Modeling & Simulation to Improve the Design of Clinical Trials

Transcription

1 Modeling & Simulation to Improve the Design of Clinical Trials Gary L. Rosner Oncology Biostatistics and Bioinformatics The Sidney Kimmel Comprehensive Cancer Center at Johns Hopkins Joint work with Lorenzo Trippa & Peter Müller 1

2 Outline History of modeling and simulation in the design of clinical trials Pharmacokinetics & pharmacodynamics Sensitivity of operating characteristics to violations of assumptions. Some examples in oncology - Decision-theoretic Bayesian designs 2

3 History of M & S to Aid Design Simulation of clinical trial for training Instant Experience Maxwell C, Domenet JG, & Joyce CR. (1971). J Clin Pharmacol New Drugs 11, Simulation for drug development Apply pop n PK & PD to aid design choice - Route of admin, dosage, schedule Evaluation of operating chars for designs Complex study designs; Bayesian designs 3

4 Stochastic Components of Trials Modeling & simulation allow for Between-pt heterogeneity - in outcomes (likelihood) - in covariates - in latent traits - in treatment effects Within-patient heterogeneity over time 4

5 Decision Theory & Clinical Trials Why Decision Theory in Design? Clinical trial design involves decisions - Sample size - Duration of follow-up - Stopping rules during study (all phases) - Whether to run the study in the 1 st place Why not make decisions explicit and coherent? 5

6 Decision Theory Consider Set of possible actions: A - E.g., stop the study; move to phase 3 Set of possible outcomes: Y Parameters characterizing stochastic nature: Θ - E.g., treatment effect, model parameters Utility function: u(a) 6

7 Bayesian Optimal Design Bayes action is action that maximizes utility, accounting for all sources of uncertainty Uncertainty in parameters p(θ) Variation in data resulting from action U(a) = Y Θ u(y)p a(y θ)p(θ)dθdy p a (y θ) - Choose: a = arg max U(a) 7

8 Dose Optimization AUC too low! TARGET AUC too high! 8

9 Bayesian Optimal Design Let u( y,d,θ) = L( AUC), AUC = f ( d,v,k) Pick dose d* that minimizes [ ] = u y,d,θ E u( y,d,θ) ( ) p ( y,θ) d dy dθ Specifically: [ ] = L[ AUC( θ,d) ] p( θ D) E u( y,d,θ) dθ 9

10 Asymmetric Loss Function Want AUC in optimal range Loss function L( auc) = L L + ( auc, AUC ) ll 0 ( auc, AUC ) ul if auc < AUC ll if AUC ll < auc < AUC ul if auc > AUC ul 10

11 Posterior for Pt s PK Parameters 1 st pt in 3 rd study, accounting for studies 1 & 2 11

12 Optimal Dose w.r.t. Posterior Loss Function Expected Loss 12

13 Principal Objective of Phase II Determine if a drug has sufficient activity to warrant further investigation Explanatory rather than pragmatic Not clinical efficacy analysis - Clinical Efficacy Activity - Activity Clinical Efficacy 13

14 Phase II as Screening for Activity New treatments arise Which treatments deserve further study? Costs - Patients - Money - Resources Goal is to make a decision! 14

15 Measure Activity: Targeted Agents Cytostatic vs. Cytotoxic Stable disease as goal Problematic but possibly useful Need randomization Phase II for activity may have to enroll more patients than in past Ratain & Eckhardt, JCO 22:4442-5,

16 Phase II Study Needs Good accrual Ability to determine if active drug Avoidance of bias caused by heterogeneity w/o control group, may see low PD proportion by chance Monitoring 16

17 Randomized Discontinuation Design Kopec et al. (1993) J Clin Epidemiol Two-phases Open treatment phase - Find pt s best dose, remove pts not adhering, etc. Randomization of responders in our version Those who tolerate drug or Those who appear to get better - Double-blind RCT 17

18 When Useful? Study long-term, non-curative therapies Small treatment effect likely Want to minimize placebo use Want to minimize effect of heterogeneity Not estimating size of trt effect in larger pt pop n - Explanatory vs. pragmatic: Activity, not clinical effectiveness - Can significantly improve prob detect active trt Fedorov and Liu,

19 Example: A CALGB Study Renal-Cell Carcinoma: Variable growth rates Expect some will exhibit SD over short follow-up (e.g., 6-12 mos) without effective trt Heterogeneity affects study - Randomized trial should account for heterogeneity Heterogeneity affects estimated trt effect Stadler, Rosner, Small et al

20 CALGB RDT for Metastatic RCC Anti-Angiogenic Agent CAI N (OR) CAI R CAI 16 wks Stable disease? Y A N D O M CAI 16 wks Placebo* N (PD) Out I Z E 16 wks * PD switch to CAI Stadler, Rosner, Small et al

21 CALGB Design Primary analysis: Compare CAI to placebo after 16 weeks (RCT) Same % PD? Information: % SD after 1st 16 weeks % SD/PD for placebo % SD/PD for CAI RCT 21

22 Operating Characteristics Simulate to answer what if questions Model tumor growth rates (or PFS) Degrees of between-patient heterogeneity Different treatment effects - %SD & %PD at key decision points Different lengths of study phases - Sample size enrolled? - Prob. correctly or incorrectly reject? 22

23 Simulations (2000): Oper Chars Exponential tumor growth Random growth rate lognormal Effect of CAI on growth Person-specific growth rate RelVol i (t) = exp[grorate i (1 CAI eff ) t] CAI eff =1 no growth CAI eff =0 no effect 23

24 Prior for Growth Parameters Assumed Initial Risk of PD Initial risk of PD = 70% (historical information) - Beta(7,3) Pr(PD 90%) = 0.05 Pr(PD 50%) =

25 Operating Chars:No CAI Effect 50/grp 100/grp Prob. Reject 5.0% 4.4% Avg. Number in 1 st Phase (no stopping rule) Avg. Number in 1 st Phase (with stopping rule) % Stopping Before 0% 0% Randomized Phase % Stopping 1/2 through Rand. Phase 1.2% 3.7% 25

26 Op Chars: Mostly (86%) Cytostatic 50/grp 100/grp Prob. Reject 88% 99% Avg. Number in 1 st Phase (no stopping rule) Avg. Number in 1 st Phase (with stopping rule) % Stopping Before 7% 38% Randomized Phase % Stopping 1/2 through Rand. Phase 58% 86% 26

27 Sensitivity to Prior Assumptions If tumors grow faster (~90% 16 wks)? Stop in 1st phase ~ 25% More patients in 1st phase Fewer rejections in randomized phase (< 5%) If tumors grow slower (~50% 16 wks)? Do not stop in 1st phase Correct type I error Changing length of phases incorrect growth rate 27

28 Optimal RDD Apply decision theory to help choose design Length of open-label phase Length of follow-up after randomization Number of patients to enroll - Function of number enrolled, durations of two phases, difference to detect, etc. Trippa, Rosner, & Müller (2012) Biometrics 28

29 Tumor Growth Model Gompertzian tumor growth Ferrante et al. (2000). Biometrics 56, dx t /dt = a X t b X t log(x t ),X 0 = x 0 Stochastic differential equation Allow for between-patient heterogeneity Allow for deviation from expected trajectory dx it = {(a i X it b i X it log (X it )}dt + σ i X it dw it Gompertz diffusion process, GP(a, b, σ...) X i0 = x i0, t [0,T] Wiener process 29

30 Tumor Growth under Control Trt Historical data on M pts under control trt { X 0 1t 0, X 0 1t 1,..., X 0 1t k }, { X 0 2t 0,..., X 0 2t k },...,{ X 0 Mt 0,..., X 0 Mt k } Prior for growth curves is mixture of processes Each component of the mixture corresponds to one of the M historical patients - With a Gompertz process, GP(a, b, σ...) {Xit} 0 t 0 1 M GP(ã j, M b j, sig j Xit 0 0 = X jt 0 0,...,Xit 0 k = X jt 0 k ) j=1 30

31 Model Growth Change at Rand Suppose pt randomized to placebo at T a i Subj-specific & now piecewise constant b i a i (t) = a 0 i I(t <T)+a 1 i I(t T ) b i (t) = b 0 i I(t <T)+b 1 i I(t T ) Can now project future tumor growth p X t+s X t,a,b,σ 1 log(xt+s ) e bs log(x t ) (1 e bs )(a σ 2 /2)/b 2 exp X t+s σ 2 (1 e 2bs )/b 31

32 Example Simulation Underlying growth trajectories based on priors w/o historical data X t X t t Control t Treatment 32

33 Using Historical Information for Prior Random sample of historical measurements at discrete times Interpolation via Gompertz tumor growth with random params based on historical data Makes robust against possible deviations from model assumptions Elicit trt effect ψ i & show resulting predictions Different params in model (piecewise fn) if trt changes at time T 33

34 Prior Elicitation & Specification !!!!!!!! Upper curve: Obs d historical tumor growth data w/ control Lower curve: Imputed measurements & trajectory w/new trt! t 34

35 Prior Elicitation & Specification Xt !!!!!!!! ! Upper curve: Obs d historical tumor growth data w/ control. Lower curve: Imputed meas ts & trajectory w/new trt. t t ii (i) (ii) (iii) Median trajectories, 80% & 50% confidence band under two alternative priors for trt effect ψ i t 35

36 Using Historical Information for Prior Simulate study to show how assumed trt effect changes operating characteristics 3 pivotal probs determine op char (for fixed N) and expert judgement on them is available p e = prob pt is eligible for randomization p 0 & p 1 = prob response for trt 0 & trt 1, resp. We only need elicitation for trt effect ψ i ψ i is a function of 3 model parameters 36

37 Summary of Prior Modify parametric stochastic process Include subj-specific params (stoch diff eq n) Predict tumor growth trajectories via obs ns at discrete times in historical cohort Model future trajectories with control therapy by resampling imputed historical trajectories Model future trajectories with trt based on a single parameter that alters the trajectories 37

38 Formulate as Decision Problem Have dist n of data (tumor growth) Have dist n of model parameters Action space: alternative designs Total no. of pts enrolled: N Duration of open-label phase: T 1 Duration of follow-up post-randomization: T 2 38

39 Utility Function Balance Costs of the study: N no. pts in 1st phase n no. pts in 2nd phase c 1 cost of enrolling pts c 2 cost of treating in 2nd phase c 3 cost per unit time of retaining pt - C(d, φ, X) =c 1 N + c 2 n + c 3 (T 1 N + T 2 n) Benefits from subsequent phase III trial: - B(d, θ, Y )=I {S d (Y ) R d } E {log(1 + ψ N+1 ) θ} u(d, θ, Y )=B(d, θ, Y ) C(d, θ, Y ) 39

40 Implementation Simulate clinical trial for various values of 3 parameters: (N,T 1,T 2 ) n T 1 is a function of and dist n of tumor growth across pop n - Pt continues if SD (tumor within bounds) Monte Carlo optimization Approximate expected utility by smoothing simulation results across various (N,T 1,T 2 ) 40

41 U U U U U Monte Carlo Approximation Sims yield (d h,u h ),h=1,...,h for different (N,T N=246 1,T 2 ) c 1 =c 2 =0.001, c 3 = , TN=208 1 =66, T 2 =111 c 1 =c 2 =0.002, c 3 = , N=260 T 1 =66, T 2 =108 c 1 =0 u(d, θ, Y ) N= T 2 T c 1 =c 2 =0.001, c 3 = , T 1 =66, T 2 =111 N= T 1 T U T c 1 =c 2 =0.002, c 3 = , N= N T 1 =66, T 2 =108 T U T c 1 =0.004, c 2 = c 3 =0, T 1 =69, T 2 = N T T u(d, θ, Y ) T1 U T N N T1 0.4 U T T1 U N N c = c =0.001,c = c = c =0.002,c = c =0.004,c = c =0 41

42 Example Solutions , c 3 20 c= , 1 = c 2 =0.001,c 3 =3.3 c =c 2 =0.002, c 3 = , T 1 =66, T 2 =111 N=260 N=208 T N 1 =66, T 2 =108 u(d, θ, Y ) U U T U N d = (216, 60, 102) N T1 T2 U 0.4 U T N c 1 = c 2 =0.002,c 3 = T1 T1 U c 1 =0.004, c 1 c 2 = =0.004,c c 3 =0, 2 = c 3 =0 T T 1 =69, T 2 =115 d = (204, N 62, 108) U N c 1 = c 2 =0.001,c 3 = c 1 = c 2 =0.002,c 3 = c 1 =0.004,c 2 = c 3 =0 d = (152, 56, 105) Decision maker favors RDD over 2-arm RCT ( ) and over no study ( ) d =(216, 60, 102) d =(152, 56, 105) d =(204, 62, 108) T 1 =0 N =0 igure 2: Optimal RDDs. Orthogonal sections of expected utility surfaces corresponding to alternative utility function T1 42

43 Provide Table of Operating Chars Historical information from CALGB Costs are c 1 = c 2 =0.001,c 3 = Action space: D = d =(N,T 1,T 2 ) (0, 1,...,300) 3 Expected growth at 4 months: 12% (trt) vs 24% (control) - Optimal design: d =(N = 221,T1 = 72,T2 = 145) 43

44 Operating Characteristics at optimal d =(N = 221,T 1 = 72,T 2 = 145) Prob reject no effect under diff probs (prop n) susceptible, π 44

45 Summary Modeling and simulation are important tools for study design Possible outcomes & op chars from simulated trials allow investigators to understand aspects of complex designs M&S aid application of decision-theoretic considerations Utility function to capture competing needs Simulation-based optimization useful 45