Transcript presentation_Anna5-26-2009-9-36-25

Bayesian Adaptive Designs for Dose
Escalation Studies
Midwest Biopharmaceutical Statistics Workshop
Anna McGlothlin
20 May 2009
Contents
Traditional Dose Escalation Design
(and its shortcomings)
The Continual Reassessment Method
An example trial using CRM
Simulation and Operating Characteristics
Overview of other novel designs
Summary
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Motivation
Why should you care about novel designs for dose
escalation studies?
Traditional designs are not reliable
for selecting the correct maximum
tolerated dose
Wrong dose carried
forward to future trials.
Standard design tends to treat a high
percentage of patients at doses
outside of the therapeutic range.
Novel designs are
better for patients!
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Dose Escalation Studies
Typically small, uncontrolled* studies.
GOAL: Determine the maximum tolerated dose (MTD),
and/or a recommended Phase II dose.
Two Approaches:
1.
Algorithm-based designs
–
–
2.
3+3 (or the more general A+B)
MTD is identified as the dose with fewer than some proportion of
dose limiting toxicities (e.g. <1/3).
Model-based designs
–
MTD is estimated as a quantile of the dose-toxicity curve
* Design may be modified to allow for a control. This will be briefly discussed later.
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Standard 3 + 3 Design
Note: DLT = Dose Limiting Toxicity
Enter 3 patients at dose level i
0/3 DLT’s
1/3 DLT’s
> 1/3 DLT’s
Add 3 patients to dose level i
1/6 DLT’s
Escalate to dose level i + 1
> 1/6 DLT’s
Stop and declare dose level i – 1
as the MTD
One common variation allows de-escalation, as in the following example.
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Example: 3 + 3 Design
Escalation History
Design: 3+3
no toxicity
toxicity
6
Dose Level
5
4
3
2
1
0
5
10
15
20
25
At the end of the
trial, dose level 3
is declared the
MTD.
Patient Number
Scenario: Curve2
Simulation Number: 080
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Problems with Standard 3 + 3 Design
The 3 + 3 design tends to treat a
high proportion of patients at low,
possibly ineffective dose levels.
1.0
0.2
0.4
0.6
0.8
If a dose has true DLT rate
of 25%, there is a 60%
chance that the algorithm
will escalate to a higher
dose for the next cohort.
0.0
The probability of stopping at an
incorrect dose level is higher than
generally believed (Reiner,
Paoletti, O’Quigley 1999).
Probability of Escalation
There is no statistical estimation of
the MTD.
Probability of Escalation for 3 + 3 Design
0.0
This design uses data only from
the most recent cohort, and
ignores data from previous
cohorts.
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0.2
0.4
0.6
0.8
1.0
True Pr(DLT)
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Model-based Designs
Model-based designs use a statistical model to describe the relationship
between dose and outcome:
Continual Reassessment Method (CRM)
• O’Quigley, Pepe, Fisher (1990)
• Faries (1994)
• Goodman, Zahurak, Piantadosi (1995)
Escalation with Overdose Control (EWOC)
• Babb, Rogatko, Zacks (1998)
Joint Toxicity/Efficacy
• Braun (2002)
• Thall and Cook (2004)
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Continual Reassessment Method
1.
Start with a prior estimate of
Pr(DLT) for each dose level.
2.
Select a mathematical model to
describe the relationship between
dose and Pr(DLT).
3.
Describe uncertainty about the
model by a prior distribution.
4.
After each patient, update the
model, and estimate the
probability of toxicity for each
dose level.
5.
Treat the next patient at the dose
whose estimate is closest to
some pre-specified target (say,
25%).
6.
Stop when a maximum sample
size is reached.
Reference: O’Quigley, Pepe, and Fisher (1990)
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Statistical Models for CRM
Let the toxicity response be yj ~ Binomial(nj, pj) for doses j = 1, …, J.
The following models are commonly used with CRM:
Hyperbolic Tangent:
 tanh(xˆ j )  1 

p j  
2


Logistic:
 pj
log
1 p
j

Power:


     xˆ j


p j  xˆ j 

Prior for β: Unit Exponential, Uniform, Gamma, etc.
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1.0
Transformation of Dose Levels
These single-parameter curves
are only defined over a restricted
set of x’s.
2
0.6
0.0
0.2
The x-hat values are calculated to
give the defined prior probabilities
on the dose-toxicity curve,
assuming that β = 1 (its prior
mean)
-3
-2
-1
0
1
2
3
x
1
2
3
4
5
6
Prior
0.01
0.05
0.15
0.30
0.45
0.65
X-hat
-2.30
-1.47
-0.88
-0.42
-0.10
0.31
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0.4
Probability of DLT
Therefore, the doses must be
transformed to ensure that they lie
in the appropriate range.
Dose Level
tanh x
0.8
p
11
Modifications to the CRM
To address concerns surrounding the original implementation of CRM,
several modifications have been proposed, including:
1.Always start at the lowest dose level.
2.Limit the escalation increment.
3.Escalate by cohorts rather than single patients.
4.Definition of MTD:
•
•
Dose whose Pr(DLT) is closest to target, or
Highest dose where Pr(DLT) is below target
5.Early stopping rules
•
•
•
Stop if CRM recommends a dose level at which XX number of cohorts have already been
treated.
Stop if any dose has probability > XX of being the MTD.
Stop if the (1 – α)*100% credible interval for MTD is sufficiently narrow.
Notable references: Faries (1994); Goodman, Zahurak, Piantadosi (1995)
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A Hypothetical Trial
Consider a dose escalation study with the following design characteristics:
• Cohort Size = 3 subjects
• Maximum Sample Size = 10 cohorts (30 subjects)
• 6 Dose Levels
• Doses must be explored in sequential order (no skipping), starting with the
lowest dose.
• MTD is defined as the dose level at which the probability of DLT is nearest to
25%.
• Early Stopping Rule: Stop if 3 cohorts have been treated at a dose, and CRM
predicts the same dose for the next cohort
• Model: Hyperbolic tangent; Unit exponential prior for β
• Prior Probability of DLT at each dose:
Dose Level
Pr(DLT)
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2
3
4
5
6
0.01
0.05
0.15
0.30
0.45
0.65
13
Hypothetical Trial
Escalation History
Design: CRM
cohort
6
no toxicity
toxicity
Dose
Level
# of
DLTs
Estimated Pr(DLT) per dose
1
2
3
4
5
6
5
Dose Level
4
prior
---
---
0.010
0.050
0.150
0.300
0.450
0.650
1
1
0
0.045
0.093
0.173
0.280
0.395
0.573
2
2
0
0.012
0.038
0.096
0.192
0.306
0.498
3
3
0
0.006
0.023
0.070
0.161
0.278
0.482
4
4
0
0.002
0.010
0.038
0.104
0.202
0.396
5
5
2
0.007
0.031
0.097
0.211
0.344
0.550
6
4
1
0.007
0.031
0.097
0.214
0.351
0.561
7
4
1
0.011
0.043
0.124
0.254
0.396
0.602
3
2
1
5
10
15
20
Patient Number
Scenario: Curve2
Simulation Number: 061
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Simulation Overview
The preceding slide demonstrated the performance of CRM for a single
hypothetical trial.
We now ask the question: “How does the method perform on average?”
This question is addressed by simulation:
1.
2.
3.
Assume we know the ‘true’ curve
Conduct a hypothetical trial using data generated from the true curve
Repeat many times
Operating Characteristics:
1.
2.
3.
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How often is each dose level chosen as MTD at the end of the trial?
Average sample size (overall and per dose level)
Etc.
15
Simulation Scenarios
Virtual Subject Response
Three different curves to represent
a possible dose-toxicity curve:
Curve1
Curve2
Curve3
•
MTD1 = Dose Level 2
• MTD2 = Dose Level 4
• MTD3 = Dose Level 5
True Pr(DLT)
0.6
For each scenario, simulate 1000
trials.
0.4
0.2
Summarize each scenario and
compare to standard design.
0.0
Prior probabilities:
1
2
3
4
5
6
Dose Level
Dose Level
Pr(DLT)
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2
3
4
5
6
0.01
0.05
0.15
0.30
0.45
0.65
16
Simulation Results
Selection Percentage of MTD
Scenario: Curve1
3 + 3 chooses lowest
dose in over 50% of
simulated trials!
True Pr(toxicity)
Target Pr(toxicity)
CRM
3+3
0.8
Average
Trial Size
Probability
of correct
MTD
CRM
16.21
0.59
3+3
12.58
0.41
0.6
Percent
Design
0.4
0.2
1
2
3
4
5
6
Dose Level
Results are from 1000 replications.
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Simulation Results
Again, 3 + 3 often chooses a
dose that is below the true MTD.
Selection Percentage of MTD
Scenario: Curve2
True Pr(toxicity)
Target Pr(toxicity)
CRM chooses the correct dose ~
53% of the time.
0.8
Average
Trial Size
Probability
of correct
MTD
CRM
21.10
0.53
3+3
18.87
0.35
0.6
Percent
Design
CRM
3+3
0.4
0.2
1
2
3
4
5
6
Dose Level
Results are from 1000 replications.
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Simulation Results
Selection Percentage of MTD
Scenario: Curve3
True Pr(toxicity)
Target Pr(toxicity)
CRM
3+3
0.8
Average
Trial Size
Probability
of correct
MTD
CRM
23.02
0.60
3+3
21.22
0.32
0.6
Percent
Design
0.4
0.2
1
2
3
4
5
6
Dose Level
Results are from 1000 replications.
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Simulation Results
Average Number of Patients
Scenario: Curve3
CRM treats higher
proportion of patients at
doses close to the MTD.
CRM
3+3
0.25
Number of Patients
0.20
0.15
0.10
0.05
1
2
3
4
5
6
Dose Level
Results are from 1000 replications.
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CRM with No Early Stopping
The probability of selecting the correct dose improves when the CRM continues
to the maximum trial size with no early stopping.
Selection Percentage of MTD
Scenario: Curve1
True Pr(toxicity)
Target Pr(toxicity)
Selection Percentage of MTD
Scenario: Curve3
Early Stopping
No Early Stopping
True Pr(toxicity)
Target Pr(toxicity)
Selection Percentage of MTD
Scenario: Curve2
Early Stopping
No Early Stopping
True Pr(toxicity)
Target Pr(toxicity)
0.6
0.6
0.6
Percent
0.8
Percent
0.8
Percent
0.8
0.4
0.4
0.4
0.2
0.2
0.2
1
2
3
4
5
6
1
2
Dose Level
3
4
5
6
1
2
Dose Level
Results are from 1000 replications.
Early Stopping
No Early Stopping
Results are from 1000 replications.
3
4
5
6
Dose Level
Results are from 1000 replications.
Average Trial Size:
Design
Curve 1
Curve 2
Curve 3
Early Stopping
16.21
21.10
23.02
No Early Stopping
30.00
30.00
30.00
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CRM vs. 3+3
1.
The standard design is easy to understand and implement.
2.
The 3+3 design tends to choose a dose below the true MTD.
3.
The CRM tends to treat patients at doses close to the MTD,
whereas 3+3 treats a higher proportion of patients at low, possibly
ineffective, doses.
4.
The CRM provides a statistical estimate of the MTD, and allows for
uncertainty around this estimate.
5.
CRM can target any relevant DLT rate.
6.
CRM incorporates available data from all cohorts, while the 3 + 3
design uses information from only the most recent cohort.
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Operational considerations
Allow sufficient time prior to protocol approval to conduct simulations
and assess operating characteristics.
Statistician will need timely access to data during the trial in order to
update the model.
Model updates can be performed prospectively – Given the current
data, what will the model-predicted dose be if:
The next patient has a DLT?
The next patient has no DLT?
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Other Dose Escalation Designs
Two-sample curves
Examples:
0.0
2.
TRT+SOC and TRT alone
Different dosing schedules
0.2
1.
0.4
Probability of toxicity
Suppose there are two distinct,
but related, populations.
0.6
0.8
1.0
1. Two-sample CRM
It may be reasonable to assume
that there is some information
common to both populations.
The dose-toxicity curves may be
modeled to account for this
shared information.
-8
-6
-4
-2
0
x
Logistic:
p jk 
bk e
 xˆ j
1  bk e
 xˆ j
b1  1, b2  b
;
Hyperbolic Tangent:
 tanh(xˆ j )  1 

p jk  
2


  bk
b1  0, b2  b
;

Reference: O’Quigley, Shen, Gamst (1999).
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 tanh(xˆ j  bk )  1 
 ;
p jk  
2


b1  0, b2  b
24
Other DE Designs (continued)
2. Escalation with Overdose Control (EWOC)
Model:
e0 1x
p
1  e0 1x
Reparameterization:
0 
1
logit(0 )  xminlogit( p)
  xmin
1 
1
logit( p)  logit(0 )
  xmin
where:
γ = MTD
ρ0 = Pr(DLT) at xmin
Marginal posterior cdf of the MTD: Πk(x)
Escalation Scheme: The kth patient is allocated to dose xk  k 1 () so that the
posterior probability of exceeding MTD is equal to the “feasibility bound,” α.
References: Babb J, Rogatko A, Zacks S (1998); Chu et al. (2009)
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Other DE Designs (continued)
3. Bivariate CRM
Suppose that interest lies in two outcomes: toxicity and efficacy.
Joint model:
f ( y, z)  k ( p1, p2 , ) p1y (1  p1 )1 y p2z (1  p2 )1 z  yz (1  )1 yz
Where:
p1 and p2 are the probabilities of toxicity and efficacy respectively
• y and z binary indicators of toxicity and efficacy
• k(p1,p2,ψ) is a normalizing constant
• ψ is the probability of combined toxicity and efficacy.
•
Two-stage design:
1. Estimate MTD using the previously described CRM.
2. Then subjects are allocated to the dose whose probability of efficacy
is closest to some pre-defined target.
Reference: Braun (2002); Alternative design for toxicity and efficacy outcomes: Thall and Cook (2004).
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Other DE Designs (continued)
1.0
Cumulative Logits
0.8
0.7
0.6
0.5
Probability of Toxicity
0.4
0.3
0.2
no or minor toxi
city
moderatetoxicity
severe toxicity
verysevere toxicity
0.1
1
2

Y 
3
4
2)
3)
4)
0.0
Suppose that toxicity is measured on an
ordinal scale:
Pr(Y
Pr(Y
Pr(Y
0.9
4. CRM for Ordered Outcomes
1
2
3
4
5
6
7
8
9
Dose Level
Define MTD as the dose for which Pr(grade 3 or above) is closest to some prespecified target (say, 25%).
Use information from lower grade toxicities to improve estimation of MTD.
Alternatively, Bekele and Thall (2004) propose a design to incorporate information
from multiple ordinal toxicities, weighted according to importance.
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Summary
Traditional designs for dose escalation are not optimal for selection of
MTD, and may expose a high proportion of patients to low doses.
Novel designs such as CRM are under-utilized, and should be
considered for dose escalation studies.
Novel designs have been proposed to address different trial objectives
(efficacy/toxicity, two-samples, etc.)
Simulations are vital to understanding the operating characteristics of
the trial design.
The most common implementation of CRM is for phase 1 oncology
trials. But its use should not be confined to just one therapeutic area.
•
Trial design may be modified to allow a control arm:
– Within each cohort, randomized subjects to TRT dose or placebo.
– The placebo information may be incorporated into the dose-toxicity model.
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Key References
Bekele BN, Thall PF (2004). Dose-finding based on multiple toxicities in a soft tissue sarcoma trial. Journal of the American
Statistical Association, 99: 26-35.
Babb J, Rogatko A, Zacks S (1998). Cancer phase I clinical trials: Efficient dose escalation with overdose control. Statistics
in Medicine, 17: 1103-1120.
Braun TM (2002). The bivariate continual reassessment method: extending the CRM to phase I trials of two competing
outcomes. Controlled Clinical Trials, 23: 240-256.
Chu P-L, Yong L, Shih WJ (2009). Unifying CRM and EWPC designs for phase I cancer clinical trials. Journal of statistical
planning and inference, 139: 1146-1163.
Faries D (1994). Practical modifications of the continual reassessment method for phase I cancer clinical trials. Journal of
Biopharmaceutical Statistics, 4:147-164.
Goodman SN, Zahurak ML, Piantadosi S (1995). Some practical improvements in the continual reassessment method for
phase I studies. Statistics in Medicine, 14:1149-1161.
Heyd JM, Carlin BP (1999). Adaptive design improvements in the continual reassessment method for phase I studies.
Statistics in Medicine, 18:1307-1321.
Ishizuka N, Ohashi Y (2001). The continual reassessment method and its applications: a Bayesian methodology for phase I
cancer clinical trials. Statistics in Medicine, 20:2661-2681.
Lasonos A (2008). A comprehensive comparison of the CRM to the standard 3+3 dose escalation scheme in Phase I dose
finding studies. Clinical Trials, 5: 465-477.
O’Quigley J, Pepe M, Fisher L (1990). Continual reassessment method: A practical design for phase I clinical trials in
cancer. Biometrics, 46:33-48.
O’Quigley J, Shen Z, and Gamst A (1999). Two-Sample Continual Reassessment Method. Journal of Biopharmaceutical
Statistics, 9: 17-44.
Rogatko, et al. (2007). Translation of Innovative Designs Into Phase I Trials. Journal of Clinical Oncology, 25: 4982-4986.
Thall PF, Cook JD (2004). Dose-Finding Based on Efficacy-Toxicity Trade-Offs. Biometrics, 60: 684-693.
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