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University of Maryland Baltimore

Experimental Therapeutics Symposium 2009

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- 1. Eﬃcient Design and Analysis of Combination Experiments to Improve Early Stage Clinical Development Hong-Bin Fang, Ph.D. Division of Biostatistics University of Maryland Greenebaum Cancer Center and Department of Epidemiology and Preventive Medicine Baltimore, MD 21201, USA Email: hfang@som.umaryland.edu •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 2. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 3. Outline • Introduction • Existing Methods for Combination Study Design • Maximal Power Experimental Design • Existing Analysis Methods for Synergy • Statistical Analysis of Interaction Index • Conclusion and Further Research •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 4. Introduction: Combination Therapy • Reduce the frequency of acquired resistance • Achieve greater eﬃcacy with lower doses and reduced toxicity • Achieve enhanced potency (or sensitization) exploring synergistic ac- tivities • Provide a ﬁrmer basis for potential clinical trials Joint action is divided into three types: 1. Independent joint action 2. Simple similar (additive) action 3. Synergistic/antagonistic action Refs.: Berenbaum MC. (1989). Pharmacological Reviews 41: 93-141. Greco WR, et al. (1995). Pharmacological Reviews 47, 331-385. Fitzgerald et al.(2006). Nature Chemical Biology 2, 458 - 466 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 5. Interaction Index To assess the interaction of two drugs at the combination of (xA, xB ), Berenbaum (1977) deﬁned an interaction index (τ ), xA xB τ= + XA XB XA and XB are the doses of drugs A and B that when administered alone yield the same eﬀect as does the combination (xA, xB ). • τ = 1, A and B are additive at (xA, xB ); Loewe Independence (Additivity) • τ < 1, A and B are synergistic at (xA, xB ); • τ > 1, A and B are antagonistic at (xA, xB ). Refs.: Berenbaum MC. (1977). Clin. Exp. Immunol. 28: 1-18. Berenbaum MC. (1989). Pharmacological Reviews 41: 93-141. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 6. Bliss Independence (Additivity) • The joint eﬀect of inhibitors A and B is the product of the eﬀect of each EAB = EA ∗ EB • Assumption: Inhibitors can bind simultaneously and mutually nonex- clusively through distinct mechanism •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 7. Fitzgerald et al. Nature 2006 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 8. Statistical Approaches for Design • Statistical (instead of mechanistic) evaluation is very valuable in clinical trials because it is impractical for a measure of success (such as synergism between two drugs) to change with every biochemical advance. • Variation: the administration of precisely the same dose to aliquots (or virtually genetically identical animals) may result in diﬀerent levels of dose eﬀect. • Sample Size: how many samples are needed, namely, how to identify doses in the combinations and how many replicates at each combina- tion and how to analyze the data produced •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 9. Current Methods of Design • Equal regression lines (Finney, 1971) • An optimal design by ﬁxing the total dose for speciﬁc models (Abdelbasit and Plackett, 1982) • Fixed ratio/ray design (Tallarida et al., 1992) • Checkerboard (Lattice) design (Martinez-Irujo et al., 1996) • A D-optimal design for in vitro combination studies in linear models (Greco et al., 1995) but n too large Refs.: Finney DJ (1971). Probit Analysis. Cambridge University Press. Abdelbasit KM, Plackett RL (1982). Biometrics 38, 171-179. Greco WR, et al. (1995). Pharmcological Reviews 47, 331-385. Martinez-Irujo, et al.(1997). Biochemical Pharmacology 51, 635-644. Tallarida RJ, et al.(1997). Life Science 61, 417-425. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 10. Fixed Ratio/Ray Design • Assume that two drugs are synergistic, additive or antagonistic for all doses of a ﬁxed ratio; • Suboptimal dose allocations result in false synergistic combinations or miss an apparent interaction at a particular combination; • Statistical power to detect additivity is undermined substantially. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 11. Maximal Power Design Statistical Model Single dose eﬀect: y = fA(XA), y = fB (XB ) −1 Potency of B relative to A: ρ(XB ) = fA fB (XB )/XB The joint model at the combination x is assumed y = fA(xA + ρ(XB )xB ) + g(xA, xB ) + ε • g(xA, xB ) = 0, A and B are additive at x; • g(xA, xB ) > 0, A and B are synergistic at x; • g(xA, xB ) < 0, A and B are antagonistic at x. Experimental design is based on testing the additive action of drugs A and B H0 : g = 0 versus H1 : g = 0. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 12. Statistical Inference After transformation, the additive model y = fA(xA + ρ(XB )xB ) = (or ≈)g(z1) + g(z2) (i) (i) Let z(i) = (z1 , z2 )T , i = 1, 2, . . . , m, y = (y11, · · · , y1k , · · · , ymk )T , (i) (i) Z: m × 2 matrix, its ith row: (g1(z1 ), g2(z2 )) Then, under H0, yT (J − V )y/(m − 2) F = T ∼ Fm−2,m(k−1) y (I − J)y/(mk − m) where U = I 1 k , V = U Z(Z T U T U Z)−1Z T U T , J = U (U T U )−1U T •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 13. Statistical Inference If z(1), . . . , z(m) are uniformly scattered in S, under H1, the F statistic has a noncentral F -distribution with degrees of freedom m − 2 and m(k − 1) and the noncentrality parameter, mk δ= 2 g 2(z)dz, σ S is maximized. Thus, the power for detecting de- partures from additivity of two drugs is maximized when m mixtures z(1), . . . , z(m) are uniformly scat- tered in S. Refs.: Wiens (1991). Statistics & Probability Letters, 12: 217-221. Tan et al.(2003). Statistics in Medicine, 22: 2091-221. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 14. Sample Size Determination Given the type I error rate: α, power: 1 − β the smallest meaningful diﬀerence: η the measurement variation: σ 2 the sample sizes can be obtained from the noncen- tral F -distribution function, ∞ e−δ/2(δ/2)k (m − 2)x P (F ≤ x) = P Fm−2+2k,n−m ≤ , k! m − 2 + 2k k=0 δ = nη 2/σ 2. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 15. Sample Size Calculation α = 0.05, 1 − β = 0.8, d = η 2/σ 2 # of Replications n0 1 2 3 4 5 6 d = 0.1 − − 139(556) 87(435) 61(366) 46(322) d = 0.2 − 78(234) 42(168) 27(135) 19(114) 14(98) d = 0.3 107(214) 40(120) 21(84) 14(70) 10(60) 7(49) d = 0.4 68(136) 25(75) 14(56) 9(45) 3(18) 3 (21) d = 0.5 48(96) 18(54) 10(40) 6(30) 3(18) 3(21) d = 0.8 24(48) 9(27) 4(16) 3(15) 3(18) 3(21) d=1 18(36) 6(18) 3(12) 3(15) 3(18) 3(21) •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 16. Experimental Design I The sample sizes are based on the F-test to detect departures from additive action with 80% power at a signiﬁcance level of 0.05 • Choose the dose range of signiﬁcance: e.g., ED20- ED80 (based on pharmacology) • Choose the meaningful diﬀerence in the dose- eﬀect outcome to be detected: η • Choose the number of replicates at each combi- nation • The variance is estimated based on the pooled variations from the two single drug experiments. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 17. Experimental Design II • The additive model depends on the individual dose-response; • Diﬀerent individual dose-responses result in dif- ferent experimental designs; We have considered three classes of drugs classiﬁed by the shape of individual dose-response curves: • both log-linear; • both linear; • linear + log-linear. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 18. Log-linear + Log-linear Let the dose-response of drugs A and B be y(xA) = αA+βA log xA, y(xB ) = αB +βB log xB The potency ρ of B relative to A is βB /βA−1 ρ(xB ) = ρ0xB , ρ0 = exp[(αB − αA)/βA]. The additive model at combination (xA, xB ): y(xA, xB ) = αA + βA log(xA + ρxB ) (βB −βA)/βA ρ = ρ0 ρ−1x A + xB . •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 19. Log-linear + Log-linear Ray Design: Using lattice points undermines the power to detect the additivity; Maximum Power Design: Using uniformly scattered points achieves maximum power to detect departures from additivity. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 20. Results of Simulation Studies Number of combinations 9 16 Maximum Discrepancy (CL2) 0.003583 0.001190 Power Type I error 0.0501 0.0493 Design Power 0.7949 0.8345 Discrepancy (CL2) 0.022719 0.017986 Ray/Lattice Type I error 0.0532 0.0501 Design Power 0.6540 0.4062 Discrepancy (CL2) average: 0.035943 0.020812 SD: 0.0173668 0.0147279 Monte Type I error average: 0.0503 0.0502 Carlo SD: 0.00311 0.00227 method Power average: 0.5330 0.7439 SD: 0.31456 0.26184 y = 20 log(z1 ) + 70 log(z2 ) + g(z1 , z2 ) + ε, g(z1 , z2 ) = 50(z1 − 2) sin(z1 ) cos(z2 ), ε ∼ N (0, σ 2 ) and σ 2 = 250. η 2 = 100, with 10,000 replications •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 21. Example: SAHA + Ara-C against K562 y = 43.69 − 10.93 log(S), y is the viability y = 35.98 − 8.25 log(C) Shiozawa et al. (2009). CCR. 15:1698-1707 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 22. Example: SAHA + Ara-C against K562 Mixtures SAHA and Ara-C for combination experiment Exper. SAHA Ara-C Exper. SAHA Ara-C # (µM) (µM) # (µM) (µM) 1 0.706 0.101 7 1.427 3.193 2 3.035 0.099 8 3.402 2.681 3 2.160 2.030 9 1.008 0.924 4 0.635 5.122 10 3.139 0.781 5 0.393 0.727 11 4.524 0.724 6 0.118 2.684 dose range: ED20-ED80; η = 15%(viability); 5 replicates; σ 2 = 804.564 ˆ •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 23. Analysis of Synergy Estimation of the Interaction Index Surface (i) (i) yij : the jth response at (xA , xB ), (i) (i) With the single dose-response curves, the interaction indexes at (xA , xB ) are (i) (i) xA xB τij = + exp{(yij − αA)/βA} exp{(yij − αB )/βB } j = 1, . . . , k, i = 1, . . . , m. The method of two-dimensional B-splines (thin plate splines) is employed to estimate the interaction index surface τ = h(xA, xB ), Ref.: Fang, et al. Stat. Medicine 2008 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 24. SAHA: Estimation of relative potency 0.842 The potency of Vorinostat relative to Etoposide is ρ(XB ) = 0.368XB , which is non-constant and depends on dose. The predicted additive model is y(xA, xB ) = 41.52 − 13.02 log(xA + ψ(xA, xB )xB ), where ψ(xA, xB ) is determined by ψ(xA, xB ) = 0.368(ψ −1(xA, xB )xA + xB )0.842. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 25. Example: SAHA + Ara-C against K562 Dose-response surface Observations: 66(11 combinations with 5 replicates at each combination) Maximum viability: 82.81%; Minimum viability: 17.72%; Mean: 30.67%; Standard deviation: 13.40. Output: F9,55 = 8.14, p-value< 0.0001 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 26. Example: SAHA + Ara-C against K562 Interaction Index Surface •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 27. Linear + Log-linear Let z1 = xA, z2 = βAξ(xA, xB )/[βB ψ(xA, xB )xB ], then, the additive model becomes y(xA, xB ) = αA + βAz1 + βB z2. The m experimental points should be uniformly scattered in the tetragon, z1 : a < αA + βAz1 + βB z2 < b, z1 > 0, z2 > 0 z2 for given a and b. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 28. Example: LY-168 with Sorafnib against WM164 y = 111.85 − 9.56xA, y is the viability y = 101.91 − 31.17xB •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 29. dose range: ED20-ED80; η = 15%(viability); 6 replicates; σ 2 = 1352.724 ˆ •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 30. Example: LY-168 + Sorafnib against WM164 Dose-response surface Observations: 133(19 combinations with 6 replicates at each combination) Maximum viability: 93.58%; Minimum viability: 3.53%; Mean: 30.45%; Standard deviation: 25.347. Output: F17,114 = 162.9696, p-value< 0.0001 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 31. Example: LY-168 + Sorafnib against WM164 Interaction Index Surface •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 32. Follow-up in Early Clinical Stage Trials Three cases classiﬁed by the shape of individual dose-response curves considered: • both log-linear (SAHA + Ara-C; Clini- cal trial ongoing); • both linear (potentially resurrect a promising drug combination) ; • linear + log-linear (In vivo experiment ongoing). •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 33. Conclusion • Maximal power design produces data eﬃciently. It is optimal in that statistical power to detect departures from additivity is maximized. • The F test can be used to test departures from additivity and the thin plate splines to estimate the interaction index surface eﬀectively with data generated by the MP design. • SYNSTAT R package at www.umgcc.org/research/biostatistics.htm •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
- 34. Acknowledgments Ming Tan, PhD, UMGCC Guo-Liang Tian, PhD, University of Hong Kong Dr. Douglas D. Ross’s Lab, U Maryland School of Medicine Dr. Wei Li’s Lab, U Tennessee College of Pharmacy Dr. Pei Feng’s Lab, U Maryland Dental School Dr. Peter Houghton, St Jude Children’s Research Hospital •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

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