Ming T Tan, PhD University of Maryland Greenebaum Cancer Center D.O.E. Presentation 7/11/2006 Optimized Experimental Design for Detecting Synergy in Drug Combination Studies

Combination Studies l Motivation:  achieve greater efficacy with lesser toxicity  provide a firmer basis for potential clinical trials l Joint action is divided into three types: 1. Independent joint action 2. Simple similar (additive) action 3. Synergistic/antagonistic action

Current Design Methods l Equally regression lines (Finney, 1971) l An optimal design by fixing the total dose for specific models (Abdelbasit & Plackett, 1982) l Fixed ratio design (Tallarida et al., 1992) l An optimal design for in vitro combination studies (Greco et al., 1994) but the sample size too large l Uniform design based on the log-linear single dose-response (Tan et al., 2003)

Model Formulation The individual dose-response curves of drugs A and B are assumed: and The potency of B relative to A is For the additive action of A and B, the regression line for the mixture is H 0 : f = 0 versus H 1 : f 0. The general nonparametric model for the combinations of A and B is assumed where the function is unspecified, Then, testing the additive action of A and B is equivalent to testing

Model Assumptions With an invertible transformation: we have the additive structure and the model becomes where g 1 and g 2 are linearly independent, and satisfies the orthogonal condition:

Model Formulation (cont’) Hence, testing the additive action of A and B is equivalent to testing H 0 : g = 0 versus H 1 : g  0 if g = 0, additive action in the mixture of A and B is implied; if g > 0, synergism in the mixture of A and B is implied; if g < 0, antagonism in the mixture of A and B is implied. The meaningful amount (  >0) of synergism or antagonism to be detected is specified by:

Statistical Inference Suppose that are m mixtures of A and B in domain S n i experiments at the dose-level, y ij are the corresponding responses, y : n  1 vector with elements y ij ordered lexicographically, I : the unit matrix, 1 k : the k  1 vector of one, Z : m  2 matrix with i -th row Under H 0,

Statistical Inference (cont’) Under H 1 : with where  is the design measure, which is a probability measure with mass p i = n i /n, i=1,2,…,m. If  is uniformly distributed on S, then and  is maximized, Uniform design measure  maximizes the minimum power of the F-test for the additive action of drugs A and B.

Uniform Scattered Points on C 2 F m (x) be the empirical distribution function of P m, F(x) be the uniform distribution function in C 2. Discrepancy: Let U m,2 =(u ij ) be an m  2 matrix, each column is a permutation of { 1,2, …,m }. Define v ij =(u ij - 0.5)/m, i=1, …,m; j=1,2, and V m,2 can be considered as m points on C 2. Then the uniform design is to choose the m points so that the discrepancy of V m,2 is the smallest over all of possible V m,2.

Example: Uniform Scattered Points on C 2 A uniform design with 7 experimental units on C 2 : Discrepancy: (smallest) The 7 experiment units: (0.071, 0.643), (0.214, 0.214), (0.357,0.929), (0.5, 0.5), (0.643, 0.071), (0.786, 0.786), (0.929, 0.357).

Uniform Experimental Design Suppose the experimental domain ( v i1, v i2 ) T, i=1, …,m; are m UD points on C 2. The m Uniform design points on S are i=1, …,m Then, m mixtures {(x Ai, x Bi ): i=1, …,m} of drugs A and B for experiment can be obtained by the inverse transform.

Sample Size Determination l The sample sizes (number of experimental units) to detect a given meaningful synergism or antagonism can be calculated in term of the noncentral F-distribution. l When the alternative hypothesis H 1 holds, the statistic Given: type I error rate (  ), power level (1-  ), measurement variation (  2 ), smallest meaningful difference (  ), the sample sizes can be obtained by

Typical Designs: # of Experiments (  = 0.05, 1-  =0.80) Number of Replications d = (234) 42 (168) 27 (135) 19 (114) 14 (98) d = (214) 40 (120) 21 (84) 14 (70) 10 (60) 7 (49) d = (136) 25 (75) 14 (56) 9 (45) 3 (18) 3 (21) d = (96) 18 (54) 10 (40) 6 (30) 3 (18) 3 (21) d = (48) 9 (27) 4 (16) 3 (15) 3 (18) 3 (21)

Layout of Experimental Design 1.Analyze single agent data and estimate single agent dose effects; 2.Choose the meaningful difference (  2 ) and calculate the number of mixtures m ; The variation (  2 ) is estimated by the pooled variance from the two single agent experiments. 3.Define the domain S ( e.g. ED20 to ED80 ) and find m mixtures of compounds A and B; 4.Upon completion of the experiment, use the F statistic to test the hypothesis of the additive action of A and B; 5.If p- value > 0.05, no detectable synergism; 6.Otherwise, we conclude the two compounds are synergistic or antagonistic. Using regression model selection methods to further diagnose the type of joint action of the two compounds

Designs for Specific Dose-Effects (i) Linear dose-response curves The single dose-respose curves are E is the fractional effect The curves are produced in the low dose/low effect region for many agents: ionising radiation, enzyme inhibitors, mutagens, agents causing chromosomal abnormalities and environmental carcinogens. The potency of B relative to A: Model for the additive action: In combination experiment, the m mixtures should be uniformly scattered in the domain

Designs for Specific Dose-Effects (ii) Linear-log dose-response curves The single dose-respose curves are The potency of B relative to A: Model for the additive action: In combination experiment, m combinations should be uniformly scattered in the domain Specially, when  A =  B, the total doses and the mixing proportions scatter uniformly on the domain. (Tan, et al. 2003)

Designs for Specific Dose-Effects (iii) Sigmoid dose-response curves The single dose-respose curve or where M is the median effective dose or concentration and m is a constant giving the order of sigmoidicity of the curve. The curves have been used in enzymology, also in studies of drugs causing muscle contraction, neuronal activators,inhibitors of cell proliferation and tumor promoters, and in general toxicology. The experimental design is the same as in the case (ii).

Designs for Specific Dose-Effects (iv) Simple exponential dose-response curves, survivor multiplication The single dose-response curve S is the survival rate. Exponential dose-response curves are typically found in cell survival experiments with ionising and non-ionising radiation and other agents with dammage DNA, particularly alkylating agents. Since the experimental design is same as in the case (i).

Example: SAHA+VP-16 Goal: to determine the effect of SAHA combined with VP-16 against the cell line HL-60 SAHA: 60 observations, dose range from 0.03  M to 10  M mean viability: 67.57%, standard deviation: the dose-response curve VP-16: 60 observations, dose range from  M to 3  M mean viability: 64.05%, standard deviation: The potency of VP-16 relative to SAHA is 9.12.

SAHA+VP-16: Experimental Design The pooled variance: the meaningful difference: 15% viability type I error rate: 0.05, power: 80%. Sample size: 90 (18 mixtures with 5 replications) The U-type matrix with 18 experiments (discrepancy: ) Dose range:  M to  M according to VP-16 (the viability is from 25% to 85%)

SAHA+VP-16: Mixtures for combination experiment Exper.#SAHA (  M ) VP-16 (  M ) Exper.#SAHA (  M ) VP-16 (  M )

SAHA+VP-16: Results of Combination Experiment Dose ranges: SAHA: 0.405~  M, VP-16: 0.031~2.35  M 90 observations: viability: 16.27%~113.06% mean:29.27% standard error: F-test: F(3, 85)=60.63 p-value< We reject H 0 : SAHA with VP-16 against HL-60 has the additive action

SAHA+VP-16: Synergy Analysis The response regression model of combination study (1) The expected response model for the additive action of SAHA and VP-16 should be (2) Compare the equations (1) and (2): · If (1)-(2) = 0, SAHA and VP-16 is additive; · If (1)-(2) < 0, SAHA and VP-16 is synergistic; · If (1)-(2) > 0, SAHA and VP-16 is antagonistic.

SAHA+VP-16: Contour Plot

Conclusions  Based on the single dose-effect curves, we proposed an optimal design for testing the additive joint action of two compounds in vitro or in vivo.  under a general dose-response model, the maximum information is extracted by having the experimental points scattered uniformly on the two-dimensional experimental domain.  The design is optimal: Minimizing the variability in modeling synergism and extracting maximum information on the joint action of two compounds.  The number of experimental units (or runs) are quite feasible in vitro or in vivo.

Acknowledgements l Ming Tan, PhD l Guoliang Tian, PhD l Peter Houghton, PhD l Douglas Ross, M.D., PhD l Ken Shiozawa, PhD.