Efficient Design and Analysis of
 Combination Experiments to Improve
    Early Stage Clinical Development

            Hong-...
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Outline
• Introduction
• Existing Methods for Combination Study Design
• Maximal Power Experimental Design
• Existing Anal...
Introduction: Combination Therapy
  • Reduce the frequency of acquired resistance
  • Achieve greater efficacy with lower do...
Interaction Index
To assess the interaction of two drugs at the combination of (xA, xB ),
Berenbaum (1977) defined an inter...
Bliss Independence (Additivity)
• The joint effect of inhibitors A and B is the product of the effect of
  each
            ...
Fitzgerald et al. Nature 2006
                 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Statistical Approaches for Design
• Statistical (instead of mechanistic) evaluation is very valuable in
  clinical trials ...
Current Methods of Design
 • Equal regression lines (Finney, 1971)
 • An optimal design by fixing the total dose for
   spe...
Fixed Ratio/Ray Design
• Assume that two drugs are synergistic, additive or antagonistic for all
  doses of a fixed ratio;
...
Maximal Power Design
                 Statistical Model
Single dose effect: y = fA(XA), y = fB (XB )
                      ...
Statistical Inference
After transformation, the additive model
 y = fA(xA + ρ(XB )xB ) = (or ≈)g(z1) + g(z2)
             ...
Statistical Inference
If z(1), . . . , z(m) are uniformly scattered in S, under
H1, the F statistic has a noncentral F -di...
Sample Size Determination
Given the type I error rate: α, power: 1 − β
the smallest meaningful difference: η
the measuremen...
Sample Size Calculation
                        α = 0.05, 1 − β = 0.8,
d = η 2/σ 2                         # of Replicatio...
Experimental Design I
The sample sizes are based on the F-test to detect
departures from additive action with 80% power at...
Experimental Design II
• The additive model depends on the individual
  dose-response;
• Different individual dose-response...
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 p...
Log-linear + Log-linear
Ray Design: Using lattice points undermines the power to detect the
additivity;
Maximum Power Desi...
Results of Simulation Studies
         Number  of combinations                                         9                  ...
Example: SAHA + Ara-C against K562




  y = 43.69 − 10.93 log(S), y is the viability




           y = 35.98 − 8.25 log(...
Example: SAHA + Ara-C against K562
      Mixtures SAHA and Ara-C for combination experiment

            Exper. SAHA Ara-C...
Analysis of Synergy
Estimation of the Interaction Index Surface
                                    (i)   (i)
 yij : the j...
SAHA: Estimation of relative potency
                                                                  0.842
The potency o...
Example: SAHA + Ara-C against K562
                      Dose-response surface




Observations: 66(11 combinations with 5...
Example: SAHA + Ara-C against K562
      Interaction Index Surface




                  •First •Prev •Next •Last •Go Back...
Linear + Log-linear
Let
z1 = xA,    z2 = βAξ(xA, xB )/[βB ψ(xA, xB )xB ],
then, the additive model becomes
        y(xA, x...
Example: LY-168 with Sorafnib against WM164




       y = 111.85 − 9.56xA, y is the viability




               y = 101....
dose range: ED20-ED80; η = 15%(viability); 6 replicates; σ 2 = 1352.724
                                                  ...
Example: LY-168 + Sorafnib against WM164
                      Dose-response surface




      Observations: 133(19 combin...
Example: LY-168 + Sorafnib against WM164
         Interaction Index Surface




                     •First •Prev •Next •L...
Follow-up in Early Clinical Stage Trials
Three cases classified by the shape of individual
dose-response curves considered:...
Conclusion
• Maximal power design produces data efficiently.
  It is optimal in that statistical power to detect
  departure...
Acknowledgments
Ming Tan, PhD, UMGCC
Guo-Liang Tian, PhD, University of Hong Kong
Dr. Douglas D. Ross’s Lab, U Maryland Sc...
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  1. 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. 2. •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
  3. 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. 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. 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. 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. 7. Fitzgerald et al. Nature 2006 •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
  8. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 26. Example: SAHA + Ara-C against K562 Interaction Index Surface •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
  27. 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. 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. 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. 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. 31. Example: LY-168 + Sorafnib against WM164 Interaction Index Surface •First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
  32. 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. 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. 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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