Practical Application of the Practical Application of the CContinual ontinual RReassessment eassessment MMethod ethod to a Phase I Dose-Finding Trial in to a Phase I Dose-Finding Trial in

Japan: Japan: EastEast meets meets WestWest

Satoshi MoritaSatoshi Morita Dept. of Biostatistics and Epidemiology, Dept. of Biostatistics and Epidemiology,

Yokohama City University Medical CenterYokohama City University Medical Center

Why a phase I dose-finding study Why a phase I dose-finding study of of CEXCEX in Japan? in Japan?

CCyclophosphamide, yclophosphamide, EEpirubicin, pirubicin, XXelodaeloda

Capecitabine (Xeloda) was/is a novel oral Capecitabine (Xeloda) was/is a novel oral fluoropyrimidine derivative with high single-agent fluoropyrimidine derivative with high single-agent anti-tumor activity in metastatic breast cancer (BC).anti-tumor activity in metastatic breast cancer (BC).

A research team from the EORTC conducted a A research team from the EORTC conducted a phase I dose-finding study to determine the phase I dose-finding study to determine the recommended dose of CEX. recommended dose of CEX. (Bonnefoi, et al., 2003)(Bonnefoi, et al., 2003)

Japanese patients/doctors would need CEX as a Japanese patients/doctors would need CEX as a treatment option.treatment option.

Why CEX trial in Japanese patients?Why CEX trial in Japanese patients?A concern was raised over possible differences in A concern was raised over possible differences in

the tolerability of CEX between Caucasians and the tolerability of CEX between Caucasians and Japanese.Japanese.

In many cases,In many cases,

EORTCEORTCBonnefoi et al., 2001Bonnefoi et al., 2001

JapanJapanIwata et al. 2005Iwata et al. 2005

ExEx. FEC (5-FU, Epi, CPA). FEC (5-FU, Epi, CPA)

Recommended dose(s)Caucasians Japanese>

The Japanese CEX phase I trialThe Japanese CEX phase I trialMorita et al.(2007) & Iwata et al.(2007)Morita et al.(2007) & Iwata et al.(2007)

To answer this question, we conducted a phase I To answer this question, we conducted a phase I dose-finding study of CEX in Japanese patients dose-finding study of CEX in Japanese patients ((J-CEXJ-CEX) from Dec., 2003 to Feb., 2006.) from Dec., 2003 to Feb., 2006.

Based on the prior information:Based on the prior information:- The EORTC CEX study (3+3 cohort design)- The EORTC CEX study (3+3 cohort design)- The previous studies for other combinations - The previous studies for other combinations such as FEC, CAF, etc,such as FEC, CAF, etc,

we applied CRM!!we applied CRM!!

Dose levels in the CEX studiesDose levels in the CEX studies

Dose Dose LevelLevel

44 3 3 2 2 1100

CPACPA(mg/m(mg/m22))

600600

EPIEPI(mg/m(mg/m22))

1001009090909075757575

CapeCape(mg/m(mg/m22/day)/day)

1800180018001800165716571657165712551255

Japanese (5 levels)Japanese (5 levels)

DLTDLT

2/22/2

9/159/15

1/31/3

1/31/3

CPACPA(mg/m(mg/m22))

600600

EPIEPI(mg/m(mg/m22))

100100

25002500

21002100

18001800

15001500

EORTC (4 levels)EORTC (4 levels)CapeCape

(mg/m(mg/m22/day)/day)

: Recommended dose level

: starting dose level

CRM in J-CEXCRM in J-CEXOne-parameter logistic modelOne-parameter logistic model

DLT = Grade 3,4 hematologic / non-hematologic toxicity DLT = Grade 3,4 hematologic / non-hematologic toxicity or grade 3 hand-foot syndromeor grade 3 hand-foot syndrome

A target Pr(DLT) = 0.33

Pr(DLT|dose j) =

(xj, ) = exp( xj)

1 + exp( xj)for j=0,…,4, with fixed > 0,

Implementation of CRM in J-CEXImplementation of CRM in J-CEXA dose-escalation/de-escalation rule:

Each cohort is treated at the dose level with an estimated Pr(DLT | x, Data) closest to 0.33 and NOT exceeding 0.40. Pick x to minimize |E[(x,)|Data] – 0.33| Untried dose is not skipped when escalating.

A trial stopping rule:The trial is to be stopped if level 0 is considered too toxic: Pr(DLT | dose 0, Data) > 0.40.

Nmax = 22 treated in cohorts of 3Start with the 1st cohort of 1 patient at dose level 1.

Setting up a CRM in J-CEXSetting up a CRM in J-CEXStep 1. Obtain pre-study point estimation of Pr(DLT) at Step 1. Obtain pre-study point estimation of Pr(DLT) at

each dose level from clinical oncologists,each dose level from clinical oncologists,

2. Pre-determine the intercept 2. Pre-determine the intercept

3. Specify a prior distribution function of the slope 3. Specify a prior distribution function of the slope

4. Specify a numerical value of 4. Specify a numerical value of xxjj, , jj = 0,…,4,= 0,…,4,

5. Specify the hyperparameters of the prior of 5. Specify the hyperparameters of the prior of pp(())in terms of how informative in terms of how informative pp(() ) is. is.

Step 3: Prior of the slope, Step 3: Prior of the slope,

For computational convenience and to For computational convenience and to constrain the slope constrain the slope to be positive, to be positive, ,,

One more restrictionOne more restriction a=b E()=1, Var()=1/a

～ Ga(a,b) with E()=a/b and Var()=a/b2

Fixing the prior mean dose-toxicity curve regardless of magnitude of prior confidence.

Step 5: Specify the hyperparameter, Step 5: Specify the hyperparameter, aThe hyperparameter The hyperparameter aa determines the credible determines the credible

interval of the dose-toxicity curve.interval of the dose-toxicity curve.Making several patterns of graphical Making several patterns of graphical

presentations, and asking the oncologists, presentations, and asking the oncologists, “which depicts most appropriately your pre-study “which depicts most appropriately your pre-study perceptions on dose-toxicity relationship?”,perceptions on dose-toxicity relationship?”,

a=8 a=5a=2

a=5

We set a = 5.

In the first cohort (patient),…In the first cohort (patient),…

Level 1Level 1(1(1 pt)pt)

DLT1DLT1 例例HFS(G3)HFS(G3)

C: 600C: 600E: 75E: 75X: 1657X: 1657

The dose-toxicity curve after updating the The dose-toxicity curve after updating the prior curve with toxicity data from the 1prior curve with toxicity data from the 1stst pt pt

0 1 2 3 4

Dose level for the 2nd cohort

Results: Dose escalation history Results: Dose escalation history and toxicity responseand toxicity response

Level 1Level 1(1(1 pt)pt)1 DLT1 DLT

HFS (G3)HFS (G3)

C: 600C: 600E: 75E: 75X: 1657X: 1657

Level 0Level 0(3 pts)(3 pts)No DLTNo DLT

C: 600C: 600E: 75E: 75X: 1255X: 1255

Level 1Level 1(3(3 pts)pts)No DLTNo DLT

C: 600C: 600E: 75 E: 75 X: 1657X: 1657

Level 2Level 2(3 pts)(3 pts)No DLTNo DLT

C: 600C: 600E: 90E: 90X: 1657X: 1657

Level 3Level 3((6 pts))2 DLTs2 DLTs

Anorexia(G3)Anorexia(G3)Mucositis(G3)Mucositis(G3)

C: 600C: 600E: 90E: 90X: 1800X: 1800

Posterior mean dose-toxicity curve and Posterior mean dose-toxicity curve and its 90% CI after treating 16 patientsits 90% CI after treating 16 patients

0 1 2 3 4

Posterior density functions of Posterior density functions of Pr(DLT | x, Data) estimated at each of the five dose levelsestimated at each of the five dose levels

Selected as RD

f [(xj, )|data] = p(|data) dd

Concern & Question I hadConcern & Question I hadWe made many “arbitrary choices” when We made many “arbitrary choices” when

designing the study, especially eliciting the designing the study, especially eliciting the prior from the oncologists.prior from the oncologists.Based on the EORTC study, using graphical Based on the EORTC study, using graphical presentations,……, BUT, still arbitrary!!presentations,……, BUT, still arbitrary!!

My concern was…‘didn’t Ga(5,5) dominate the posterior inferences after enrolling the first two / three cohorts?’

My question was…‘how could we determine the strength of the prior relative to the likelihood?’.

Fundamental question in Fundamental question in Bayesian analysisBayesian analysis

The amount of information contained in The amount of information contained in the prior?the prior?

Priorp(θ)(((

(((

(((

(((

Trans-Pacific Research Project!!Trans-Pacific Research Project!!December 2005 ~December 2005 ~

MDACC, HoustonJapan

Time difference15 hours

Prior effective sample sizePrior effective sample size

These concerns may be addressed by quantifying These concerns may be addressed by quantifying the prior information in terms of an equivalent the prior information in terms of an equivalent number of hypothetical patients, i.e., a prior number of hypothetical patients, i.e., a prior effective sample size (ESS). effective sample size (ESS).

A useful property of prior ESS is that it is readily A useful property of prior ESS is that it is readily interpretable by any scientifically literate interpretable by any scientifically literate reviewer without requiring expert mathematical reviewer without requiring expert mathematical training.training.

This is important, for example, for consumers of This is important, for example, for consumers of clinical trial results.clinical trial results.

Work together as a teamWork together as a team

Peter (Müller)

Peter (Thall)

Paper?

You all right?

You all right?

The answer seems The answer seems straightforwardstraightforward

For many commonly used models,For many commonly used models,e.g., beta distributione.g., beta distribution

Effective sample size1.5 + 2.5 = 4

3 + 8 = 11

16 + 19 = 35

Be (1.5,2.5)

Be (16,19)

Be (3,8)

For many parametric Bayesian For many parametric Bayesian models, however…models, however…

How to determine the ESS of the prior is How to determine the ESS of the prior is NOT obvious.NOT obvious.E.g., usual normal linear regression modelE.g., usual normal linear regression model

22

10

210

210

inverse~ normal, bivariate~,

,,

)( ,)(

YVarXYE

General approach to determine General approach to determine the ESS of prior the ESS of prior pp(() )

Morita, Thall, Müller (2008) Biometrics

1) Construct an “ε-information” prior q0(θ)

2) For each possible ESS m = 1, 2, ..., consider a sample Ym of size m

3) Compute posterior qm(θ|Ym) starting with prior q0(θ)

4) Compute distance between qm(θ|Ym) and p(θ)

5) The value of m minimizing the distance is the ESS

Definition of Definition of εε-information prior-information prior

　　　　　 　　　　　 has the same mean and correlations as has the same mean and correlations as , while inflating the variances, while inflating the variances 　　　　　　　　　　　　　　　　　　　　　　

00~θθq

00~θθq θθ ~p

θθ ~p

The basic idea isThe basic idea isTo find the sample size To find the sample size m, that would be , that would be

implied by implied by normal approximationnormal approximation of the of the prior prior pp((θθ) and the posterior ) and the posterior qqmm((θθ||YYmm).).

This led us to use This led us to use the second derivative of the second derivative of the log densitiesthe log densities to define the distance. to define the distance.

MM

m

m=1m=1

………………

Distance between p and qDistance between p and qmm

Difference of the traces ofDifference of the traces of  the two information the two information matrices, evaluated at thematrices, evaluated at the  prior mean: prior mean:

0 , ,, , , ( , )p qm p q D D m

2

2

,

})~(log{

jjp

pD

θθθ 2

02

,

}),~(log{,,

j

mmmjq

qmD

YθθYθ

d

j jpp DD1 ,, θθ

mmm

d

j mjq

q

dfmD

m,D

YYYθ

θ

1 ,

,

,,

DEFINITION of ESSDEFINITION of ESS

The effective sample size The effective sample size ((ESSESS)) of of with respect to the likelihood with respect to the likelihood is the (interpolated) integer is the (interpolated) integer mm that minimizes that minimizes

the distance between p and qthe distance between p and qmm 0,,, qpm θ

θθ ~p θYmmf

AlgorithmAlgorithm

Step 1. SpecifyStep 1. Specify 0 0q

Step 3. ESS is the interpolated value of Step 3. ESS is the interpolated value of m minimizing minimizing 0( , , , )m p q

Step 2. Compute for eachStep 2. Compute for each 0( , , , )m p q 0, ,m M

analytically or using simulation-based numerical approximation

.~~ where

,,1,,,,1

2

b/aβ

XXXmD iim

i immq

XY

Step 1:Step 1:

Step 2: Step 2:

Assume a uniform distribution for Xi

Use simulation to obtain

ESS = 2.1

10,000, with ,/~

,/~~Specify 00 ccbcaGaq

J-CEXJ-CEX

ly,analytical 1~~ Compute 2 aDp

5~~ ba

A computer program, ESS_RegressionCalculator.R,

to calculate the ESS for a normal linear or logistic regression model is available from the website http://biostatistics.mdanderson.org /SoftwareDownload.

In the context of dose-finding studies, In the context of dose-finding studies,

Prior assumptions (arbitrary choices) include Prior assumptions (arbitrary choices) include - one- / two-parameter model, - one- / two-parameter model, - priors of the intercept and slope parameters, - priors of the intercept and slope parameters,

- numerical values for dose levels, etc.- numerical values for dose levels, etc.

It may be interesting to discuss the impact of prior It may be interesting to discuss the impact of prior assumptions in terms of prior ESS and other assumptions in terms of prior ESS and other criteria…in order to obtain a “criteria…in order to obtain a “sensible priorsensible prior”. ”.

→ → One of the on-going projects!!One of the on-going projects!!

Thank you for your kind attention!!

Back-upBack-up

Step 1: Pre-study point estimation of Step 1: Pre-study point estimation of Pr(DLT | dose j)

Dose levelDose level 0 0 1 1 2 2 3 3 4 4Elicited Pr(DLT)Elicited Pr(DLT) .05.05 .10.10 .25.25 .40.40 .60.60

Step 2: Intercept Step 2: Intercept = 3

refrecting oncologists’ greater confidencein higher than lower dose levels.

= 3 = -3

Step 4: Dose levels, Step 4: Dose levels, xBased on the elicited Based on the elicited Pr(DLT | dose j), specify the numerical values xxjj, , jj = 0,…,4.= 0,…,4.““Backward fitting” Backward fitting” (Garrett-Mayer,2006,Clinical Trials)(Garrett-Mayer,2006,Clinical Trials)

(xj, ) = exp( xj)

1 + exp( xj)

Prior dose-toxicity curve and Prior dose-toxicity curve and its 90% credible interval its 90% credible interval

0 1 2 3 4

In the context of dose-finding studies, In the context of dose-finding studies,

Prior assumptions (arbitrary choices) include Prior assumptions (arbitrary choices) include - one- / two-parameter model, - one- / two-parameter model, - priors of the intercept and slope parameters, - priors of the intercept and slope parameters, - numerical values for dose levels, etc. - numerical values for dose levels, etc.

It may be interesting to discuss the impact of prior It may be interesting to discuss the impact of prior assumptions in terms ofassumptions in terms of1) prior ESS,1) prior ESS,2) prior predictive probabilities: 2) prior predictive probabilities:

PrPr[[((xx,,))>0.99>0.99] & ] & PrPr[[((xx,,)<)<0.010.01],],3) the sensitivity to dose selection decision,3) the sensitivity to dose selection decision,

in order to obtain a “in order to obtain a “sensible priorsensible prior”.”.

ESS of a beta distributionESS of a beta distribution

Saying Saying BeBe((aa, , bb) has ESS = ) has ESS = aa + + bb

implicitly refers to the fact that implicitly refers to the fact that

θ θ ~ ~ BeBe((aa, , bb) and ) and YY | | θθ ~ ~ binbin((nn, , θ) θ) impliesimplies

θθ | | YY 〜〜 BeBe((aa++YY, , bb++nn--YY) )

which has ESS which has ESS = = aa++bb++nn

ESS of a beta distribution (cont’d)ESS of a beta distribution (cont’d)

SayingSaying BeBe((aa,,bb) has ESS = ) has ESS = a a ++ bb

implictly refers to an earlierimplictly refers to an earlier

BeBe((cc,,dd) prior with very small ) prior with very small cc++dd ==εε

and solving for and solving for

mm = = aa++b b –– ((cc++dd) =) = aa++bb – – εε

for a very small value for a very small value εε > 0> 0

Prior ESS of a beta distributionPrior ESS of a beta distribution- Beta-binomial case -- Beta-binomial case -

Be(a,b) Be(c,d)

Be(c+Y,d+m-Y)

p(θθ) q0(θθ)

qm(θθ|Ym)where c+d = is very small

Be(a,b) has a prior ESS = a + b

Solving for m = a+b – (c+d) = a+b –