Auxiliary-variable-enriched Biomarker Stratified Design

Transcription

1 Auxiliary-variable-enriched Biomarker Stratified Design Ting Wang University of North Carolina at Chapel Hill 8th May, 2017 A joint work with Xiaofei Wang, Haibo Zhou, Jianwen Cai and Stephen L George 1 / 42

2 Outline 1 Introduction to Biomarker Stratified Design (BSD) 2 Auxiliary-variable-enriched Biomarker Stratified Design (ABSD) 3 Numerical results 4 Example 5 Conclusion 2 / 42

3 Outline 1 Introduction to Biomarker Stratified Design (BSD) 2 Auxiliary-variable-enriched Biomarker Stratified Design (ABSD) 3 Numerical results 4 Example 5 Conclusion 3 / 42

4 Precision medicine and biomarker Precision Medicine: tailor medical treatment to the individual characteristics of each patient Biomarker: clinical-pathological, molecular or genetic profiles 4 / 42

5 Examples of targeted therapies Cetuximab: treatment of colorectal cancer in patients with wild-type K-ras Trastuzumab: treatment of HER-2-positive breast cancer Gefitinib: treatment of NSCLC in patients with EGFR mutation 5 / 42

6 Biomarker-driven clinical trials Targeted design (biomarker-positive only) Biomarker Stratified Design or All-comer Design Hybrid Designs See Korn and Freidlin (2016) for a comprehensive review of these designs. 6 / 42

7 Targeted design Treatment A Assess Biomarker Biomarker Biomarker Negative Positive Randomize Treatment B Off study 7 / 42

8 Biomarker Stratified Design (BSD) Treatment A Assess Biomarker Biomarker Biomarker Negative Positive Randomize Treatment B Treatment A Randomize Treatment B 8 / 42

9 Biomarker Stratified Design (BSD) Advantages: unbiased estimates of treatment effects across different biomarker-defined subgroups and for the entire randomly assigned population Disadvantages: low prevalence of biomarker-positive patients ascertaining the true biomarker status is costly 9 / 42

10 Outline 1 Introduction to Biomarker Stratified Design (BSD) 2 Auxiliary-variable-enriched Biomarker Stratified Design (ABSD) 3 Numerical results 4 Example 5 Conclusion 10 / 42

11 Our proposal Motivation: enrichment through auxiliary variables Auxiliary variables: inexpensive to obtain relative to the true biomarker (e.g. smoking status, histology, gender, race, etc.) positively correlated to the true biomarker 11 / 42

12 Auxiliary-variable-enriched Biomarker Stratified Design (ABSD) Treatment A Assess Auxiliary Variable Auxiliary Negative Positive Auxiliary Randomize Randomize to the trail π 0 1-π 0 Treatment B Randomize to treatments Off study Treatment A Treatment B 12 / 42

13 Auxiliary-variable-enriched Biomarker Stratified Design (ABSD) Advantages: all patients are screened by auxiliary variables only randomized patients will be tested for true biomarker status after selection. potentially increase efficiency of testing interaction effect and treatment effect on biomarker-positive patients 13 / 42

14 Concepts and notation Proportional hazard model: λ(t) = λ 0 (t)exp(β 1 M + β 2 D + β 3 MD) (1) λ 0 (t) baseline hazard function M biomarker indicator (0 marker-negative; 1 marker-positive) D treatment indicator(0 control; 1 experimental treatment) MD interaction term 14 / 42

15 Testing different treatment parameters Table 1: Summary of treatment parameters of interest Scenario H 0 v.s. H a Interpretation 1 θ 1 = 0 θ 1 0 a Treatment effect on M+ 2 θ 2 = 0 θ 2 0 b Interaction effect 3 θ 1 = 0 θ 1 0 Treatment on M+ AND θ 2 = 0 θ 2 0 Interaction 4 θ 1 = 0 θ 1 0 Teatment on M+ AND θ o = 0 θ o 0 c Overall treatment effect a θ 1 = β 2 + β 3 in equation (1) b θ 2 = β 3 in equation (1) c θ o = pθ 1 + (1 p)θ 3 where θ 3 = β 2 in equation (1) 15 / 42

16 Concepts and notation M M the true biomarker expensive M+ low prevalence an auxiliary variable for M predictive of M (positively correlated to M) easily and cheaply assessed Enrich the study by increasing the proportion of M+ and decreasing the proportion of M through M Measure M after selection 16 / 42

17 Concepts and notation S = 1 if selected; S = 0 if otherwise π 1 = Pr(S = 1 M = 1); π 0 = Pr(S = 1 M = 0) r = Pr(M = 1); r = Pr( M = 1) PPV = Pr(M = 1 M = 1) relationship with ρ = Corr(M, M): PPV = r + ρ r(1 r) r(1 r) r (2) 17 / 42

18 Concepts and notation Let r = Pr(M = 1 S = 1), then M+ low prevalence: π 1 = 1, then r = rπ 1PPV + π 0 (r rppv ) rπ 1 + (1 r)π 0 (3) r = r + r(ppv r)(1 π 0) r(1 π 0 ) + π 0 (4) 18 / 42

19 Goal Our goal is to minimize the number of randomized patients n and thus reduce the total trial cost under different hypothesis testing scenarios for biomarker-stratified trials. To find the minimum n satisfying the requisite power, we will find the optimal probability π / 42

20 Scenario 1: Testing the treatment effect on M+ patients In equation (1), the treatment effect on M+ patients is θ 1 = β 2 + β 3. H 0 : θ 1 = 0 v.s. H a : θ 1 = θ 1 The power can be written as ˆθ 1 Pw = 1 Pr( se(ˆθ 1 ) z α/2 θ 1 = θ1) (5) The variance of ˆθ 1 is given as (George and Desu, 1974; Schoenfeld, 1983), var(ˆθ 1 ) = = 2 m 1C m 1E nr ( ) (6) p 1C p 1E 20 / 42

21 Scenario 1: Testing the treatment effect on M+ patients From equation (4), r can be written as r = PPV PPV r r 1 π 0 + (1 r) (7) By the expected accrual time T = ns a, p 1j can be written as (George and Desu, 1974), p 1j = 1 a n s λ ij e τλ ij (1 e nsλ ij /a ), j = C, E (8) Further, the expected number of screened patients necessary to achieve a fixed number (n) of randomized patients is n s = n r + (1 r)π 0 (9) π 0 = 0 is the optimal solution for minimizing var(ˆθ 1 ). 21 / 42

22 Scenario 2: Testing the interaction effect In expression (1), the interaction effect between treatment and biomarker is θ 2 = β 3. H 0 : θ 2 = 0 v.s. H a : θ 2 = θ 2 The power can be written as ˆθ 2 Pw = 1 Pr( se(ˆθ 2 ) z α/2 θ 2 = θ2) (10) The variance of ˆθ 2 is (Peterson and George, 1993) var(ˆθ 2 ) = 1 m 1C + 1 m 1E + 1 m 0C + 1 m 0E = 2 nr ( ) + p 1C p 1E n(1 r ) ( ) p 0C p 0E (11) 22 / 42

23 Scenario 2: Testing the interaction effect Equation (4), Equation (9), And, p ij = 1 r = rppv + π 0(r rppv ) r + (1 r)π 0 n s = n r + (1 r)π 0 a n s λ ij e τλ ij (1 e nsλ ij /a ), i = 1, 0, j = C, E (12) Let n H2 (π 0 ) represent the minimum number of randomized patients for required power of the interaction effect. Search all π 0 s to find the minimum of n H2 (π 0 ). 23 / 42

24 Scenario 3: Testing the treatment effect on M+ patients and the interaction effect Let n H1 (π 0 ) and n H2 (π 0 ) represent the minimum number of randomized patients for Scenario 1 and 2 respectively when Pr(S = 1 M = 0) = π 0. Search all π 0 s to find the minimum of max(n H1 (π 0 ), n H2 (π 0 )), This is just the minimum number of patients for testing both hypotheses. 24 / 42

25 Scenario 4: Testing the treatment effect on M+ patients and the overall treatment effect Define θ o = rθ 1 + (1 r)θ 3 where θ 1 = β 2 + β 3 is the treatment effect on M+, θ 3 = β 2 is the treatment effect on M, r = Pr(M+) H 0 : θ o = 0 v.s. H a : θ o = θo The power can be written as ˆθ o Pw = 1 Pr( se(ˆθ o ) z α/2 θ o = θo) (13) 25 / 42

26 Scenario 4: Testing the treatment effect on M+ patients and the overall treatment effect The variance of ˆθ o is var(ˆθ o ) = r 2 var(ˆθ 1 ) + (1 r) 2 var(ˆθ 3 ) + 2r(1 r)cov(ˆθ 1, ˆθ 3 ) = r 2 var(ˆθ 1 ) + (1 r) 2 var(ˆθ 3 ) = r 2 ( ) + (1 r) 2 ( ) m 1C m 1E m 0C m 0E = 2r 2 nr ( (1 r)2 ) + p 1C p 1E n(1 r ) ( ) p 0C p 0E (14) 26 / 42

27 Scenario 4: Testing the treatment effect on M+ patients and the overall treatment effect Let n H1 (π 0 ) and n H3 (π 0 ) represent the minimum number of randomized patients for Scenario 1 and overall treatment effect respectively when Pr(S = 1 M = 0) = π 0. Search all π 0 s to find the minimum of max(n H1 (π 0 ), n H3 (π 0 )), This is just the minimum number of patients for testing both hypotheses. 27 / 42

28 Cost comparison of design C D C f C M C s T f The cost of ABSD: The cost of BSD: C ABSD = (C D + C M )n ABSD + C f T f,absd + C s n s C BSD = (C D + C M )n BSD + C f T f,bsd the treatment cost of each patient the cost of follow up in unit time the cost of testing biomarker for each patient the screening cost for each patient Expected total follow up time To directly compare the two designs, we consider three ratios ξ patient = n ABSD n BSD, ξ cost = C ABSD C BSD, ξ screening = n s n BSD 28 / 42

29 Outline 1 Introduction to Biomarker Stratified Design (BSD) 2 Auxiliary-variable-enriched Biomarker Stratified Design (ABSD) 3 Numerical results 4 Example 5 Conclusion 29 / 42

30 Numerical setting treatment effect on M+ patients: P w1 = 90%, α 1 = interaction effect: P w2 = 90%, α 2 = overall treatment effect: P w3 = 90%, α 3 = τ = 1; r = r = 0.05, 0.1, 0.15; C D = 5000, C f = 200, C M = 1000, C s = 100; λ 0c = 0.8, λ 1c = 1.6, λ 1e = 1; λ 0e = 0.7 quantitative interacton; λ 0e = 1.1 qualitative interaction. 30 / 42

31 Minimization of n for required powers of treatment effect on M+ patients and interaction effect (a) (b) patientratio r, r ~ = 0.05 r, r ~ = 0.1 r, r ~ = 0.15 costratio r, r ~ = 0.05 r, r ~ = 0.1 r, r ~ = PPV PPV (c) (d) patientratio r, r ~ = 0.05 r, r ~ = 0.1 r, r ~ = 0.15 costratio r, r ~ = 0.05 r, r ~ = 0.1 r, r ~ = PPV PPV Figure 1: Under Scenario 3, for different PPV and different pairs of r = r, (a) Patient ratio when λ 0e = 0.7; (b) Cost ratio when λ 0e = 0.7; (c) Patient ratio when λ 0e = 1.1; (d) Cost ratio when λ 0e = / 42

32 Minimization of n for required powers of treatment effect on M+ patients and overall treatment effect (a) (b) patientratio r, r ~ = 0.05 r, r ~ = 0.1 r, r ~ = 0.15 costratio r, r ~ = 0.05 r, r ~ = 0.1 r, r ~ = PPV PPV (c) (d) patientratio r, r ~ = 0.05 r, r ~ = 0.1 r, r ~ = 0.15 costratio r, r ~ = 0.05 r, r ~ = 0.1 r, r ~ = PPV PPV Figure 2: Under Scenario 4, for different PPV and different pairs of r = r, (a) Patient ratio when λ 0e = 0.7; (b) Cost ratio when λ 0e = 0.7; (c) Patient ratio when λ 0e = 1.1; (d) Cost ratio when λ 0e = / 42

33 Outline 1 Introduction to Biomarker Stratified Design (BSD) 2 Auxiliary-variable-enriched Biomarker Stratified Design (ABSD) 3 Numerical results 4 Example 5 Conclusion 33 / 42

34 Gefitinib or Carboplatin-Paclitaxel in pulmonary adenocarcinoma Consider the study of Gefitinib or Carboplatin-Paclitaxel in Pulmonary Adenocarcinoma in North America. M: EGFR mutation Among North American, r 10% M: Predictive score considering adenocarcinomas, no history of smoking, females and Asian descent r 15%, PPV 60% a = 10 patients/month, τ = 2 years 34 / 42

35 Gefitinib or Carboplatin-Paclitaxel in Pulmonary Adenocarcinoma H 0 :θ 1 = 0 vs.h a :θ 1 0 H 0 :θ 1 = 0 vs.h a :θ 1 0 or H 0 :θ 2 = 0 vs.h a :θ 2 0 H 0 :θ o = 0 vs.h a :θ o 0 Based on the information provided in Mok et al. (2009), ABSD: n = 157, n s = 1047 BSD: n = 926 Comparison: ξ patient = 0.168, ξ cost = 0.421, ξ screening = / 42

36 Outline 1 Introduction to Biomarker Stratified Design (BSD) 2 Auxiliary-variable-enriched Biomarker Stratified Design (ABSD) 3 Numerical results 4 Example 5 Conclusion 36 / 42

37 Conclusions The ABSD design always improves trial efficiency while retaining the features of BSD design. Compared to a conventional all-comer BSD, the ABSD design always reduces the number of randomized patients and the total cost of the trial, especially for testing the treatment effect on M+ patients or the interaction effect, when the prevalence rate r is small. 37 / 42

38 References I B. Freidlin and E. L. Korn. Biomarker enrichment strategies: matching trial design to biomarker credentials. Nature Reviews Clinical Oncology, 11(2):81 90, B. Freidlin, L. M. McShane, and E. L. Korn. Randomized clinical trials with biomarkers: design issues. Journal of the National Cancer Institute, S. L. George and M. Desu. Planning the size and duration of a clinical trial studying the time to some critical event. Journal of chronic diseases, 27(1-2):15 24, S. L. George and X. Wang. Targeted clinical trials. In D. Harrington, editor, Designs for Clinical Trials: Perspectives on Current Issues, pages Springer, E. L. Korn and B. Freidlin. Biomarker-based clinical trials. In S. L. George, X. Wang, and H. Pang, editors, Cancer Clinical Trials: Current and Controversial Issues in Design and Analysis, pages Chapman and Hall/CRC, T. S. Mok, Y.-L. Wu, S. Thongprasert, C.-H. Yang, D.-T. Chu, N. Saijo, P. Sunpaweravong, B. Han, B. Margono, Y. Ichinose, et al. Gefitinib or carboplatin paclitaxel in pulmonary adenocarcinoma. New England Journal of Medicine, 361(10): , B. Peterson and S. L. George. Sample size requirements and length of study for testing interaction in a 2 k factorial design when time-to-failure is the outcome. Controlled clinical trials, 14(6): , C. N. Prabhakar. Epidermal growth factor receptor in non-small cell lung cancer. Translational lung cancer research, 4(2): , Apr D. A. Schoenfeld. Sample-size formula for the proportional-hazards regression model. Biometrics, 39: , R. Simon and A. Maitournam. Evaluating the efficiency of targeted designs for randomized clinical trials. Clinical Cancer Research, 10(20): , T. Wang, X. Wang, H. Zhou, J. Cai, and S. L. George. Auxiliary-variable-enriched biomarker stratified design. Manscuript in preparation, / 42

39 The End 39 / 42

40 Expected total follow-up time For ABSD: T f = i=0,1 j=c,e 1 ) + ] + a r i (1 r ) 1 i λ ij 2 For BSD: T f = i=0,1 j=c,e T total 1 ) + ] + a r i (1 r ) 1 i λ ij 2 { r i (1 r ) 1 i min{(t total 1 ) +, n s 2 λ ij a }a 1 +[ n s λ ij a (T total [ 1 2 [n s a (T total 1 λ ij ) + ] + + (T total n s a )]} { r i (1 r ) 1 i min{(t total 1 ) +, n BSD 2 λ ij a }a 1 +[ n BSD ( λ ij a [ 1 2 [n BSD a (T total 1 λ ij ) + ] + +(T total n BSD a )]} 40 / 42

41 The impact of mis-specified PPV on power Figure 3: A: Powers of treatment effect on true M+ patients for different true PPV s given different specified PPV s when r = r = 0.15; B: Powers of interaction effect for different true PPV s given different specified PPV s when r = r = / 42

42 Adaptive method involving Bayesian algorithm Update PPV after every group of k patients has been observed with M+ d the number of groups Initial PPV Beta(α 0, β 0 ) m d = N(M = 1, M = 1 d) B(dk, PPV ) PPV d Beta(α 0 + m d, β 0 + dk m d ) PPV d = α 0 + m d α 0 + m d + β 0 + dk m d = α 0 + m d α 0 + β 0 + dk (15) 42 / 42