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Resolving Issues with the Simple Continual Reassessment Method Andy Grieve, Ph.D. SVP Clinical Trials Methodology Innova8on Centre Ap8vSolu8ons, Cologne, Germany andy.grieve@ap8vsolu8ons.com
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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
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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
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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
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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)
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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 )
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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
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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
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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 ?
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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
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Resche-Rigon, Zohar and Chevret Example
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 0.123 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
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One Parameter Logistic
α = 3 α = 2 α = 1 α = 0 α = -1 α = -2 α = -3
𝑝↓𝑗
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Resche-Rigon, Zohar and Chevret Example
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 0.123 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
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Resche-Rigon, Zohar and Chevret Example
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 0.323 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 4 1 Success 0.107 0.307 0.387 0.583 0.784 0.842 5 1 Success 0.087 0.280 0.360 0.561 0.771 0.832 6 1 Success 0.074 0.259 0.339 0.544 0.761 0.824 7 1 Success 0.064 0.243 0.323 0.530 0.753 0.818 8 1 Success 0.056 0.230 0.309 0.518 0.745 0.812 9 1 Success 0.050 0.218 0.297 0.507 0.739 0.807
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
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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)
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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
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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 )
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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)
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Magnitude of Bias
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 0.123 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
0.001 0.05 0.10 0.30 0.60 0.70 Estimated Failure Probabilities
1 2 3 4 5 6 1 3 Failure 0.323 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)
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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 )
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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 )
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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.
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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
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Neuenschwander, Branson & Gsponer SIM, 2008
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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Neuenschwander, Branson & Gsponer SIM, 2008
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)
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Neuenschwander, Branson & Gsponer SIM, 2008
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
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Truncated Bivariate Normal Distribution Determined by Stochastic Approximation
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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
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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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