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Dose-Finding with Two Agents in Phase I Oncology Trials
Thall, Millikan, Mueller & Lee, Biometrics, 2003
Outline:
- The Two-Agent Problem
- Probability Model
- Prior Elicitation
- A Two-Stage Design
- Illustration
The Two-Agent Problem
- Study two agents used together in a phase I
clinical trial, with dose-finding based on Toxicity
- Prior information on each agent used alone in
previous trials is available
- Goal: Find one or more dose pairs of the two
agents used together - for future clinical use and/or
study in a randomized phase II trial
Difficulties in Two-Agent Phase I Trials
• Synergy little is known a priori about actual clinical effects of the two agents used together
• The set of possible dose pairs is much larger than the usual interval of doses in the single-agent case
Difficulties in Two-Agent Phase I Trials
• Due to synergy, little is known a priori about actual clinical effects of the two agents used together
• Dose-finding must be sequential and adaptive for ethical reasons
• Sample sizes typically are very small
• Patient heterogenEity may be substantial
Previous Approaches to the Problem:
1) Select a combination based on “Total Equivalent Dose” (Simon and Korn;1990,1991)
2) Use a single-agent method (e.g. the CRM, isotonic regression) on a “staircase” of dose pairs
A
B
Gem/CTX Trial (R. Millikan, P.I.)
- 2 patients per cohort
- 20 patients in Stage 1 (10 cohorts)
- 40 patients in Stage 2 (20 cohorts)
- Stage 1 doses :
{(144, 72), (300, 150), … (1200, 600)}
mg/m2 (gemcitabine, cyclophosphamide)
- Target toxicity probability PTOX*= 0.30
0
200
400
600
800
1,000
1,200
1,400 0
400
600
900
0
10
20
30
40
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60
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80
P(tox)
Cyclophosphamide
Gemcitabine
A Hypothetical Dose-Toxicity Surface
0 200 400 600 800 1,000 1,200 1,4000
200
400
500
600
700
900
1,000
020
40
60
80 Cyclophosphamide
Gemcitabine
“Isotoxic” Dose Pair Contours in the Gemcitabine-Cyclophosphamide Plane
A New Two-Stage Method
1) Information on the single-agents used alone is obtained from
Historical data or Elicited from the physician
2) Nothing is assumed, quantitatively, about synergistic effects of the two agents used together
Probability Model
Prob(Toxicity) as a function of the combination contains the two single-agent Toxicity probabilities
as sub-models
Model Parameters = (1,2 , 3)
1 = Parameters for agent 1 alone
2 = Parameters for agent 2 alone
3 = Parameters for synergistic effects
Probability Model
x = (x1, x2) = doses of the two agents
xProb(Toxicity | x, )
1x11Prob(Toxicity | x1, 1)
2x22Prob(Toxicity | x2, 2)
x1 and x2 are standardized to [0, 1]
Probability Model
• Informative Priors on the single-agent parameters, 1 and 2 , are obtained from historical data or elicited from the physician
• An Uninformative Prior is used for the parameters, 3 , characterizing synergistic effects of the two agents used together
Single-Agent Prior Elicitation Algorithm
1. What is the highest dose having negligible (<5%) Toxicity?
2. What dose has the targeted (30%) Toxicity?
3. What dose above the target has unacceptably high (60%) Toxicity?
4. At what dose above the target are you nearly certain (99% sure) that Toxicity is above the target (30%) ?
Dose-Finding Algorithm: Preliminaries
1) Determine cohort size, and sample sizes for each of the two stages
2) Determine a set D1 of dose pairs x = (x1,x2) and fixed diagonal line L1 for
dose-finding in Stage 1
3) Elicit a target Prob(Toxicity, x) = from the physician
L2 (data) = Dose pair contour where
mean{Prob(Toxicity, x)|data} =
For the Gem/CTX Trial :
- 2 patients per cohort
- 20 patients in stage 1 (10 cohorts)
- 40 patients in stage 2 (20 cohorts)
Stage 1 doses
D1 = {(.12, .12), (.25, .25), … (1,1)}
{(144, 72), (300, 150), … (1200, 600)}
mg/m2 (gemcitabine, cyclophosphamide)
- Target Toxicity probability = .30
Dose-Finding Algorithm
Stage 1 : Treat each cohort at the dose pair on L1 having mean Prob(toxicity) closest to the target (Ptox=.30). After the first toxicity, say at x*, add all pairs on L1 below x* and pairs
midway between those above x* .
Stage 2 : Alternate cohorts between pairs on the upper left and lower right portions of L2
Dose-Finding Criteria in Stage 2
Choose the dose pair for the next cohort to:
1) Maximize the amount of Information
2) Maximize Cancer-Killing Potential
The algorithm optimizes these two criteria separately, and then chooses the
average of the two optimal dose pairs
Cancer Killing Potential
Moving from xn* = (xn,1
*, xn,2*) to x = (x1, x2)
on L2 change in cancer killing potential is
K(x, xn*) = (x1 -xn,1
*) + (x2 -xn,2*)
where = cancer-killing effect of 1 unit change in agent 1 relative to 1 unit change in agent 2. On L2, one summand of K(x, xn
*) is >0 and the other is <0 Choose x to maximize K(x, xn
*)
Information
Fisher Information Matrix :
I(x, ) = [(x, )(j) (x, )(k)/(x, ){1- (x, )}]
where (x, )(j) = ∂(x, )/ ∂j
Posterior Mean Information About (x, ) :
In(x) = E [ log{det I(x, )} | datan ]
Computer Simulation Results: Average
| P(Tox | Selected Dose) – PTOX* |
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Scen 1 Scen 2 Scen 3 Scen 4
0
5
10
15
20
25
30
0-10 11-20 21-30 31-40 41-50 51-60 61-70
Scenario 1
# Treated # Toxicities
True Prob(Toxicity)
Computer Simulation Results
0
5
10
15
20
0-10 11-20 21-30 31-40 41-50 51-60 61-70
Scenario 2
# Treated # Toxicities
True Prob(Toxicity)
Computer Simulation Results
0
5
10
15
20
25
0-10 11-20 21-30 31-40 41-50 51-60 61-70
Scenario 3
# Treated # Toxicities
True Prob(Toxicity)
Computer Simulation Results
05
101520253035
0-10 11-20 21-30 31-40 41-50 51-60 61-70
Scenario 4
# Treated # Toxicities
True Prob(Toxicity)
Computer Simulation Results
Concluding Remarks
- A 2-stage, outcome-adaptive, Bayesian method for dose-
finding with two agents in a phase I clinical trial
- In Stage 2, dose pairs are chosen to maximize
Cancer-Killing Potential and/or Information
- Several dose pairs may be selected for future study
- Free state-of-the art Computer Software available