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

(1)

HAL Id: hal-01737106

https://hal-univ-rennes1.archives-ouvertes.fr/hal-01737106

Submitted on 18 Jul 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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

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

To cite this version:

A. Toumazi, E. Comets, C. Alberti, T. Friede, F. Lentz, et al.. dfpk An R-package for Bayesian dose-

finding designs using pharmacokinetics (PK) for phase I clinical trials. Computer Methods and Pro-

grams in Biomedicine, Elsevier, 2018, 157, pp.163-177. �10.1016/j.cmpb.2018.01.023�. �hal-01737106�

(2)

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^,^∗

aINSERM, UMRS 1138, Team 22, CRC, University Paris 5, University Paris 6, Paris, France

bINSERM, CIC 1414, University Rennes-1, Rennes, France

cINSERM, IAME UMR 1137, University Paris Diderot, Paris, France

dINSERM, UMR 1123, Hôpital Robert-Debré, APHP, University Paris 7, Paris, France

eDepartment of Medical Statistics, University Medical Center Göttingen, Göttingen, Germany

fFederal Institute for Drugs and Medical Devices, Bonn, Germany

gStatistics and Epidemiology, Division of Health Sciences, Warwick Medical School, The University of Warwick, UK

a rt i c l e i n f o

Article history:

Received 30 June 2017 Revised 11 January 2018 Accepted 24 January 2018

Keywords:

Dose-ﬁnding

Maximum tolerated dose Pharmacokinetics Phase I clinical trials R package

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. Inpreviouspa- pers,weproposedseveralBayesianadaptivepharmacokinetics-baseddose-findingdesignsinsmallpop- ulations.Theobjectiveofthisworkistoimplementthesedose-findingmethodsinanRpackage,called

dfpk.

Methods: AllmethodsweredevelopedinasequentialBayesiansettingandBayesianparameterestima- tioniscarriedoutusingtherstanpackage.Allavailablepharmacokineticsandtoxicitydataareusedto suggestthedoseofthenextcohortwithaconstraintregardingtheprobabilityoftoxicity.Stoppingrules arealsoconsideredforeachmethod.Theggplot2packageisusedtocreatesummaryplotsoftoxicities orconcentrationcurves.

Results: Forallimplementedmethods,dfpkprovidesafunction(nextDose)toestimatetheprobability ofeﬃcacyandtosuggestthedosetogivetothenextcohort,andafunctiontoruntrialsimulationsto design atrial (nsim). The sim.data functiongeneratesateachdose thetoxicity valuerelatedto a pharmacokineticmeasureofexposure,theAUC,withanunderlyingpharmacokineticonecompartmental modelwithlinearabsorption.It isincludedasanexamplesincesimilar data-framescanbegenerated directlybytheuserandpassedtonsim.

Conclusion: Thedevelopeduser-friendlyRpackagedfpk,availableontheCRANrepository,supportsthe designofinnovativedose-ﬁndingstudiesusingPKinformation.

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

∗ Corresponding author..

E-mail address: moreno.ursino@inserm.fr (M. Ursino).

1The last two authors contributed equally as co-senior authors.

PKdatacollectedduringsuchstudyprovidesthedescriptionofthe dose-concentration relationships[2]. Nevertheless, these two ap- proaches are oftenconductedandreported independently indif- 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 theinfor- https://doi.org/10.1016/j.cmpb.2018.01.023

(3)

mationcollectedduringthetrial,andtotrytoutilizethePKmea- 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),whichprovidestheBayesianadap- tivePK-baseddose-ﬁndingdesigns insmallpopulations proposed byUrsinoetal.[8]throughthefreelyavailableRsoftware[9].The sixmethods detailedin[8]havebeenimplementedin

dfpk

^.^For

eachofthem,twofunctionsareprovided: (i) afunctionto determine the next recommended dose (during the trial) or the recommended 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-toxicityre- sponsearealsoprovidedbythepackage.

Thepaperisorganisedasfollows.Section2introducesthesta- tisticalmethodsproposed byUrsinoetal.[8],alongwiththede- scriptionofthesuggestedscenariostobesimulated.Section3out- 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,...,d_k

}

^be ^the ^set ^of ^K ^possible ^doses ^with

d₁<<d_kandd_[_i_]∈Dbethedoseadministeredtotheithsubject (i=1,...,n,wherendenotesthesamplesize)andy_ibeabinary variablewhichtakesvalue 1iftheithsubjectshowsaDLT(dose- limitingtoxicity)and0otherwise.Moreover,letz_ibethelogarithm 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 dosesalreadyevalu- atedandthe firstdoselevelimmediatelyabove.Thefinal recom- mendedMTD isgiven bythe dosethat wouldhavebeenadmin- isteredfor the (ⁿ+1)^st ^subject ênrolledⁱⁿ ^the ^trial.^Finally, ^we addedinallmethodsthesamestoppingrule:iftheposteriorprob- abilityoftoxicityofthefirstdoseisgreaterofaspecifiedthresh- old,thennodoseissuggestedandthetrialisstopped.

Eachmethodisseparatedfromtheothers.Weadoptedthecon- ventionofstartingthesubscriptionof

β

^parameter^from⁰^for^each

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

2.1.1. PKCOV

PKCOVisamodiﬁcationofthemethodproposedbyPiantadosi andLiu[5]whosuggestedtousetheAUCasacovariateforp_T,the probabilityoftoxicity, throughthelogit link.Therefore,thedose-

toxicitymodelis

logit

(

^pT

(

^dk,

2

) ∀

^dk∈D, (3) whereLisathresholdsetbeforestartingthetrialandthehatde- 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.

³^z, ⁽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 available,m₂=20andm₃=10aregoodstartingvaluesforasensitiv- ityanalysis.The probability oftoxicity associatedwith eachdose isobtainedby usingthe estimatedparametersofeach regression modelinthefollowingexpectedvalueformula:

P

(

^y=1

|

^dk,

β

⁼

β

^ˆ^,

ν

⁼

ν

^ˆ

)

=E

1 1+e^β^ˆ²⁻^β^ˆ³^z

=

1

1+e^β^ˆ²⁻^β^ˆ³^z

g

(

^z

)

^dz, ⁽⁵⁾

(4)

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

PKPOP,avariationofPKLOGIT,arisesbyreplacingzwithz_k,_pop inEq.(4),wherez_k,_popisthemeanvalueofthelogarithmofAUC atdosed_kpredictedbyEq.(2).Inotherwords,wereplacetheob- servedAUCvalue forthepatientwiththepopulationmeanvalue.

Then, theprobability oftoxicityateach doseiscomputedinvert- ing Eq.(4),usingtheestimatedparameters

β

ˆ2 and

β

ˆ3 along with z_k,_poppredictedbyEq.(2).

PKTOXisessentiallythePKLOGITmethodwithaprobitregres- sionmodelreplacingthelogisticregressioninEq.(4),thatis p_T

(

ⁱⁿ

thesamewayofEq.(4).

2.2. Simulatingdatafortrialdesign

Thepackageimplementsseveralexamplesreproducingthesce- nariosproposedinUrsinoetal.[8]toevaluatethemethodperfor- manceusingsimulateddata.Inthesesettings,toxicityislinkedto a PK measure ofexposure,namely toAUC, andwe useda simulation settingsimilar to the one in[12].A ﬁrst orderabsorption, linear,one compartmentPK modelwasusedtosimulatePKdata.

Inthismodel,theconcentrationattimetafteradministrationofa dosed_kofthedrugattime0,canbe writtenasa functionofka, theabsorptionrateconstantfororaladministration,CL,theclear- ance ofelimination, andV,thevolumeofdistribution,asfollows:

c

(

^t

)

=dk

V ka

ka−CL/V

e⁻⁽^CL^/^V⁾^t−e⁻^k^a^t

, (8)

α

^AUC≥

τ

T.Choosingthedifferentparameters leadstoseveralsce- narios.Theprobabilityoftoxicityiscomputedas:

pT

(

^dk

)

=

^log^d^k

⁻^log

τ

^T⁻^log^CL

ω

CL² +

ω

²_α

. (9)

3. R-functions

Package

dfpk

împlements âll ^the dose-finding methods described 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. Overall presentation of the main R functions nextDose (Fig. 1A) and nsim (Fig. 1B) available in the package dfpk, indicating the corresponding inputs and outputs.

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,^and^used ⁱⁿ^the

nsim

^function ⁽ⁱⁿ^Fig. ¹^B.),

which will perform n simulated clinical trials. Also in this case, plotswithgraphicalrepresentationssupportthenumericresults.

Bayesianparameterestimationiscarried outusingthe

rstan

packagewhilethe

ggplot2

^package^is^used^to^create^plots.^Three

S4 classesare implemented in thepackage, “dose” ,“doseﬁnding”

and“scen”,inorderto store theoutputs ofthe mainR functions

nextDose

^,

nsim

^and

sim.data

^,respectively.Theclassesarede- 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:

In the remainder of this section, we present the R functions availableinthepackagealongwiththerequiredinputparameters andexamples.Amoreextensivedemonstrationanddocumentation canbeaccessedfromtheon-lineusermanualontheCRANserver byinstallingthepackageordirectlywithintheRconsole.

(5)

Table 1

Arguments for the functions nextDoseand the dose-ﬁnding models.

model The dose-ﬁnding model chosen between “pktox”, “pkcov”, “pkcrm”, “pklogit”,

“pkpop” and “dtox”.

y A binary vector of toxicity outcomes from previous patients.

AUCs A vector with the computed AUC values of each previous patient for PKTOX, PKCRM, PKLOGIT and PKPOP.

doses A vector with the doses panel.

x A vector with the dose level assigned to previous patients.

theta The toxicity target.

options List with the Stan model’s options.

prob The threshold of the posterior probability of toxicity for the stopping rule in the selected model; defaults to 0.9.

betapriors A vector with the value for the prior distribution of the regression parameters in the selected model.

thetaL A second threshold of AUC in the PKCRM model only; defaults to theta for PKCRM and NULL for the models PKTOX, PKCOV, PKLOGIT, PKPOP & DTOX.

p 0 The skeleton of CRM for PKCRM; defaults to NULL.

L The AUC threshold to be set before starting the trial for PKCRM; defaults to NULL.

deltaAUC A vector of the difference between computed individual previous patients’ AUC and the AUC of population at the same dose level (deﬁned as an average);

argument for PKCOV; defaults to NULL.

CI A logical constant indicating the estimation of the 95% credible intervals (CI) of the probability of toxicity at each dose level for the selected model; defaults to TRUE.

Table 2

Input arguments required by each dose-ﬁnding method in the nextDose function.

Method Required arguments Optional arguments

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

(A)PKTOXmodelInthefollowingexample,wesupposed thataclinicaltrialisstill ongoingand15patientshavebeenenrolledsofar.

Forthiscase,PKTOXwasselectedasthedose-ﬁndingmodel.The

nextDose

functionrequiresthefollowinginputarguments:the panel ofdosesofthedrug,thetargettoxicityprobability,

θ

ând^theôptions^for^the^Stan^modelâsâ^list,^containing^the^numberôf

chains,thenumberofiterationsandthewarm-upiterations.

(6)

Othernecessaryinputargumentsarethevectorofdoselevelsassignedtothepatients(asintegervector),thatisdenotedby

x

^,^the

AUCsvaluesandthevectoroftoxicityoutcomes(0/1areaccepted),denotedas

y

,foreachenrolledpatient.

Theusercanchangethepriordistributionparameterofthetoxicity-AUCregressionbyaddingtheparameter

betapriors

.Ifitis notspeciﬁed,thevaluesuggestedbyUrsinoetal.[8]areusedbydefault.

ThedefaultchoicesofbetapriorsforPKTOXmodelarethefollowing:

β

| ν

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

InthissectionasecondillustrationisshownusingthePKCRMdose-ﬁndingmethodinthefunction

nextDose

^.^In^this^way,^we^can

highlightthedifferencesintherequiredinputs/outputs.Inthiscase,inadditiontotheinputargumentsthatareusedinthePKTOX

(7)

Fig. 2. (A) Plot of the data for the ﬁrst 15 patients in the study and, (B) the plot of the posterior distributions given this data of the probability of toxicity at each dose according to the PKTOX method (including the mean estimation along with the 95% 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)

The

sim.data

^function^generatesând^stores^PKând^toxicity^dataⁱⁿôrder^to^beûsed^for^simulationâccording^to^the^model^described

inSection2.2.Theinitialstepconsistsingeneratingthepatients’ responsesatalldosesforeachtrial.Thefunction

sim.data

takesthe trial’ssettingsandreturnsalistofdataincludingthesubject’sPKparametersalongwiththesimulatedtoxicityobservations.

(8)

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

PKparameters

) andtheinter-individualvariability(IIV)fortheclearanceandthevolumeofdistribution(

omegaIIV

). Then,for each doselevel(

doses

^), ^we^computed ^the^desiredprobability oftoxicity (

preal

^), ^and^for^all ^patients⁽

N

⁾ ^and^simulated^data ⁽

TR

^),^it

computestheconcentrationmeasurementsatthespeciﬁedtimepoints(

timeSampling

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

sigma

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

limitTox

^and^the^patient’ssensitivityparameter

omegaAlpha

^.^Finally,^a^default^value^of

therandomnumbergenerator(

seed

α

ÂUCs. ^Since ^we âre ûsing^S4 ^classes,^theûser^can êasily ^create^his/herôwn ^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.

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:

Eachtrial’sresultisstoredinalistoftheR“scen”object,

gen.scen

,whichregroupsthesubject’sPK parameters,theconcentration measurements forall patientsandthesimulatedtoxicities valuesateachdoselevel.Here, we providesome

sim.data

^results^for^the

ﬁrsttrial:

(9)

BasedontheaboveAUCwiththesensitivityparametervaluesandthechosentoxicitythreshold

τ

T=10.96,weobserveforallpatients (onerow=onepatient)inthesecondtrial,thetoxicityoutcomesateachdoselevel(onedose=onecolumn)asfollows:

(10)

Fig. 3. Plot of the concentration of the drug vs time for 12.6, 34.65, 44.69, 60.8, 83.69 and 100.37 mg with k a= 2 h ¹, CL = 10 Lh ¹and V = 100 L .

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

⁾

The function

nsim

simulatesasingleornprospectiveclinical trials.Inthesimulatedtrial,thedoseisescalated stepwise cohortby cohort until the ﬁrst toxicity response is observed and then the chosen dose-ﬁnding method design is applied (two-stage design) as suggestedinUrsinoetal.[8].

(11)

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

^,â^vector^containing^theîndexôf^real^blood^sampling^thatênables^theûser^to

useallconcentrationpoints,previouslysimulated,oronlyasubsetofthemand(f)

AUCmethod

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

nsim

^,^using^only^theconcentrationsamplesselectedby

icon

^.

Bydefault,allsimulatedsamplesareused.Similarlytothe

nextDose

function,theuserneedstochooseoneofthedose-ﬁndingmodels whichareavailableinthepackage.

Asanexample,10trials(i.e.

TR

=10)weresimulatedwith30patients(i.e.

N

=30)pertrial,usingthefunction

nsim

.Atthebeginning, the

simulatedData

îs^defined⁽ⁱⁿ^this^case,^weûse^the^simulated^data

gen.scen

^generatedⁱⁿ^the^Section^3.2.1^),representingthetrue toxicityprobabilitiesofeachtrial,wheresixdoselevelsareconsidered.Thedoselevelsshouldbeenteredinavector,asfollows:

Thetargettoxicityprobability

theta

îs^set^to^0.2,^meaning^that^the^MTDîs^definedâs^the^dose^for^whichât^most^20%ôf^dose^limiting

toxicity(DLT)responsesoccur.Forthisexample,thePKTOXmethodisselectedandthe95%CIoftheprobabilityoftoxicityateachdose ischosentobeestimatedandincludedintheresultssincetheinputargument

CI

issetto

TRUE

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

After settingtheindexofrealbloodsampling(i.e.

icon

⁾^and^entering^thecorrespondingmodel’sinputparameters,wecallthe

nsim

functionasfollowing:

The resultis saved inthe R object,named

simResult

,of a S4 class “dosefinding”.The output can be divided in three parts.The firstpartshowsadatasummary ofthechosendose-findingmodel(PKTOXinthiscase),thesecondparttheStanoptionsandfinallythe dose-findingresultsincludingthepercentageofthedose-allocationandthepercentageoftheMTDselection.Thegeneratedoutputisas follows:

(12)

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

representstheplotselec- 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

^.^Note^that,^if^thesimulation’sre- 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%CIofprobabilityoftox- 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

^package^implements^novel^methods^fordose-ﬁnding phase I clinical trials incorporating PK in the dose-toxicity relationships[8]. In thispackage, each method can be used during a prospectiveadaptive trial,where the doseforthe next cohort of patientsdependsontheoutcomesofthe previouscohorts, inor- dertoestimatetherecommendeddoseforfurtherclinicaltrials.It can alsobe used to performsimulations beforethe beginning of trialinordertostudytherobustnessofthemethodtothediffer- entparameterssettingchoices.Running simulationsisalsouseful tocalibratesome parameters,butittakestime,thereforewesug- 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 throughsim- 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.OurRpackagecansupportin- terdisciplinarytrialteamsinimplementinginnovativedose-ﬁnding designusingPKinformationinphaseIstudies.

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

The choice of the prior distributions is crucial. In this package, we set as default the prior distributions suggested in [8].

(13)

Fig. 4. (A) Plot of the dose escalation (for each patient) in the trial, (B) Boxplot of the sampling distribution of the probability of toxicity at each dose over the total number of trials, and (C) Plot of the posterior distributions of the probability of toxicity at each dose (including the estimation along with the lines of the 95% CI), according to the PKTOX method.

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,wesug- gesttouseadedicatedserver.Moreover,runninginparallelallthe scenariosandnotsequentiallycanalsoreducethetimeofthesim- ulations.

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

Table A.1

Simulation’s estimated times running 10 or 1,0 0 0 trials, in minutes and hours respectively, using different dose-ﬁnding methods under several settings for Bayesian algorithm (number of chains and iterations).

Bayesian settings TR = 10 TR = 1,0 0 0

Methods Methods

PKTOX PKCRM PKTOX PKCRM chains = 4, iter = 40 0 0 18 min 10 min ≈ 32 h ≈ 17 h chains = 4, iter = 60 0 0 26 min 14 min ≈ 45 h ≈ 25 h chains = 3, iter = 40 0 0 13 min 7 min ≈ 23 h ≈ 13 h chains = 6, iter = 40 0 0 26 min 14.5 min ≈ 44 h ≈ 24 h

AppendixB.S4classes

Oneofthebigadvantagesof

dfpk

^packageîsîts^flexible^frame-

work based on the S4 classes andmethods structure. S4 classes haveallowedustoconstructrich andcomplicateddatarepresen- 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

nextDose

,

sim.data

and

nsim

, respectively. You can click on the corresponding help pages as backgroundinformation for the nextsteps.

(14)

Table B.1

The required slots (i.e. arguments) in the dose class.

N The total number of enrolled patients.

y A binary vector of toxicity outcomes from previous patients; 1 indicates a toxicity, 0 otherwise.

AUCs A vector with the computed AUC values of each patient.

x A vector with the dose levels assigned to the patients in the trial.

theta The toxicity target.

options A list of Stan model’s options.

newDose The next recommended dose (RD) level; equals to 0 if the trial has stopped, according to the stopping rules.

pstim The estimated mean probabilities of toxicity.

pstimQ1 The 1st quartile of estimated probability of toxicity.

pstimQ3 The 3rd quartile of estimated probability of toxicity.

parameters The Stan model’s estimated parameters.

model A character string to specify the selected dose-ﬁnding model used in the method.

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

^.^We^can^lookⁱⁿ^detailât^the^structureôf^the^classâs^follows:

where,

ClassNewDose

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

ⁱⁿ^the^object

nextD

^.^To^present

theresultswecanuseeitherthemethod

show

or

print

asfollows:

Bothmethodsgiveaniceandsimplepresentationoftheoutcomethatisstoredintheobject

nextD

.Inanycasewhereuserwantsto changehowtheresultsarepresented,she/hecaneasilydoitbysettingher/hisown

show

^or

print

^methodⁱⁿ^the^class

dose

^.^Similarly,

a

plot

genericfunctionisdeﬁnedforthisclassandcanbeeasilyaccessedthroughthecommand:

2.

scen-class

(15)

Table B.2

The required slots (i.e. arguments) in the scenclass.

PKparameters Subject’s pharmacokinetic’s (PK) parameters from the population distributions deﬁned by the population mean.

nPK The length of time points.

time The sampling time points.

idtr The id number of the corresponding simulated dataset.

N The total sample size per trial.

doses A vector of the doses panel.

preal The prior toxicity probabilities.

limitTox The toxicity threshold.

omegaIIV The inter-individual variability for the clearance and the volume of distribution.

omegaAlpha The patient’s sensitivity parameter.

conc The concentration computed at the PK population values.

concPred The concentration values with proportional errors for each patient at each dose.

tox The toxicity outcome.

tab A summary matrix containing the sampling time points at the ﬁrst row followed by concPred, parameters and alphaAUC. It used by the simulation function nsim.

parameters The simulated PK parameters of each patient.

alphaAUC A vector with the computed AUC values of each patient.

A

scen

isaS4classtosaveandshowadatasetsimulatedbythefunction

sim.data

.Wecanlookindetailatthestructureofthe class

scen

^as^follows:

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

scen-class

^by^a^similar^way^using^the^command

new

ⁱⁿ^the^following^way:

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

Lastly,athirdS4classisavailableinthepackagedfpk,called

dosefinding

^,^whichîs^created^to^storeâll^thedose-findingresultsthat aresimulatedthroughtheRfunction

nsim

.

(16)

Table B.3

The required slots (i.e. arguments) in the dosefinding class.

pid Patient’s ID provided in the study.

N The total sample size per trial.

time The sampling time points.

conc The estimated concentration values for each patient at each dose.

p0 The skeleton of CRM for PKCRM.

L The AUC threshold to be set before starting the trial for PKCRM.

nchains The number of chains for the Stan model.

niter The number of iterations for the Stan model.

nadapt The number of warmup iterations for the Stan model.

newDose The next maximum tolerated dose (MTD) if TR = 1 otherwise the percentage of MTD selection for each dose level after all trials starting from dose 0; equals to 0 if the trial has stopped before the end, according to the stopping rules.

MTD A vector containing the next maximum tolerated doses (MTD) of each trial (TR); equals to 0 if the trial has stopped before the end, according to the stopping rules.

MtD The ﬁnal next maximum tolerated (MTD) dose after all the trials.

theta The toxicity threshold.

doseLevels A vector of dose levels assigned to patients in the trial.

toxicity The estimated toxicity outcome.

AUCs A vector with the computed AUC values of each patient.

TR The total number of trials to be simulated.

preal The prior toxicity probabilities.

pstim The estimated mean probabilities of toxicity.

pstimQ1 The 1st quartile of the estimated probability of toxicity.

pstimQ3 The 3rd quartile of the estimated probability of toxicity.

model A character string to specify the selected dose-ﬁnding model used in the method.

seed The seed of the random number generator that is used at the beginning of each trial.

It’sstructureisgivenbelow:

where,21differentslotsareavailable.TableB.3deﬁnesallthepossibleslots.

Identicallywiththe

dose

and

scen

S4classes,theslotscanpickedusingthe@“operator” andthegenericfunctionsandmethodscan beappliedinthesameway.

References

[1] S. Chevret , Statistical Methods for Dose-Finding Experiments of Statistics in Practice, John Wiley and Sons, Chichester, West Sussex, England, 2006 . [2] H. Derendorf , L.J. Lesko , P. Chaikin , W.A. Colburn , P. Lee , R. Miller , R. Pow-

ell , G. Rhodes , D. Stanski , J. Venitz , Pharmacokinetic/pharmacodynamic mod- eling in drug research and development, J. Clin. Pharmacol. 40 (12) (20 0 0) 1399–1418 .

[3] E. Comets , S. Zohar , A survey of the way pharmacokinetics are reported in published phase i clinical trials, with an emphasis on oncology, Clin. Pharma- cokinet. 48 (6) (2009) 387–395 .

[4] F. Bretz , J.C. Pinheiro , M. Branson , Combining multiple comparisons and mod- eling techniques in dose-response studies, Biometrics 61 (3) (2005) 738–748 . [5] S. Piantadosi, G. Liu, Improved designs for dose escalation studies using phar-

macokinetic measurements, Stat. Med. 15 (15) (1996) 1605–1618, doi: 10.1002/

(SICI)1097-0258(19960815)15:15 1605::AID-SIM325 3.0.CO;2-2 .

[6] S. Patterson, S. Francis, M. Ireson, D. Webber, J. Whitehead, A novel Bayesian decision procedure for early-phase dose-ﬁnding studies, J. Biopharm. Stat. 9 (4) (1999) 583–597, doi: 10.1081/BIP-100101197 .

[7] J. Whitehead, Y. Zhou, L. Hampson, E. Ledent, A. Pereira, A bayesian approach for dose-escalation in a phase i clinical trial incorporating pharmacodynamic endpoints, J. Biopharm. Stat. 17 (6) (2007) 1117–1129 . PMID: 18027220 doi:

10.1080/10543400701645165 .

[8] M. Ursino, S. Zohar, F. Lentz, C. Alberti, T. Friede, N. Stallard, E. Comets, Dose- ﬁnding methods for phase I clinical trials using pharmacokinetics in small pop- ulations, Biom. J. (2017), doi: 10.10 02/bimj.20160 0 084 .

[9] R Core Team, R: a language and environment for statistical computing, in: R Foundation for Statistical Computing, Vienna, Austria, 2013 . ISBN 3-90 0 051- 07-0, http://www.R-project.org/ .

[10] J. Whitehead, S. Patterson, D. Webber, S. Francis, Y. Zhou, Easy-to-implement Bayesian methods for dose-escalation studies in healthy volunteers, Biostatis- tics 2 (1) (2001) 47–61, doi: 10.1093/biostatistics/2.1.47 .

[11] J. O’Quigley , M. Pepe , L. Fisher , Continual reassessment method: a practical design for phase 1 clinical trials in cancer, Biometrics 46 (1) (1990) 33–48 . [12] G. Lestini, C. Dumont, F. Mentré, Inﬂuence of the size of cohorts in adaptive

design for nonlinear mixed effects models: an evaluation by simulation for a pharmacokinetic and pharmacodynamic model for a biomarker in oncology, Pharm. Res. 32 (10) (2015) 3159–3169, doi: 10.1007/s11095- 015- 1693- 3 . [13] E. Zhang, H.G. Zhang, Cran task view: clinical trial design, monitoring,

and analysis, 2016, ( https://cran.r-project.org/web/views/ClinicalTrials.html ).

Accessed: 2017-10-22.