors.PublishedbyElsevierB.V. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Dose-findingstudiesandpharmacokinetics(PK)arecarriedout atthefirstphasesofclinicalevaluationofanewdruginhumans. Drug safetyisevaluatedin thedose-findingstudy,whichaims at identifyingthemaximumtolerateddose(MTD)[1].Meanwhile,the

∗ _{Correspondingauthor..}

E-mailaddress:[email protected] (M.Ursino).

1_{Thelasttwoauthorscontributedequallyasco-seniorauthors.}

PKdatacollectedduringsuchstudyprovidesthedescriptionofthe dose-concentration relationships[2]. Nevertheless, these two ap-proaches are oftenconductedandreported independently in dif-ferentsectionsinpublicationsreportingtrialresults[3].Identifying therightdoseorsetofdosesatanearlystageiscrucial:selecting too toxic doses can result in patient overdosing, while selecting an inefficientdose increasesthe likelihood that the drug willbe found to be ineffectivein subsequentclinical evaluation [4]. Par-ticularlyinthecaseofsmallpopulations,suchasrarediseasesor paediatrics, itshould beusefulto take intoaccount all the

infor-https://doi.org/10.1016/j.cmpb.2018.01.023

mationcollectedduringthetrial,andtotrytoutilizethePK mea-surementswithinthedose-findingdesign.Onlyfewattemptshave beendescribed inthe literature so far and, usually,the methods were built fora very specific situation [5–8]. Moreover, no soft-wareimplementationsarepubliclyavailable.

In thisarticle wepresent thenewR package

dfpk

(short for dose-findingPharmacokinetics),whichprovidestheBayesian adap-tivePK-baseddose-findingdesigns insmallpopulations proposed byUrsinoetal.[8]throughthefreelyavailableRsoftware[9].The sixmethods detailedin[8]havebeenimplementedin

dfpk

.For eachofthem,twofunctionsareprovided: (i) afunctionto deter-mine the next recommended dose (during the trial) or the rec-ommended MTD (at the end of the trial) and (ii) a function to runsimulations of phase Istudies to designa newtrial. Interac-tive graphicalrepresentations ofthe dose-concentration curve, of thedoseallocationprocessinthetrialandofthedose-toxicity re-sponsearealsoprovidedbythepackage.

Thepaperisorganisedasfollows.Section2introducesthe sta-tisticalmethodsproposed byUrsinoetal.[8],alongwiththe de-scriptionofthesuggestedscenariostobesimulated.Section3 out-linesthe structure ofthe package andthe main functionsof the paper(

sim.data

,

nextDose

and

nsim

)withpracticalexamples. Section4&5includeconclusion,discussionandrecommendations.

2. Computationalmethods

The presentsection briefly reviews the methods proposed by Ursino et al. [8]to perform dose-finding takinginto account PK measurements.Wethendescribethescenariossimulatedin[8]in orderto evaluatetherobustnessofthemethod,whichhavebeen addedasexamplesinthedfpkpackage.

2.1.Dose-findingmethods

Let D=

{

d1,...,dk

}

be the set of K possible doses with

d1<<dkandd[i]∈Dbethedoseadministeredtotheithsubject (i=1,...,n,wherendenotesthesamplesize)andyibeabinary variablewhichtakesvalue 1iftheithsubjectshowsaDLT (dose-limitingtoxicity)and0otherwise.Moreover,letzibethelogarithm oftheareaunderthecurve(AUC)oftheconcentrationsofdrugin bloodplasmaagainsttime,fortheithpatient.

All methods share the same fundamental idea for the dose-escalation rule: the dose chosen for the next cohort enrolled is the one with probability of toxicity nearest to the target

θ

se-lected by the trial investigators. A no-skipping rule is given: if some doses have not yet been tested, the dose is chosen from

D∗_⊂D,a subset ofD whichcontainsall the dosesalready evalu-atedandthe firstdoselevelimmediatelyabove.Thefinal recom-mendedMTD isgiven bythe dosethat wouldhavebeen admin-isteredfor the

(

n+1

)

st subject enrolledin the trial.Finally, we addedinallmethodsthesamestoppingrule:iftheposterior prob-abilityoftoxicityofthefirstdoseisgreaterofaspecified thresh-old,thennodoseissuggestedandthetrialisstopped.

Eachmethodisseparatedfromtheothers.Weadoptedthe con-ventionofstartingthesubscriptionof

β

parameterfrom0foreach method.Therefore,evenifthe parametersshare thesamenames acrossmodels,theyhavedifferentinterpretations.Inthefollowing, webrieflydescribehowtheprobabilityoftoxicityisestimatedand computedineachmethod.

2.1.1. PKCOV

PKCOVisamodificationofthemethodproposedbyPiantadosi andLiu[5]whosuggestedtousetheAUCasacovariateforpT,the probabilityoftoxicity, throughthelogit link.Therefore,the

dose-toxicitymodelis

logit

(

pT

(

dk,

zdk,β

)

)

=−

β0

+

β1

log

(

dk

)

+

β2

zdk

∀

dk∈D, (1)

where

β

=

(

β

1,

β

2

)

,

β

0 isaconstantselectedthroughasensitivity analysisorwithpriorinformation,

z_dkisthedifferencebetween thelogarithmofpopulationAUCatdosedkandz,thelogarithmof AUCofthesubjectatthesamedose.Independentuniform distri-butionsareselectedaspriordistributionsfor

β

1and

β

2.Indetail,

β

1∼U

(

max

(

0,m1−5

)

,m1+5

)

,wherem1reflectstheprior infor-mationontheparameterandthelengthofthedomainofthe dis-tributioncangoupto10,and

β

2∼U(0,5).Both

β

0andm1should beselectedusingprior information,such asfrompreclinicaldata, and sensitivity analysisshould be done. The estimated probabil-ityof toxicity versus doseis obtained by invertingEq. (1),using

β

1=

β

ˆ1,theestimatedparameter,and

zdk=0.

2.1.2. PKLIMandPKCRM

PKLIM isa modification ofthe methodproposed by Patterson etal.[6]andWhiteheadetal.[10].AnormalPK-toxicitymodelis used:

zi

|

β

,

ν

∼N

β0

+

β1

logdi,

ν

2

, (2)

where

β

₌

(

β

0,

β

1

)

are the regression parameters, and

ν

is the standard deviation. A bivariate normal distribution and a beta distribution are chosen for

β

and

ν

, respectively, that is,

β

∼ N

(

m,

ν

2

₍

_g

₎₎

_and

ν

_{∼Beta(1,1).Therefore,ahierarchicalprior} dis-tributionisgivento

β

,wherem=

(

−logClpop,1

)

andgshouldbe chosenusingpriorinformation.Forinstance,Clpopdenotesthe at-tendedvalueoftheclearanceatpopulationlevel,andgreflectsthe precision. Theprobability oftoxicity ofeachdoseiscomputedas

P

(

z>L

|

dk,

β

=

β

ˆ,

ν

=

ν

ˆ

)

∀

dk∈D, (3)

whereLisathresholdsetbeforestartingthetrialandthehat de-notestheposteriormeansoftheparameters.Sinceanassumption underlyingthemodelisthatDLTsarebasedontheAUCexceeding some threshold,the methodcould be applicable only when such a thresholdisknown.In ordertoavoidthisproblem, thePKCRM methodwasproposed,whichisthecombinationofPKLIMandthe CRM[11] usingapower workingmodelandnormalprior onthe parameter.InPKCRMthedoserecommendedforthenextsubject isthelowestofthedosesrecommendedbythetwomethods.

Notethat although the samenotationhas beenused for con-venience,theparameters

β

0 and

β

1 are differentinthedifferent models.

2.1.3. PKLOGIT,PKPOP,PKTOX

PKLOGIT,inspiredbyWhiteheadetal.[7],combinestwo regres-sions to compute the probability oftoxicity versus the dose. The firstoneisthesameasEq.(2),thatiszversusdose.Inthesecond,

z isusedasacovariateina logisticregressionmodelforpT.This meansthatnowtheprobabilityoftoxicityisdescribedintermof AUCandnotanymoreintermofdose.Therefore,wehavethat

logit

(

pT

(

z,

β

))

=−

β

2+

β

3z, (4)

where

β

2 and

β

3 have independent uniform prior distributions, thatis,

β

2∼U(0,m2) and

β

3∼U(0,m3),withm2≥m3,andvalues can bechosen usingprior information.Ifnoinformationis avail-able,m2=20andm3=10aregoodstartingvaluesfora sensitiv-ityanalysis.The probability oftoxicity associatedwith eachdose isobtainedby usingthe estimatedparametersofeach regression modelinthefollowingexpectedvalueformula:

P

(

y=1

|

dk,

β

=

β

ˆ,

ν

=

ν

ˆ

)

=E

1

1+eβˆ2−βˆ3z

=

₁

1+eβˆ2−βˆ3z

where g(z) represents the distribution of the logarithm of AUC giventhedosedkobtainedfromEq.(2).

PKPOP,avariationofPKLOGIT,arisesbyreplacingzwithzk,pop inEq.(4),wherez_k,popisthemeanvalueofthelogarithmofAUC atdosedkpredictedbyEq.(2).Inotherwords,wereplacethe ob-servedAUCvalue forthepatientwiththepopulationmeanvalue. Then, theprobability oftoxicityateach doseiscomputed invert-ing Eq.(4),usingtheestimatedparameters

β

ˆ2 and

β

ˆ3 along with

zk,poppredictedbyEq.(2).

PKTOXisessentiallythePKLOGITmethodwithaprobit regres-sionmodelreplacingthelogisticregressioninEq.(4),thatis

pT

(

z,

β

)

=

(

−β2+

β3

z

)

, (6)

with

represents the standard cumulative normal distribution. As intheprevious models,independentuniformdistributions are chosen as prior distributions for the parameters. The probabil-ity oftoxicity versus doseis then computedin the sameway of Eq.(5)usingtheprobitregressioninsidetheintegral.

2.1.4. DTOX

DTOX follows the usual wayof estimating pT versus dose di-rectlywithoutthePK measurementsandisincludedtocheckthe behaviourofthisstandardmethod.Thedose-toxicitymodelis:

pT

(

dk,

β

)

=

(

−β0+

β1

log

(

dk

))

∀

dk∈D. (7)

Independentuniformbivariatepriordistributionischosenfor

β

in thesamewayofEq.(4).

2.2. Simulatingdatafortrialdesign

Thepackageimplementsseveralexamplesreproducingthe sce-nariosproposedinUrsinoetal.[8]toevaluatethemethod perfor-manceusingsimulateddata.Inthesesettings,toxicityislinkedto a PK measure ofexposure,namely toAUC, andwe useda simu-lation settingsimilar to the one in[12].A first orderabsorption, linear,one compartmentPK modelwasusedtosimulatePKdata. Inthismodel,theconcentrationattimetafteradministrationofa dosedkofthedrugattime0,canbe writtenasa functionofka, theabsorptionrateconstantfororaladministration,CL,the clear-ance ofelimination, andV,thevolumeofdistribution,asfollows:

c

(

t

)

=dk

V ka

ka−CL/V

e−(CL/V)t_−e−kat

, (8)

IndividualCLandVare sampledfromlog-normaldistributions withmeanCLpop

(

Lh−1

)

andVpop(L),respectively,andstandard de-viation

ω

CL=

ω

V,while ka isfixed inthis studyaslimited infor-mationconcerningtheabsorptionphasewasavailableinthe orig-inaldataset toestimate its inter-individualvariability.In orderto link PK to the toxicity profile of patients, a sensitivity parame-ter

α

,comingfromalog-normaldistributionwithmeanequalto 1 andstandard deviation

ω

_α, anda threshold

τ

T are introduced. Weassumedthatapatient incursadoselimitingtoxicity(DLT)if

α

AUC≥

τ

T.Choosingthedifferentparameters leadstoseveral sce-narios.Theprobabilityoftoxicityiscomputedas:

pT

(

dk

)

=

logdk−log

τ

T−logCL

ω

2 CL+

ω

2α

. (9)

3. R-functions

Package

dfpk

implements all the dose-finding methods de-scribed inSection 2. InFig.1,an overallexplanation ofthemain functions inthe

dfpk

package is given. The aim of the package is to assist the design ofphase I clinical trials. During the trial, the patient data already accrued (“Trial data” in Fig. 1A) can be used in the

nextDose

function in order to determine the next

Fig.1. OverallpresentationofthemainRfunctionsnextDose(Fig.1A)andnsim (Fig.1B)availableinthepackagedfpk,indicatingthecorrespondinginputsand out-puts.

recommended dose, or the MTD at the end of trial. Plots are alsoavailable aftertheestimationprocess. Whenplanninga new trial, datasets containing PK and toxicity measurements can be simulateddirectly by the user (“Simulated data”) or through the

sim.data

function,andused inthe

nsim

function (inFig. 1B.), which will perform n simulated clinical trials. Also in this case, plotswithgraphicalrepresentationssupportthenumericresults.

Bayesianparameterestimationiscarried outusingthe

rstan

packagewhilethe

ggplot2

packageisusedtocreateplots.Three S4 classesare implemented in thepackage, “dose” ,“dosefinding”

and“scen”,inorderto store theoutputs ofthe mainR functions

nextDose

,

nsim

and

sim.data

,respectively.Theclassesare de-taileddescribedintheAppendixB.

Thepackage

dfpk

isavailableontheCRANarchiveandcanbe easily installed on the fly through the URL https://cran.r-project. org/web/packages/dfpk. Once the package is installed, it can be loadedwiththecommand:

Table1

ArgumentsforthefunctionsnextDoseandthedose-findingmodels.

model Thedose-findingmodelchosenbetween“pktox”,“pkcov”,“pkcrm”,“pklogit”, “pkpop” and“dtox”.

y Abinaryvectoroftoxicityoutcomesfrompreviouspatients.

AUCs AvectorwiththecomputedAUCvaluesofeachpreviouspatientforPKTOX, PKCRM,PKLOGITandPKPOP.

doses Avectorwiththedosespanel.

x Avectorwiththedoselevelassignedtopreviouspatients. theta Thetoxicitytarget.

options ListwiththeStanmodel’soptions.

prob Thethresholdoftheposteriorprobabilityoftoxicityforthestoppingrulein theselectedmodel;defaultsto0.9.

betapriors Avectorwiththevalueforthepriordistributionoftheregressionparameters intheselectedmodel.

thetaL AsecondthresholdofAUCinthePKCRMmodelonly;defaultstothetafor PKCRMandNULLforthemodelsPKTOX,PKCOV,PKLOGIT,PKPOP&DTOX. p0 TheskeletonofCRMforPKCRM;defaultstoNULL.

L TheAUCthresholdtobesetbeforestartingthetrialforPKCRM;defaultsto NULL.

deltaAUC Avectorofthedifferencebetweencomputedindividualpreviouspatients’AUC andtheAUCofpopulationatthesamedoselevel(definedasanaverage); argumentforPKCOV;defaultstoNULL.

CI Alogicalconstantindicatingtheestimationofthe95%credibleintervals(CI)of theprobabilityoftoxicityateachdoselevelfortheselectedmodel;defaults toTRUE.

Table2

Inputargumentsrequiredbyeachdose-findingmethodinthenextDosefunction.

Method Requiredarguments Optionalarguments

pktox y,AUCs,doses,x,theta,prob,options,CI betapriors pkcov y,AUCs,doses,x,theta,deltaAUC,prob,options,CI betapriors pkcrm y,AUCs,doses,x,theta,p0,L,prob,thetaL,options,CI betapriors pklogit y,AUCs,doses,x,theta,prob,options,CI betapriors pkpop y,AUCs,doses,x,theta,prob,options,CI betapriors dtox y,doses,x,theta,prob,options,CI betapriors

3.1.Dose-findingmethods(

nextDose

)

The

nextDose

functionisusedtoperformparameterestimationateachstepduringadose-findingtrial.It givestherecommended dosetoadministertothenext cohortofpatients, orthefinal estimatedMTD ifappliedattheendofthetrial.Itcan beusedduringan ongoingclinicaltrialorwithasimulateddataset,asdescribedlaterintheSection3.2.

The description ofthe input argumentsused in the

nextDose

functionare provided in Table1. Theuser hasto choosethe dose-finding methodfrom the available set in the

model

parameter. Then, he/she should provide the parameters required by the selected method,asspecifiedinTable2,whiletheothersaresetautomaticallytoNULL.Anyargumentnotspecifiedbytheuserwillbesettothe correspondingdefaultchoice[8].The numberofchainsanditerationsforthe Bayesianalgorithmcan bechanged usingtheappropriate

rstan

options.

3.1.1. Demonstration

Othernecessaryinputargumentsarethevectorofdoselevelsassignedtothepatients(asintegervector),thatisdenotedby

x

,the AUCsvaluesandthevectoroftoxicityoutcomes(0/1areaccepted),denotedas

y

,foreachenrolledpatient.

Theusercanchangethepriordistributionparameterofthetoxicity-AUCregressionbyaddingtheparameter

betapriors

.Ifitis notspecified,thevaluesuggestedbyUrsinoetal.[8]areusedbydefault.

ThedefaultchoicesofbetapriorsforPKTOXmodelarethefollowing:

β

|

ν

∼N

(

m,

ν

×g×diag

(

1,1

)

)

,

ν

∼Beta

(

1,1

)

,

m=

−log

CLpop

,1

,

β

2 ∼U

(

0,beta2mean

)

,

β

3 ∼U

(

0,beta3mean

)

, (10)

whereClpop isthepopulation clearanceandthedefaultchoicesareClpop=10,g=10000,beta2mean= 20andbeta3mean= 10. Detailsaboutthedefaultchoicesofallthedose-findingmodelsareavailableintheuser-manualontheCRAN.

Theseargumentsareusedinthe

nextDose

functiondependingonthechosenmodel,PKTOX,usingthefollowingsyntax:

omittingalldefaultparameters.

Theresultsarestoredina“dose”object.Theoutputbelowreportspartoftheresultsdisplayed,thatisthenumberofpatientswho arecurrentlyenrolled,theselecteddose-findingmodel,andtheobserveddoselevelsofthedrug:

Accordingto theseresults,thenext recommendeddoselevelis5,whichwouldbe thedoseforthenext cohortofpatientsgiven thedata.Theestimatedtoxicityprobabilitiesandmodelparametersarealsoshown:

The packagealso providesa graphicalrepresentationof theresults.Forexample,usingthe genericfunction

plot()

on a “dose”

object,wecanselectifwewanttopresentgraphically:(1)thedoseallocationofthecurrentlyenrolledpatientsduringthetrialor (2)theposteriordistributions oftheprobability oftoxicityateachdosepresentingtheestimationalongwiththelinesofthe95% credibleintervals(CI),asbelow:

Fig.2presents(A)thedataforthefirst15patientsinthestudy,and(B)theplotoftheposteriordistributions giventhisdata of theprobabilityoftoxicityateachdoseaccordingtothePKTOXmethod(includingthemeanestimationalongwiththe95%CI). (B) PKCRMmodel

Fig.2. (A)Plotofthedataforthefirst15patientsinthestudyand,(B)theplotoftheposteriordistributionsgiventhisdataoftheprobabilityoftoxicityateachdose accordingtothePKTOXmethod(includingthemeanestimationalongwiththe95%CI).

method (i.e.y,AUCs, doses,x, theta, options), thePKCRMmethod requirestheskeleton ofCRM (

p0

) andtheAUC threshold(

L

), whichmustbesetbeforestartingthetrial.Inthisexampleweuse:

AftersettingalltherequiredargumentsforthechosenPKCRMmodel,wecallthenextDosefunctionasfollowing:

Theresultingobjectdisplaysasfollows:

3.2.

Generate

data

(sim.data)

Inparticular,itstartsbydrawingsubject’sPK parameters(k_α,CLandV) fromthe populationdistributionsdefinedbythepopulation mean(

PKparameters

) andtheinter-individualvariability(IIV)fortheclearanceandthevolumeofdistribution(

omegaIIV

). Then,for each doselevel(

doses

), wecomputed thedesiredprobability oftoxicity (

preal

), andforall patients(

N

) andsimulateddata (

TR

),it computestheconcentrationmeasurementsatthespecifiedtimepoints(

timeSampling

).Inaddition,weaddaproportionalerrordrawn from a normaldistribution withzero mean anda standard deviation

sigma

. Accordingto eq. (9), the function computesthe toxicity valuesforeachdoselevelusingthethreshold

limitTox

andthepatient’ssensitivityparameter

omegaAlpha

.Finally,adefaultvalueof therandomnumbergenerator(

seed

)issetat190591.

Theresultsarestoredinalistof“scen”objects,whichconsistsof

PKparameters

,

nPK

,thelengthofthetimepoints,

timeSampling

,

idtr

,

N

,

doses

,

preal

,

limitTox

,

omegaIIV

,

omegaAlpha

,

concentration

, the concentration computed at the PK population values,

concPred

,theconcentration valueswithproportional errorsforeach patientateach dose,

tox

,

tab

,a summarymatrix,used inthesimulationfunction,containingthesamplingtimepointsatthefirstrowfollowedby

concPred

,

parameters

,thesimulatedPK parameters ofeach patient,

alphaAUC

,the

α

AUCs. Since we are usingS4 classes,theusercan easily createhis/herown datasets. For example,he/shecancreateanewobjectforeachsimulatedtrial,named

UserData

,usingthecommand

new

inthefollowingway

andthen add itin alist, alongwiththe other simulateddatasets.A moreexpanded descriptionofthe “scen”class andobjectscan be accessedfromAppendixB2.

3.2.1. Demonstration

Thefollowingillustrationshowshowtogeneratethedatasetsofthefirstscenariodescribedby Ursinoetal.[8].Inthisexample,we setthenumberoftrialsto10(i.e.

TR

=10),thethresholdoftoxicityvalueto10.96,thedoselevelsto(12.6,34.655,44.69,60.807,83.689 and100.371mg)whichareusedtoobtainthetrueprobabilitiesoftoxicity,48evenlyspacedsamplingtimepointsbetween0and48h,a samplesizeof30andka=2,CL=10andV =100asillustratedbelow:

Fig.3. Plotoftheconcentrationofthedrugvstimefor12.6,34.65,44.69,60.8,83.69and100.37mgwithka=2h1,CL=10Lh1andV=100L.

Inthisexample,weused

α

₌1forallpatients,buttheusercansimulate

α

fromalog-normaldistributionwithmean₌1andstandard deviation

ω

_α.Selecting

ω

_α=0implies

α

=1asinthisexample.

Onceagain,providingtheselectedlistofthegenerated“scen”objecttothegenericplotfunction

plot

,theuserobtainsaplotofthe drug’sconcentrationintheplasmaagainstthetimet.Underthesamesettingsandoutputsasabove,theplotisimplementedasfollows: Fig.3presentsthePKconcentration curves,describedinSection 2,withthePK parametersk_α=2h−1_,CL=10Lh−1 _andV=100Lfor the6doses(12.6,34.65,44.69,60.8,83.69and100.37mg).

3.3. Dose-findingsimulation(

nsim

)

In additiontotheinput argumentsof

nextDose

,the

nsim

function hasthe followingadditionalarguments:(i)

N

,thetotalsample sizepertrial,(ii)

cohort

,thecohortsize,(iii)

TR

,thetotalnumberoftrialstobesimulated,(iv)

simulatedData

,alistforeachtrial containingprevioussimulateddatasetsasin3.2.1,(v)

icon

,avectorcontainingtheindexofrealbloodsamplingthatenablestheuserto useallconcentrationpoints,previouslysimulated,oronlyasubsetofthemand(f)

AUCmethod

,astringnumberspecifyingtheestimation methodforAUC; validchoicesare1fora“compartmental method” (see[8]fordetails)and2fornon-compartmentalmethod(defaults to2).TheestimatedAUCforeachpatientiscomputedinsidethefunction

nsim

,usingonlytheconcentrationsamplesselectedby

icon

. Bydefault,allsimulatedsamplesareused.Similarlytothe

nextDose

function,theuserneedstochooseoneofthedose-findingmodels whichareavailableinthepackage.

3.3.1. Demonstration

Asanexample,10trials(i.e.

TR

=10)weresimulatedwith30patients(i.e.

N

=30)pertrial,usingthefunction

nsim

.Atthebeginning, the

simulatedData

isdefined(inthiscase,weusethesimulateddata

gen.scen

generatedintheSection3.2.1),representingthetrue toxicityprobabilitiesofeachtrial,wheresixdoselevelsareconsidered.Thedoselevelsshouldbeenteredinavector,asfollows:

Thetargettoxicityprobability

theta

issetto0.2,meaningthattheMTDisdefinedasthedoseforwhichatmost20%ofdoselimiting toxicity(DLT)responsesoccur.Forthisexample,thePKTOXmethodisselectedandthe95%CIoftheprobabilityoftoxicityateachdose ischosentobeestimatedandincludedintheresultssincetheinputargument

CI

issetto

TRUE

.SinceoursimulationusesStanmodels, wealsoneedtospecifythemodel’soptions,asalist,containingthenumberofchains,howmanyiterationseachchainwilluseandthe numberofwarm-upiterations.Thedefaultchoiceis:

After settingtheindexofrealbloodsampling(i.e.

icon

)andenteringthecorrespondingmodel’sinputparameters,wecallthe

nsim

functionasfollowing:

Basedontheaboveexample,thenextrecommendeddoselevel is4mgwithapercentageofMTDselectionequalsto60%.

The MTD,toxicity responsesaswell asthedoseescalation for eachtrialcanbeobtainedasfollows:

Thegenericfunction

plot()

canbeusedinordertoillustrate the doseescalation duringthe trialorthe dose-toxicityresponse for each dose level. The main input argument is a “dosefinding”

object.The simulation output canbe shown graphicallywiththe command:

where

TR

representsthe numberoftrial(defaults toTR= 1) for whichwe wanttoplotthegraph,

CI

indicatesifthesimulation’s resultsincludethe estimated95% oftheprobability oftoxicity or not (defaultsto

CI

=

TRUE

) and

ask

representstheplot selec-tion index(defaultsto ask= TRUE)showinga selection menuas above:the usershould enter1 tosee thedoseescalation alloca-tion of the selected trial, 2 to create a boxplot of the sampling distribution oftheprobability oftoxicity ateach doseinthe end ofthetrialoverthetotalnumberoftrials,and3toplotthefinal posterior distributions ofthe probability oftoxicity ateach dose (the plotincludes theestimation alongwiththe linesof the95% CI for theselected trial). 0 is the command to exitand 2 isthe defaultchoicefortheinput

ask

.Notethat,ifthesimulation’s re-sults don’t include the 95% CI of probability of toxicity then the selectionmenucontainsonlythefirsttwochoices.

Fig.4(A)showsthedoseallocationplotbasedonourexample, wherethenon-toxicity responseisrepresentedasacircleandthe toxicityresponseasacross.Inaddition,Fig.4(B)and4(C)present the output plot choosing, in the menu, 2 and 3, respectively. In Fig. 4(C),the 95% CIandprior probabilities oftoxicity are repre-sentedasdotted anddashedlines,respectively. The reddot-dash lineinthelasttwoFiguresrepresentsthetoxicitythresholdwhich isusedfortheselectionoftheMTD.

Notethat, sincetheBayesianmodelsareimplementedinStan, running simulations can take very long time. Moreover, simula-tionsincludingtheestimationofthe95%credibleintervals(CI)for probability of toxicity at each dose level (i.e.

CI

=

TRUE

), can take moretime than excluding them (i.e.

CI

=

FALSE

). Inthis case,torunthe10trialsincludingthe95%CIofprobabilityof tox-icity,about30minareneededonasingleportablecomputerwith anIntelCorei5.Instead,tosimulateonlytheMTDunderthesame settings,withoutestimatingthe95%CI,about18minarerequired onthesamecomputer.Amoreexpandedcomparisonofthe

nsim

functionandfor1,000simulatedtrials,whichisoftenusedforthe simulation studies in the literature of dose-finding methods, are showninAppendixA.However,wesuggest torun simulationson adedicatedserver.

4. Conclusion

The

dfpk

packageimplementsnovelmethodsfordose-finding phase I clinical trials incorporating PK in the dose-toxicity rela-tionships[8]. In thispackage, each method can be used during a prospectiveadaptive trial,where the doseforthe next cohort of patientsdependsontheoutcomesofthe previouscohorts, in or-dertoestimatetherecommendeddoseforfurtherclinicaltrials.It can alsobe used to performsimulations beforethe beginning of trialinordertostudytherobustnessofthemethodtothe differ-entparameterssettingchoices.Running simulationsisalsouseful tocalibratesome parameters,butittakestime,thereforewe sug-gesttorunsimulationsonadedicatedserver.

Thepackage isuser-friendly andseveralflexibleinputsare al-lowed:forinstancetheusercangeneratebyher/himselfscenarios (simulated datasets)and pass them on to the function

nsim

, or changethehyper-prior parametersoftheprior distributions used inthe Bayesian regression.The package will also be updated ac-cordingusersuggestionsandneeds.

5. Discussion

Designing early phase clinical trials is of crucial importance, since all future steps in the clinical development orfailures de-pendonthesefirstresults.Statisticalcomputer programs,suchas

R

,facilitatedesignandthecheckingofperformance through sim-ulations.Anexampleofexisting Rpackagescanbe foundin[13]. However, to the best of our knowledge, there are no other soft-ware packages available that implement a formal integration of dose-findingandpharmacokinetics.OurRpackagecansupport in-terdisciplinarytrialteamsinimplementinginnovativedose-finding designusingPKinformationinphaseIstudies.

Ursino et al. [8] compared a number of dose-finding meth-ods under several scenarios, in order to verify their behaviour andcharacteristics.ThePKCRMmethodbehavesastheCRMalone whentheLisveryhigh.Ontheotherhand,itgivesthesame prob-abilityofcorrect MTD selection(as theCRM) while reducing the probability ofoverdosing, when Lis appropriately chosen. There-fore,thisdesignisrecommendedwheninpreclinicalphases non-monitorabletoxicityhasbeenobservedorinsomepediatric stud-ies,whenLcanbeeasilysetfromaliteraturereview.ThePKLOGIT andPKTOX methodsare recommended whenmore precise dose-responsecurveestimationisrequired.ComparedtotheCRMthese methodsareabletobetterestimate theprobabilityoftoxicity as-sociatedwitheachdosealongwithaccurateMTDselection.Inthis way,aricherknowledgecanbetransmittedtosubsequentphases ofclinicaldevelopment.Theothermethodshavesimilarbehaviour toany dose-toxicityregression,andcan be usedfor comparisons insimulations.

Fig.4. (A)Plotofthedoseescalation(foreachpatient)inthetrial,(B)Boxplotofthesamplingdistributionoftheprobabilityoftoxicityateachdoseoverthetotalnumber oftrials,and(C)Plotoftheposteriordistributionsoftheprobabilityoftoxicityateachdose(includingtheestimationalongwiththelinesofthe95%CI),accordingtothe PKTOXmethod.

Theseprior hyperparameters were chosen aftersensitivity analy-sistogive goodperformance inmostcases.However, wesuggest settingthepriorhyperparamentersusingpreclinicalinformationor otherexternalpertinentinformationifthisisavailable.

Acknowledgements

We thanks thethree anonymousreviewersandtheassociated editorfor their suggestions. Thisresearch wasconductedas part oftheEuropeanproject,InSPiRe,InnovativeMethodologyforSmall PopulationsResearch.AlltheauthorswerefundedbytheEuropean Union’sSeventhFrameworkProgrammeforresearch,technological developmentand demonstration under grant agreement number FPHEALTH2013–602144.

AppendixA. Simulationtimes

Simulations based on Bayesian methods can be very tedious andtimeconsuming.Therefore,for1,000simulatedtrials,we sug-gesttouseadedicatedserver.Moreover,runninginparallelallthe scenariosandnotsequentiallycanalsoreducethetimeofthe sim-ulations.

Table A.1showstheestimatedtimesforconductingthe simu-lationsof10and1,000trials,excludingthe95%credibleintervals, forthemethodsPKTOXandPKCRMunderseveralsettings(i.e.the numberofchainsanditerationsfortheBayesianalgorithm).

TableA.1

Simulation’sestimatedtimesrunning10or1,000trials,inminutesand hours respectively, using different dose-findingmethodsunder several settingsforBayesianalgorithm(numberofchainsanditerations).

Bayesiansettings TR=10 TR=1,000

Methods Methods

PKTOX PKCRM PKTOX PKCRM

chains=4,iter=4000 18min 10min ≈ 32h ≈ 17h chains=4,iter=6000 26min 14min ≈ 45h ≈ 25h chains=3,iter=4000 13min 7min ≈ 23h ≈ 13h chains=6,iter=4000 26min 14.5min ≈ 44h ≈ 24h

AppendixB.S4classes

Oneofthebigadvantagesof

dfpk

packageisitsflexible frame-work based on the S4 classes andmethods structure. S4 classes haveallowedustoconstructrich andcomplicateddata represen-tationsthatneverthelessseemsimpletotheenduser.Theclassis theabstractdefinition,whileeverytimeweactuallyuseittostore theresultsforagivendataset,wecreateanobjectoftheclass.

Three S4 classes are available, the

dose-class

, the

scen-class

and the

dosefinding-class

, in order to store, show or plot the corresponding results of the main R functions

TableB.1

Therequiredslots(i.e.arguments)inthedoseclass.

N Thetotalnumberofenrolledpatients.

y Abinaryvectoroftoxicityoutcomesfrompreviouspatients;1indicatesatoxicity,0otherwise. AUCs AvectorwiththecomputedAUCvaluesofeachpatient.

doses Avectorwiththedosespanel.

x Avectorwiththedoselevelsassignedtothepatientsinthetrial. theta Thetoxicitytarget.

options AlistofStanmodel’soptions.

newDose Thenextrecommendeddose(RD)level;equalsto0ifthetrialhasstopped,accordingtothestoppingrules. pstim Theestimatedmeanprobabilitiesoftoxicity.

pstimQ1 The1stquartileofestimatedprobabilityoftoxicity. pstimQ3 The3rdquartileofestimatedprobabilityoftoxicity. parameters TheStanmodel’sestimatedparameters.

model Acharacterstringtospecifytheselecteddose-findingmodelusedinthemethod.

Inthissection,abrieflydescriptionoftheaccessibleclassesisshown.

1.

dose-class

The

dose-class

is created to store and present the next recommended dose level in an ongoing trial through the R function

nextDose

.Wecanlookindetailatthestructureoftheclassasfollows:

where,

ClassNewDose

isaunionofclasses“numeric”,“logical” and“NULL”.

Accordinglytothestructure,the

dose

classconsistsof13slots(i.e.arguments)whicheachone hasaspecifictype. TableB.1givesa briefdefinitionofeachcorrespondingslotsinthe

dose

class.

Anobjectthatcomesfromthe

dose-class

mustcontainalltheaboveslots.Theslotsareaccessedusing@,justascomponentsofa listthatareaccessedusing$.Here,anillustrationofhowtoaccesstheslotsofanobjectisgiven.

Wesupposethat

nextD

isanobjectofthe

dose

class.Forexample,theslots

model

and

y

canbeobtainedasfollows:

where,PKTOX wastheselected dose-findingmodelandthevector (0,0,0,0,0,1,0,0,0,0,1,0,0,0,0)wasthetoxicityoutcome for eachpatientthatareusedinthefunction

nextDose

.

Oncetheclassesaredefined,weprobablywanttoperformsomecomputationsonobjects.Inmostcaseswedonotcarehowtheobject isstoredinternally,thecomputer shoulddecidehow toperformthetasks. TheS4wayofreaching thisgoalistousegeneric functions andmethoddispatch.

Basedonouraboveexample,we hypothesisedthat westoredtheoutputofthefunction

nextDose

intheobject

nextD

.Topresent theresultswecanuseeitherthemethod

show

or

print

asfollows:

Bothmethodsgiveaniceandsimplepresentationoftheoutcomethatisstoredintheobject

nextD

.Inanycasewhereuserwantsto changehowtheresultsarepresented,she/hecaneasilydoitbysettingher/hisown

show

or

print

methodintheclass

dose

.Similarly, a

plot

genericfunctionisdefinedforthisclassandcanbeeasilyaccessedthroughthecommand:

TableB.2

Therequiredslots(i.e.arguments)inthescenclass.

PKparameters Subject’spharmacokinetic’s(PK)parametersfromthepopulationdistributions definedbythepopulationmean.

nPK Thelengthoftimepoints. time Thesamplingtimepoints.

idtr Theidnumberofthecorrespondingsimulateddataset. N Thetotalsamplesizepertrial.

doses Avectorofthedosespanel. preal Thepriortoxicityprobabilities. limitTox Thetoxicitythreshold.

omegaIIV Theinter-individualvariabilityfortheclearanceandthevolumeofdistribution. omegaAlpha Thepatient’ssensitivityparameter.

conc TheconcentrationcomputedatthePKpopulationvalues.

concPred Theconcentrationvalueswithproportionalerrorsforeachpatientateach dose.

tox Thetoxicityoutcome.

tab Asummarymatrixcontainingthesamplingtimepointsatthefirstrow followedbyconcPred,parametersandalphaAUC.Itusedbythe simulationfunctionnsim.

parameters ThesimulatedPKparametersofeachpatient. alphaAUC AvectorwiththecomputedAUCvaluesofeachpatient.

A

scen

isaS4classtosaveandshowadatasetsimulatedbythefunction

sim.data

.Wecanlookindetailatthestructureofthe class

scen

asfollows:

Thanks toS4classes,theusercaneasily createhis/herowndatasets.Forexample,he/shecancreateanewobjectforeach simulated trial,namedUserData,andstoreitina

scen-class

byasimilarwayusingthecommand

new

inthefollowingway:

andthenadditinalist,alongwiththeothersimulateddatasets. AdetaileddefinitionofeachslotispresentedintheTableB.2.

Once again, user can accessto the slots andapply thegeneric functionsandmethods in this classby exactlythe same wayas in

dose-class

.Assume thatwe run 10(i.e.

TR

=10) simulateddatasetsusing thefunction

sim.data

and

simulatedData

isa

scen

objectthen:

3.

dosefinding-class

TableB.3

Therequiredslots(i.e.arguments)inthedosefindingclass.

pid Patient’sIDprovidedinthestudy. N Thetotalsamplesizepertrial. time Thesamplingtimepoints. doses Avectorwiththedosespanel.

conc Theestimatedconcentrationvaluesforeachpatientateachdose. p0 TheskeletonofCRMforPKCRM.

L TheAUCthresholdtobesetbeforestartingthetrialforPKCRM. nchains ThenumberofchainsfortheStanmodel.

niter ThenumberofiterationsfortheStanmodel. nadapt ThenumberofwarmupiterationsfortheStanmodel.

newDose Thenextmaximumtolerateddose(MTD)ifTR=1otherwisethepercentageof MTDselectionforeachdoselevelafteralltrialsstartingfromdose0;equals to0ifthetrialhasstoppedbeforetheend,accordingtothestoppingrules. MTD Avectorcontainingthenextmaximumtolerateddoses(MTD)ofeachtrial

(TR);equalsto0ifthetrialhasstoppedbeforetheend,accordingtothe stoppingrules.

MtD Thefinalnextmaximumtolerated(MTD)doseafterallthetrials. theta Thetoxicitythreshold.

doseLevels Avectorofdoselevelsassignedtopatientsinthetrial. toxicity Theestimatedtoxicityoutcome.

AUCs AvectorwiththecomputedAUCvaluesofeachpatient. TR Thetotalnumberoftrialstobesimulated.

preal Thepriortoxicityprobabilities.

pstim Theestimatedmeanprobabilitiesoftoxicity. pstimQ1 The1stquartileoftheestimatedprobabilityoftoxicity. pstimQ3 The3rdquartileoftheestimatedprobabilityoftoxicity.

model Acharacterstringtospecifytheselecteddose-findingmodelusedinthe method.

seed Theseedoftherandomnumbergeneratorthatisusedatthebeginningof eachtrial.

It’sstructureisgivenbelow:

where,21differentslotsareavailable.TableB.3definesallthepossibleslots.

Identicallywiththe

dose

and

scen

S4classes,theslotscanpickedusingthe@“operator” andthegenericfunctionsandmethodscan beappliedinthesameway.

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