Introduction to the Bayesian continual reassessment method ...biostat.mc.vanderbilt.edu/wiki/pub/Main/JoAnnAlvarez/crmPres.pdf · Introduction to the Bayesian continual reassessment

Introduction to the Bayesian continualreassessment method (CRM) for phase one clinical

trials

JoAnn Alvarez, [email protected]

Department of BiostatisticsCenter for Quantitative Sciences

Vanderbilt University School of Medicine

2013 October 11

[email protected]

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Overview

1 introduction

2 model

3 variations on CRM

J. Alvarez Bayesian CRM

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Phase 1 goal

Find the right dose


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The higher the dose,

the more efficaciousthe higher the toxicity

We balance efficacy and toxicity by choosing a targetedtoxicity level (TTL) (probability of toxicity)

Then find dose with targeted toxicity level: MTD, maximumtolerated dose


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Phase 1 design

Goal is to

get information about the best dose

treat each patient ethically

: with the treatment bestsupported by the current evidence


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Phase 1 design

Goal is to


treat each patient ethically: with the treatment bestsupported by the current evidence


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Design of phase one study involves best way to acheive:


treat each patient with the treatment best supported by thecurrent evidence

Main design issue is what dose to give each patient


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Basic idea of CRM

after each patient outcome is observed, dose-responserelationship is re-estimated

next patient is given the dose that is the current estimate ofthe MTD


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patients arrive sequentially

observation Yj on each patient is whether they have a toxicresponse


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Dose-response relationship

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

Standardized dose

P(Y

= 1

| do

se =

x)


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Let θ be the TTL.

Objective is to find corresponding dose, x∗ (MTD)


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Dose-response model

Need a model for the dose-response relationship.

Choose any one-parameter function ψ(x, a), monotonic in xand a.

We assume that there exists an a0: ψ(x∗, a0) = θ

Can think of a0 as a population parameter


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since each patient gets the current best estimate of the MTD,more information is collected for values near the true MTD

model not expected/required to be accurate at doses far fromthe MTD

model expected to perform well near the MTD.

only need one-parameter model


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Objective is to find the value of a that gives x∗

a : ψ−1(θ, a0) = x∗

This value is a0.


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The model

We assume∀θ,∀x∗, ∃!a0 : ψ(x∗, a0) = θ.


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∀θ,∀x∗, ∃!a0 : ψ(x∗, a0) = θ.

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

Standardized dose

ψ(a

, x)=

P(D

LT|d

ose

= x

)

x*

θ


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∀θ,∀x∗, ∃!a0 : ψ(x∗, a0) = θ.

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

Standardized dose

ψ(a

, x)=

P(D

LT|d

ose

= x

)

x*

θ


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∀θ,∀x∗, ∃!a0 : ψ(x∗, a0) = θ.

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

Standardized dose

ψ(a

, x)=

P(D

LT|d

ose

= x

)

x*

θ

a0 = ?


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Example dose-response curve family

ψ(x, a) =(tanh(x) + 1)a

2a

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

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0.2

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1.0

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ψ(a

, x)=

P(D

LT|d

ose

= x

)

a0 = 0.1

a0= 0.3

a0= 0.5

a 0= 1

a 0= 5


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Dose-response models

logistic

hyperbolic tangent

power


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Dose-response models

Logistic model:

ψ(x, a) =e3+ax∗

1 + e3+ax∗

−3 −2 −1 0 1

0.0

0.2

0.4

0.6

0.8

1.0

Standardized dose

P(Y

= 1

| do

se =

x)


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Suppose the jth patient has enrolled and is ready to receivethe treatment.

The first j − 1 patients have observed response data:x(1) Y1x(2) Y2x(3) Y3

......

x(j − 1) Yj−1

Want to give patient j current best guess of MTD.


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Let the current posterior for a0 be fa0(a, data)


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Ways to estimate P(DLT) for each dose

Estimate P (Y = 1|x = xi):

Plug-in estimatorUsing the data from the first j − 1 patients, calculatethe posterior mean of a0.Plug it into ψ to get an updated dose-response curve.

Mean estimatorestimate the probability of toxicity it by its mean,integrating over all possible values of a0 at each dose.

P (Y = 1|x = xi) =

∫ ∞0

ψ(xi, a)fa0(a, data) da


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Choose next dose

Now that we have updated P (Y = 1|x = xi), we have to choosethe ‘best’ dose


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Choose next dose

Patient j’s dose, x(j), will be

dose that is closest to the the current estimate of the MTD:the dose that gives θ in the current estimate of ψdose that gives the estimated P(DLT) closest to the TTL, θ

Standardized dose

P(Y

= 1

| do

se =

x)

x*

θ

x1 x2 x3 x4

●

●

●

●

●

θ1^

θ2^

θ3^

θ4^


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Choose next dose

Patient j’s dose, x(j), will be

dose that is closest to the the current estimate of the MTD:the dose that gives θ in the current estimate of ψdose that gives the estimated P(DLT) closest to the TTL, θ

Standardized dose

P(Y

= 1

| do

se =

x)

x*

θ

x1 x2 x3 x4

●

●

●

●

●

θ1^

θ2^

θ3^

θ4^


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Observe patient j’s response, Yj : either toxicity or no toxicity.

Patient j’s datum will be used to update the posteriordistribution of a0, fa0(a, data)


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Likelihood function

For one patient:

P (Yj = yj) = (ψ(xj , a))yj (1− ψ(xj , a))1−yj .

Based on the bernoulli pmf


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Bayes

posterior after observing jth patient:

fa0(a, dataj) =(ψ(xj , a))

yj (1− ψ(xj , a))1−yjfa0(a, dataj−1)∫∞0 (ψ(xj , u))yj (1− ψ(xj , u))1−yjfa0(u, dataj−1) du


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Choice of prior

Common priors for a0

gamma

uniform

lognormal


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Final estimate of MTD

Can be estimated in same way as the dose for each patient isdetermined, since each patient is treated with the currentestimate

Determined after last patient is observed, if prespecifiedmaximum n


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Variations

modified CRM

EWOC escalation with overdose control (Babb et al., 1998)

Intervals of toxicity (Neuenschwander et al., 2008)


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Reference

O’Quigley, J., Pepe, M., and Fisher, L. (1990).

Continual reassessment method: A practical design for phase 1 clinical trials in cancer.Biometrics, 46:43–48.


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Standardized doses

Calculated based on

the probabilities of toxicity that the docs think the set ofstudy doses have

the assumed dose-response model

an initial estimate of a0

This info would be based on previous studies.


Documents

Introduction to the Bayesian continual reassessment method ...biostat.mc.vanderbilt.edu/wiki/pub/Main/JoAnnAlvarez/crmPres.pdf · Introduction to the Bayesian continual reassessment