resolving issues with the simple continual reassessment method · the continuous reassessment...

Resolving Issues with the Simple Continual Reassessment Method Andy Grieve, Ph.D. SVP Clinical Trials Methodology Innova8on Centre Ap8vSolu8ons, Cologne, Germany [email protected]

© 2012 Aptiv Solutions

Outline

n  Introduction to the Continual Reassessment Method (CRM) n  Issues with the CRM

–  Rate of escalation –  Undue influence of early observations`

n  1-parameter model –  Choice of model –  Method for choosing next dose

n  2-parameter model n  Conclusions n  References

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The Background (Oncology)

n  Given several doses of a new compound, determine an acceptable dose for treating patients in future trials

n  Assumptions –  Definition of Dose Limiting Toxicity (DLT) –  Definition of Maximum Tolerated Dose (MTD)

• Prob ( DLT | MTD) = π* –  Prob (Response) é with dose A) –  Prob (Toxicity) é with dose B)

•  These conflict : A) is good; B) is bad


Standard 3+3 Method (Storer, 1989)

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n  Dose levels (Fibonacci), DLT escalation scheme specified

# Patients with DLT Next Dose Level

0/3 é To next level

1/3 3 more patients at this level

1/3 + 0/3 é To next level

1/3 + (1/3, 2/3 or 3/3) Stop: choose previous level

2/3 Stop: choose previous level

3/3 Stop: choose previous level


5

Problems with 3+3 design

n  MTD is not defined – Prob ( DLT | MTD) = π* ? n  It has a high chance of picking an ineffective dose –

(πMTD < π) – O’Quigley et al (1990) n  It doesn’t utilise all of the toxicity data – only the

information from the last 3 or 6 patients n  It has poor operating characteristics


The Continuous Reassessment Method(CRM) O’Quigley et al (1990)

n  Goal : identify a dose with the targeted toxicity (π*) as quickly as possible and focus experiment at that dose

n  Doses are pre-defined : d1, d2, …., dk

n  Outcome is binary : DLT / No DLT

n  Assumption : There exists a monotone dose-response function 𝜓(𝑑;𝜃)=𝑃𝑟𝑜𝑏(𝐷𝐿𝑇|𝑑,𝜃) depending on a single parameter θ

n  The number of patients N is fixed in advance

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CRM Original form

n  Given the doses : d1 , d2 , …., dk , define a set of probabilities p1 , p2 , …., pk

n  Define : Prob(DLT|dj,θ) = (pj)q - power model

–  This can be thought of as a local model

n  Aside – In Quigley et al(1990) dose was not necessarily predefined. –  Could be a combination of compounds whose rank order was

assumed

n  Given p1 , p2 , …., pk , d1 , d2 , …., dk can be defined by 𝑑↓𝑖 = 𝑡𝑎𝑛ℎ↑−1 (2𝑝↓𝑖 −1)


8

CRM Original form

n  A second alternative model looked at by O’Quigley et al specifies the Dose-response model as follows :

n  For some constant α - Quigley et al(1990) suggested α =3

n  The doses are based on solving the equation

𝑝𝐷𝐿𝑇 𝑑↓𝑗  = exp(𝛼+𝛽𝑑↓𝑗 )/1+exp(𝛼+𝛽𝑑↓𝑗 ) 

𝑝↓𝑗 = exp(𝛼+𝛽𝑑↓𝑗 )/1+exp(𝛼+𝛽𝑑↓𝑗 ) 


CRM – Original Form

n  A “weak” prior is assumed for θ, eg exp(-‐θ) with mean 1 –  Alternatively : Prob (DLT|dj ,θ) = (pj)exp(θ)

p(θ) ~ N(0,σ2)

n  Suppose that you have observed a sequence of doses and response pairs (di,yi={0,1}) i=1, …, N

n  Posterior distribution for θ is

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𝑝(𝜃|𝑑,𝑦)∝∏𝑖=1↑𝑁▒𝑝( 𝑦↓𝑖 | 𝑑↓𝑖 ,𝜃) 𝑒↑−𝜃 𝑑𝜃 


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n  The mean of the distribution is available to give information about θ

n  Expected probabilities (EXACT)

n  Choose as next dose the one which gives πi closest to the target π*

n  Or (APPROXIMATE): choose as next dose the one for which is closest to π*

n  Continue until a pre-specified number of patients - final dose is the estimate

CRM – Original Form

𝜋↓𝑖 =∫𝜃↑▒𝑝𝑌=1𝑑↓𝑖 ,𝜃 𝑝𝜃𝑑,𝑦 𝑑𝜃 

𝐸[𝑝↓𝑖↑exp(𝜃)  |𝐷𝑎𝑡𝑎]


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O’Quigley et al (1990) Simulation

Pat Dose Posterior Estimated Expected Response 1 2 3 4 5 6 Mean 1 2 3 4 5 6

0 0.05 0.10 0.20 0.30 0.40 0.50 1.00 0.05 0.10 0.20 0.30 0.40 0.50  1 0 1.38 0.02 0.04 0.11 0.19 0.38 0.61  2 0 1.68 0.01 0.02 0.07 0.13 0.31 0.55  3 1 0.93 0.06 0.12 0.22 0.33 0.52 0.72  4 0 1.07 0.04 0.08 0.18 0.27 0.48 0.68  5 1 0.72 0.12 0.19 0.31 0.42 0.61 0.77  6 1 0.50 0.22 0.32 0.45 0.55 0.71 0.84  7 0 0.56 0.19 0.28 0.41 0.51 0.68 0.82  8 0 0.60 0.16 0.25 0.38 0.48 0.66 0.81  9 0 0.64 0.15 0.23 0.36 0.46 0.64 0.80  10 0 0.69 0.13 0.21 0.33 0.44 0.62 0.78  11 0 0.73 0.11 0.19 0.31 0.42 0.60 0.77  12 0 0.77 0.10 0.17 0.29 0.40 0.59 0.76  13 1 0.63 0.15 0.23 0.36 0.47 0.65 0.80  14 0 0.66 0.14 0.22 0.34 0.45 0.63 0.79  15 0 0.69 0.13 0.20 0.33 0.43 0.62 0.78  16 0 0.72 0.12 0.19 0.31 0.42 0.61 0.77  17 0 0.75 0.11 0.18 0.30 0.41 0.60 0.77  18 0 0.77 0.10 0.17 0.29 0.40 0.59 0.76  19 1 0.67 0.13 0.21 0.34 0.45 0.63 0.79  20 0 0.69 0.12 0.20 0.33 0.43 0.62 0.78  21 0 0.72 0.12 0.19 0.32 0.42 0.61 0.77  22 1 0.64 0.15 0.23 0.36 0.46 0.64 0.80  23 0 0.66 0.14 0.22 0.35 0.45 0.63 0.79  24 1 0.60 0.17 0.25 0.38 0.49 0.66 0.81  25 1 0.54 0.20 0.29 0.42 0.52 0.69 0.83   0


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0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

0 5 10 15 20 25 30

S

S S S F S S F S S S F

F

F S S S

S S S

F

F

S F

S

5

6

4

3

2

1

Dose θ

Patient Number

Posterior Distributions for θ O’Quigley et al (1990)


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Neuenschwander, Branson & Gsponer SIM, 2008

1 2.5 5 10 20 30 50 100 200

Dose

0.0

0.2

0.4

0.6

0.8

1.0 P

roba

bilit

y of

DLT

DLT O

utcome N

o DLT

Power Model, target=0.3, EXACT


Resche-Rigon, Zohar and Chevret Clinical Trials, 2008

Patient Number

Dose Given Outcome

0.001 0.05 0.10 0.30 0.60 0.70 Estimated Failure Probabilities

1 2 3 4 5 6 1 3 Failure 0.474 0.757 0.798 0.857 0.899 0.911 2 1 Success 0.146 0.534 0.621 0.759 0.852 0.877 3 1 Success 0.069 0.412 0.516 0.696 0.823 0.856 4 1 Success 0.046 0.349 0.458 0.658 0.805 0.844 5 1 Success 0.035 0.311 0.421 0.632 0.792 0.835 6 1 Success 0.028 0.284 0.394 0.613 0.784 0.829 7 1 Success 0.024 0.263 0.372 0.597 0.776 0.824 8 1 Success 0.021 0.247 0.355 0.584 0.769 0.819 9 1 Success 0.018 0.234 0.34 0.572 0.764 0.815

10 1 Success 0.017 0.222 0.328 0.562 0.758 0.812 11 1 Success 0.015 0.213 0.317 0.552 0.754 0.808 12 1 S/F 13 1 S/F 14 1 S/F 15 1 S/F 16 2 S/F

One-parameter logistic, α=3, target=0.1, APPROX


Questions

n  Differences in model – power or 1-parameter logistic

n  Differences in analytic approaches – EXACT or APPROXIMATE

n  Which features if any are important ?


Resche-Rigon, Zohar and Chevret Clinical Trials, 2008

Patient Number

Dose Given Outcome



10 1 Success 0.017 0.222 0.328 0.562 0.758 0.812 11 1 Success 0.015 0.213 0.317 0.552 0.754 0.808 12 1 S/F 13 1 S/F 14 1 S/F 15 1 S/F 16 2 S/F

One-parameter logistic, α=3, target=0.1, APPROX


Resche-Rigon, Zohar and Chevret Example

Patient Number

Dose Given Outcome


1 2 3 4 5 6

1 3 Failure 0.123 0.404 0.498 0.695 0.857 0.898

2 1 Success 0.066 0.308 0.405 0.623 0.818 0.869

3 1 Success 0.038 0.242 0.336 0.566 0.785 0.845

4 1 Success 0.025 0.203 0.294 0.527 0.762 0.827

5 1 Success 0.020 0.182 0.270 0.504 0.748 0.817

6 1 Success 0.017 0.171 0.257 0.491 0.740 0.810

7 2

Power Model, target=0.1, APPROXIMATE


One Parameter Logistic

α = 3 α = 2 α = 1 α = 0 α = -1 α = -2 α = -3

𝑝↓𝑗 



Patient Number

Dose Given Outcome


1 2 3 4 5 6

1 3 Failure 0.123 0.404 0.498 0.695 0.857 0.898

2 1 Success 0.066 0.308 0.405 0.623 0.818 0.869

3 1 Success 0.038 0.242 0.336 0.566 0.785 0.845

4 1 Success 0.025 0.203 0.294 0.527 0.762 0.827

5 1 Success 0.020 0.182 0.270 0.504 0.748 0.817

6 1 Success 0.017 0.171 0.257 0.491 0.740 0.810

7 2

Power Model, target=0.1, APPROXIMATE



Patient Number

Dose Given Outcome



10 1 Success 0.046 0.209 0.287 0.498 0.733 0.803 11 1 Success 0.042 0.201 0.278 0.490 0.728 0.799 12 1 S/F 0.038 0.193 0.270 0.483 0.723 0.795 13 1 S/F 0.035 0.187 0.263 0.476 0.719 0.792 14 1 S/F 0.033 0.181 0.257 0.470 0.715 0.789 15 1 S/F 0.031 0.176 0.251 0.464 0.711 0.786 16 2 S/F 0.029 0.171 0.246 0.459 0.708 0.784

Power Model, target=0.1, EXACT


Determining Next Dose

n  Remember

n  Choose as next dose the one which gives πi closest to the target π*

n  For the power model this is:

n  Or, use

𝜋↓𝑖 =∫𝜃↑▒𝑝𝑌=1𝑑↓𝑖 ,𝜃 𝑝𝜃𝑑,𝑦 𝑑𝜃 

𝜋↓𝑖 =∫𝜃↑▒𝑝↓𝑖↑𝜃  𝑝𝜃𝑑,𝑦 𝑑𝜃=𝐸( 𝑝↓𝑖↑𝜃 )

𝜋↓𝑖 = 𝑝↓𝑖↑𝐸(𝜃) 


Convexity of Dose Response Probabilities as a Function of θ

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16 18 20

p iθ

θ

pi=0.1 (0.1) 0.9

pi = 0.9

pi = 0.1


Jensen’s Inequality for Convex Functions

n  General from of Jensen’s inequality

n  Implying for the CRM

n  Using approximation will underestimate the expected probability for a given dose and therefore larger doses can be chosen

𝜙(𝐸(𝜃))≤𝐸(𝜙(𝜃))

𝑝↓𝑖↑𝐸(𝜃) ≤𝐸(𝑝↓𝑖↑𝜃 )


Expectation of a Function Using a Taylor Expansion

n  Let 𝐸𝜃𝑋 =𝜇 be the posterior mean and expand 𝜙(𝜃) be the posterior mean and expand 𝜙(𝜃) about 𝜇

where σ2 is the posterior variance

𝜙(𝜃)=𝜙(𝜇)+(𝜃−𝜇)𝜙↑′ (𝜇) +… + (𝜃−𝜇)↑𝑘 /𝑘! 𝜙↑(𝑘) (𝜇)+… ⇒ 𝐸(𝜙(𝜃))≈ 𝜙(𝜇)+ 1/2 𝜎↑2   𝜙′′(𝜇)


Magnitude of Bias

Patient Number

Dose Given Outcome


1 2 3 4 5 6

1 3 Failure 0.123 0.404 0.498 0.695 0.857 0.898

2 1 Success 0.066 0.308 0.405 0.623 0.818 0.869

3 1 Success 0.038 0.242 0.336 0.566 0.785 0.845

Patient Number

Dose Given Outcome


1 2 3 4 5 6 1 3 Failure 0.323 0.524 0.589 0.733 0.866 0.903 2 1 Success 0.193 0.405 0.481 0.656 0.825 0.872 3 1 Success 0.137 0.345 0.424 0.612 0.801 0.854

APPROXIMATE

EXACT

𝐸(𝜙(𝜃))≈ 𝜙(𝜇)+ 1/2 𝜎↑2   𝜙′′(𝜇)


Why the Approximation ?

n  O’Quigley et al was published in 1990 n  No MCMC

–  Sample from p(θ|data) => a sample from p(Prob(DLT|di) for all i

n  O’Quigley and colleagues weren’t Bayesians and what they didn’t want to do was a lot of integrations

where wj and θj are the zeros and weights of a class of orthogonal polynomials

𝑝(𝜃|𝑑,𝑦)∝∏𝑖=1↑𝑁▒𝑝( 𝑦↓𝑖 | 𝑑↓𝑖 ,𝜃) 𝑒↑−𝜃 𝑑𝜃≈∑𝑗=1↑𝑚▒𝑤↓𝑗 ∏𝑖=1↑𝑁▒𝑝( 𝑦↓𝑖 | 𝑑↓𝑖 ,𝜃↓𝑗 )   


Why a 1-Parameter Model?

n  O’Quigley et al was published in 1990 n  No MCMC n  O’Quigley and colleagues weren’t Bayesians and what

they didn’t want to do was a lot of integrations and particularly not in two-dimensions

𝑝(𝛼,𝛽|𝑑,𝑦)∝∏𝑖=1↑𝑁▒𝑝𝑦↓𝑖  𝑑↓𝑖 ,𝑎,𝛽 𝑝(𝛼,𝛽)𝑑𝛼𝑑𝛽≈∑𝑗=1↑𝑚↓1 ▒∑𝑘=1↑𝑚↓2 ▒𝑤↓𝑗  𝑤↓𝑘 ∏𝑖=1↑𝑁▒𝑝( 𝑦↓𝑖 | 𝑑↓𝑖 ,𝛼↓𝑗 , 𝛽↓𝑘 )   


Way Forward

n  Better models

–  A 1-parameter model doesn’t have the flexibility to model dose-response data very well

–  Why not a 2-parameter model

n This is necessary but it is not sufficient n Choosing the dose

–  Basing dose choice on point estimates is inefficient –  Basing dose choice on point estimates ignores the

safety issues: Babb et al, Neuenschwander et al.


29 © Andy Grieve

Babb et al SIM, 1998

n  Escalation With Overdose Control n  2-parameter logistic + a prior density for (α,β) n  At any point in the study determine p(α,β|data) n  From p(α,β|data) calculate for each dose the probability that

the dose di exceeds the MTD n  Select as the dose for the next the maximum dose for which

Prob(di > MTD|data) < α



n  Determine the posterior probability that the DLT probability at each dose is in the range: Underdosing : 0.00-0.20 Target : 0.20-0.35 Excessive : 0.35-0.60 Unacceptable : 0.60-1.00

n  Choose the dose with the largest posterior probability

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0%

20%

40%

60%

80%

100%

1 2.5 5 10 15 20 25 30 40 50

Under-dosing (0.00-0.20) Target (0.20-0.35)

Excessive (0.35-0.60) Unacceptable (0.60-1.00)


32


n  Choice of prior distributions n  Specifically a normal prior for the log of the parameters is

determined by fixing desirable characteristics based on minimally informative distribution at fixed doses and transforming to a normal.

n  Require the specification not only of pj but an uncertainty interval for pj


Truncated Bivariate Normal Distribution Determined by Stochastic Approximation


Conclusions

n  CRM was a distinct change in emphasis in MTD studies –  Bayesian –  Estimation rather than an algorithmic approach (3 + 3 design)

n  BUT n  Because it is 1-parameter it is not flexible enough to

model real data n  By concentrating on the target dose alone it can be

unsafe. n  Two alternatives – EWOC and N-CRM


References

Babb J, Rogatko A and Zacks S (1998). Cancer phase I clinical trials: efficient dose escalation with overdose control. Statistics in Medicine, 17, 1103-1120. Garrett-Mayer E (2006). The continual reassessment method for dose-finding studies: a tutorial. Clinical Trials, 3, 57-71. Neuenschwander B, Branson M and Gsponer T (2008). Critical aspects of the Bayesian approach to phase I cancer trials. Statistics in Medicine, 27, 2420-2439. O'Quigley J, Pepe M and Fisher L (1990). Continual Reassessment Method: A Practical Design For Phase 1 Clinical Trials in Cancer. Biometrics, 46, 33-48. Resche-Rigon M, Zohar S and Chevret S (2008). Adaptive designs for dose-finding in non-cancer phase II trials: influence of early unexpected outcomes. Clinical Trials, 5, 57-71, 595–606.

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