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

Single Agent Dose-Toxicity Curve

Dose

Pro

b(T

ox)

100%

Target Prob(tox) (e.g. 30%)

MTD

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

50

60

70

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

Dose-Finding On Fixed L1 and Random L2

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]

Admissibility Conditions

Probability Model

11122

2

333

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%) ?

Elicited Doses for the Single Agents

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 ]

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

1x

2x

Computer Simulation Scenarios

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