Dose-Finding with Two Agents in Phase I Oncology Trials

1. Dose-Finding with Two Agents in Phase I Oncology Trials Thall, Millikan, Mueller & Lee, Biometrics, 2003

2. Outline: - The Two-Agent Problem - Probability Model - Prior Elicitation - A Two-Stage Design - Illustration

3. 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

4. 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

5. 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 sequentialand adaptivefor ethical reasons • Sample sizes typically are very small • Patient heterogenEity may be substantial

6. Previous Approaches to the Problem: • 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 B A

7. Single Agent Dose-Toxicity Curve

8. 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

9. A Hypothetical Dose-Toxicity Surface

10. “Isotoxic” Dose Pair Contours in the Gemcitabine-Cyclophosphamide Plane

11. 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

12. Dose-Finding On Fixed L1 and Random L2

13. Probability Model Prob(Toxicity) as a function of the combination contains the two single-agent Toxicity probabilities as sub-models Model Parametersq = (q1, q2 ,q3) q1 = Parameters for agent 1 alone q2 = Parameters for agent 2 alone q3 = Parameters for synergistic effects

14. Probability Model x = (x1, x2) = doses of the two agents • (x , q) = Prob(Toxicity| x, q) • 1(x1 , q1) = Prob(Toxicity | x1, q1) • 2(x2 , q2) = Prob(Toxicity | x2, q2) x1 and x2 are standardized to [0, 1]

15. Admissibility Conditions

16. Probability Model q1 = (a1 , b1) q2 = (a2 , b2) q3 = (a3 , b3)

17. Probability Model • Informative Priors on the single-agent parameters, q1 and q2 , are obtained from historical data or elicited from the physician • An Uninformative Prior is used for the parameters, q3 , characterizing synergistic effects of the two agents used together

19. Single-Agent Prior Elicitation Algorithm • What is the highest dose having negligible (<5%) Toxicity? • What dose has the targeted (30%) Toxicity? • What dose above the target has unacceptably high (60%) Toxicity? • At what dose above the target are you nearly certain (99% sure) that Toxicity is above the target (30%) ?

20. Elicited Doses for the Single Agents

21. 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) =p* from the physician L2 (data) = Dose pair contour where mean{Prob(Toxicity, x)|data} =p*

23. 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 probabilityp* = .30

24. 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, sayat 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

25. 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

26. 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*) + l (x2 -xn,2*) wherel= 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*)

28. Information Fisher Information Matrix: I(x, q) = [p(x, q)(j)p(x, q)(k)/p(x, q){1- p(x, q)}] where p(x, q)(j) = ∂p(x, q)/ ∂qj Posterior Mean Information Aboutp(x, q): In(x) = E [ log{det I(x, q)} | datan]