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 sequential and adaptive for ethical reasons Sample sizes typically are very small Patient h e tero g en E ity may be substantial

6. 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

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/m 2 (gemcitabine, cyclophosphamide) - Target toxicity probability P TOX * = 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 L 1 and Random L 2

13. 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

14. Probability Model x = (x 1, x 2 ) = doses of the two agents  x  Prob(Toxicity | x,  )  1  x 1  1  Prob(Toxicity | x 1,  1 )  2  x 2  2  Prob(Toxicity | x 2,  2 ) x 1 and x 2 are standardized to [0, 1]

15. Admissibility Conditions

16. Probability Model  1  1  1  2  2  2   3  3  3 

17. 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

19. 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%) ?

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 D 1 of dose pairs x = (x 1,x 2 ) and fixed diagonal line L 1 for dose-finding in Stage 1 3) Elicit a target Prob(Toxicity, x) =  from the physician L 2 (data) = Dose pair contour where mean{Prob(Toxicity, x)|data} = 

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 D 1 = {(.12,.12), (.25,.25), … (1,1)}  {(144, 72), (300, 150), … (1200, 600)} mg/m 2 (gemcitabine, cyclophosphamide) - Target Toxicity probability  =.30

24. Dose-Finding Algorithm Stage 1 : Treat each cohort at the dose pair on L 1 having mean Prob(toxicity) closest to the target (Ptox=.30). After the first toxicity, say at x*, add all pairs on L 1 below x* and pairs midway between those above x*. Stage 2 : Alternate cohorts between pairs on the upper left and lower right portions of L 2

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 x n * = (x n,1 *, x n,2 * ) to x = (x 1, x 2 ) on L 2  change in cancer killing potential is K(x, x n * ) = (x 1 -x n,1 * ) + (x 2 -x n,2 * ) where = cancer-killing effect of 1 unit change in agent 1 relative to 1 unit change in agent 2. On L 2, one summand of K(x, x n * ) is >0 and the other is <0  Choose x to maximize K(x, x n * )

28. Information Fisher Information Matrix : I(x,  ) = [  (x,  ) (j)  (x,  ) (k) /  (x,  ){1-  (x,  )} ] where  (x,  ) (j) = ∂  (x,  )/ ∂  j Posterior Mean Information About  (x,  ) : I n (x) = E [ log{det I(x,  )} | data n ]

60. Computer Simulation Scenarios

61. Computer Simulation Results: Average | P(Tox | Selected Dose) – P TOX * |

62. True Prob(Toxicity) Computer Simulation Results

63. True Prob(Toxicity) Computer Simulation Results

64. True Prob(Toxicity) Computer Simulation Results

65. True Prob(Toxicity) Computer Simulation Results

66. 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