Ming T Tan, PhD

University of Maryland Greenebaum Cancer Center

Email: mtan@umm.edu

2006 D.O.E. Presentation 7/11/2006

Optimized Experimental Optimized Experimental Design for Detecting Synergy Design for Detecting Synergy in Drug Combination Studiesin Drug Combination Studies

Combination StudiesCombination Studies Motivation:

achieve greater efficacy with lesser toxicity 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

Current Design MethodsCurrent Design Methods

Equally regression lines (Finney, 1971)

An optimal design by fixing the total dose for specific models (Abdelbasit & Plackett, 1982)

Fixed ratio design (Tallarida et al., 1992)

An optimal design for in vitro combination studies (Greco et al., 1994) but the sample size too large

Uniform design based on the log-linear single dose-response (Tan et al., 2003)

Model FormulationModel FormulationThe 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

H0: f = 0 versus H1: f 0.

The general nonparametric model for the combinations of A and B is assumed

)( AA xfy )( BB xfy

)()( 1BBAB xffx

where the function is unspecified,

Then, testing the additive action of A and B is equivalent to testing

))(( BBAA xxxfy

( ) ( , )AA A B A B

B

Xy f x x f x x

X

),( BA xxf ),0(~ 2 N

Model AssumptionsModel Assumptions

With an invertible transformation: we have the additive structure

and the model becomes

where g1 and g2 are linearly independent,

and satisfies the orthogonal condition:

),(),,( 212211 zzxzzx BA

1 1 1 2 2 2( ) ( ) ( )AA A B

B

Xf x x g z g z

X

),()()( 21222111 zzgzgzgy

),(),( 21 BA xxfzzg

T

SzgzgzzGdzdzzzgzzG

)(),(),(,0),(),( 221121212121

Model Formulation (cont’)Model Formulation (cont’)

Hence, testing the additive action of A and B is equivalent to testing H0: g = 0 versus H1: 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:

S

dzdzzzg 22121

2 ),(

Statistical InferenceStatistical InferenceSuppose that are m mixtures of A and B in domain S

ni experiments at the dose-level,

yij are the corresponding responses,

y: n 1 vector with elements yij ordered lexicographically,

I: the unit matrix, 1k: the k1 vector of one,

Z: m2 matrix with i-th row

Under H0, mnmT

T

FmnyJIy

myVJyF

,2~

)()(

)2()(

mww ,...,1

Szzw Tiii ),( 21

mi nnnminj ...;,...,1,,...,1 1

))(),(( 2211 ii zgzg

)1,...,1(1 mnndiagU

TTTT UZUZUZUZV 1)( TT UUUUJ 1)(

Statistical Inference (cont’)Statistical Inference (cont’)

S gTg bBbwdwg

n

,

1,

22

)()(

)()()(),()()( , wdwgwGbwdwGwGBSgS

T

)(~ ,2 mnmFF Under H1: with

where

is the design measure, which is a probability measure with mass pi = ni /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.

0, gb

S

wdwgn

)()(22

Uniform Scattered Points on Uniform Scattered Points on CC22

Fm(x) be the empirical distribution function of Pm,

F(x) be the uniform distribution function in C2.

Discrepancy:

Let Um,2=(uij) be an m 2 matrix, each column is a permutation

of {1,2,…,m}. Define vij=(uij - 0.5)/m, i=1,…,m; j=1,2,

and Vm,2 can be considered as m points on C2.

 Then the uniform design is to choose the m points so that the discrepancy of Vm,2 is the smallest over all of possible Vm,2.

)1,0()1,0(,..., 21 CxxP mm

)()(sup)(2

xFxFPD mCx

m

Example: Example: Uniform Scattered Points on Uniform Scattered Points on CC22

A uniform design with 7 experimental units on C2:

Discrepancy: 0.1582

(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).

T

U

3614725

76543212,7

v1

v2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Uniform Experimental DesignUniform Experimental DesignSuppose the experimental domain

(vi1, vi2)T, i=1,…,m; are m UD points on C2.

The m Uniform design points on S are

i=1,…,m

Then, m mixtures {(xAi, xBi): i=1,…,m} of drugs A and B for experiment can be obtained by the inverse transform.

HLHL ZzZZzZzzS 22211121 ,:),(

LLHii

LLHii

ZZZvz

ZZZvz

22222

11111

)(

)(

Sample Size DeterminationSample Size Determination 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.

When the alternative hypothesis H1 holds, the statistic

Given: type I error rate (), power level (1- ), measurement

variation (2), smallest meaningful difference (),

the sample sizes can be obtained by

22,2 ),(~ nFF mnm

km

xmFP

k

exFP mnkm

k

k

22

)2(

!

)2/()( ,22

0

2/

Typical Designs: # of ExperimentsTypical Designs: # of Experiments(( = 0.05, 1- = 0.05, 1- =0.80) =0.80)

Number of Replications 

1 2 3 4 5 6

d = 0.2 --78

(234)42

(168)27

(135)19

(114)14 (98)

d = 0.3107

(214)40

(120)21 (84)

14 (70)

10(60)

7(49)

d = 0.468

(136)25 (75)

14 (56)

9 (45)

3 (18)

3 (21)

d = 0.548 (96)

18 (54)

10 (40)

6 (30)

3 (18)

3 (21)

d = 0.824 (48)

9 (27)

4 (16)

3 (15)

3 (18)

3 (21)

22 /d

Layout of Experimental DesignLayout of Experimental Design1. 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-EffectsDesigns for Specific Dose-Effects(i) (i) Linear dose-response curvesLinear dose-response curves

The single dose-respose curves are E is the fractional effectThe 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

BBBAAA xxExxE )(,)(

AB /

BBAABA xxxxE ),(

mixx BiAi ,...,1),,(

21,21:),( bxbaxaxxS BABA

Designs for Specific Dose-EffectsDesigns for Specific Dose-Effects(ii) (ii) Linear-log dose-response curvesLinear-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)

BBBAAA xyxy log,log

]/)exp[(,)( 0/

0 AABBBABxx

00 )1(loglog AAAA zy AB

AB

BA

AABA

xx

xxxz

/

/ ,

miz Aii ,...,1),,(

)1,0(),( HL ZZS

Designs for Specific Dose-EffectsDesigns for Specific Dose-Effects(iii) (iii) Sigmoid dose-response curvesSigmoid dose-response curves

The single dose-respose curve

or

where M is the median effective dose or concentration andm is a constant giving the order of sigmoidicity of the curve.The curves have been used in enzymology, also in studies ofdrugs causing muscle contraction, neuronal activators,inhibitorsof cell proliferation and tumor promoters, and in generaltoxicology. The experimental design is the same as in the case (ii).

mMxxExE )/())(1/()(

)log(log))](1/()(log[ MxmxExE

Designs for Specific Dose-EffectsDesigns for Specific Dose-Effects(iv) (iv) Simple exponential dose-response curves, Simple exponential dose-response curves,

survivor multiplicationsurvivor 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).

)exp()( xxS

xxS )(log

Example: SAHA+VP-16Example: 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: 37.47 the dose-response curve

VP-16: 60 observations, dose range from 0.003 M to 3 M mean viability: 64.05%, standard deviation: 27.88

The potency of VP-16 relative to SAHA is 9.12.

)021.0log(9.1125.64 Sy

)0012.0log(9.1197.37 Vy

SAHA+VP-16: Experimental DesignSAHA+VP-16: Experimental DesignThe pooled variance: 836.176

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: 0.0779)

Dose range: 0.019M to 2.974M according to VP-16

(the viability is from 25% to 85%)

T

U

113146179112415718102135168

1817161514131211109876543212,18

SAHA+VP-16: SAHA+VP-16: Mixtures for combination experimentMixtures for combination experiment

Exper.# SAHA (M)

VP-16

(M)

Exper.# SAHA

(M)

VP-16

(M)

1 0.405 0.060 10 2.820 1.273

2 2.104 0.038 11 10.175 0.631

3 1.000 0.323 12 0.504 1.855

4 3.780 0.183 13 8.940 1.094

5 0.579 0.696 14 18.708 0.187

6 4.458 0.437 15 6.707 1.667

7 9.651 0.031 16 17.556 0.642

8 4.138 0.800 17 3.476 2.350

9 10.412 0.276 18 15.406 1.206

SAHA+VP-16:SAHA+VP-16:Results of Combination ExperimentResults of Combination Experiment

Dose ranges:

SAHA: 0.405~18.708M,

VP-16: 0.031~2.35M

90 observations:

viability: 16.27%~113.06%

mean:29.27%

standard error: 15.92

F-test: F(3, 85)=60.63

p-value<0.0001

We reject H0:

SAHA with VP-16 against HL-60 has the additive action

SAHA+VP-16: Synergy AnalysisSAHA+VP-16: Synergy AnalysisThe response regression model of combination study (1)

The expected response model for the additive action of SAHAand 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.

)log(12.6)log(14.795.33 VSy

12.90222.0

021.0)12.91(log4.5)0222.0log(9.11

VS

SVSy

SAHA+VP-16: Contour PlotSAHA+VP-16: Contour Plot

1 6 11 16

SAHA (uM)

0.1

0.6

1.1

1.6

2.1

VP

-16

(u

M)

Synergy

Synergy

ConclusionsConclusions 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.

AcknowledgementsAcknowledgements

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