The Search for SynergismThe Search for SynergismA Data Analytic Approach

L. Wouters, J. Van Dun, L. Bijnens

May 2003

Three Country CornerRoyal Statistical Society

2

OverviewOverview

Combined action of drugs

Screening for synergism

Experimental Design

Fitting concentration response curves, estimation of IC50

Graphical analysis of combined action

– isobolograms

– fraction plots

– combination index

3

Drug CombinationsDrug Combinations

Additive

Sub-additive: antagonismfight against one another

Super-additive: synergismwork together

4

Drug Combinations: Drug Combinations: Antagonism - SynergismAntagonism - Synergism

Major therapeutic areas:

– Oncology

– Infectious disease

Ideal combination:

– Synergistic for therapeutic activity

– Antagonistic for toxicity

5

Non-additivity and Statistical Non-additivity and Statistical InteractionInteraction

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8 1.0

Concentration

% E

ffect

Drug A f(x), drug B g(x)

Combination: a + b, h(a,b)

f(a) = 50 %, g(b) = 60 %additivity h(a,b) = 110 % ?Drug can be antagonistic with itself

f(a) = 0%, g(b)=0%additivity h(a,b) = 0% ?Drug can be synergistic with itself

6

Problems with Synergism - Problems with Synergism - AntagonismAntagonism

Synergism is controversial issue

Literature large but confusing

Different definitions

Different methods and experimental designs

Pharmacological - biostatistical approaches

Greco (1995) Pharmacol Rev 47: 331-385

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Sarriselkä agreement (1992)Sarriselkä agreement (1992)

Combinedeffect

Both agentsactive (Loewemodel)

Both agentsactive (Blissmodel)

Only one agentactive

Neither agentactive

> predicted Loewesynergism

Blisssynergism

Synergism Coalism

= predicted Loeweadditivity

Blissindependence

Inertism Inertism

< predicted Loeweantagonism

Blissantagonism

Antagonism -

8

Loewe AdditivityLoewe Additivity

ICx,A, ICx,B concentrations required for each drug A, B individually to obtain a certain effect x (x % inhibition)

Let Cx,A, Cx,B doses of drug A and drug B in the combination that jointly yield same effect x

Drug A has lower potency ICx,A > ICx,B

Relative potency of A: ICx,A / ICx,B

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Loewe Additivity (cont.)Loewe Additivity (cont.)

Assume constant relative potency and additivity

Combination can be expressed as equivalent concentrations of either drug :

BxBxBx

AxAAxBx

Bx

AxAx ICC

IC

ICCICC

IC

ICC ,.

,

,,,

,

,, ,

1,

,

,

, Bx

Bx

Ax

Ax

IC

C

IC

C

10

Methods Based on Loewe AdditivityMethods Based on Loewe Additivity

Isobologram

Interaction index of Berenbaum (1977)

Bivariate spline fitting method of Sühnel (1990)

Hypothesis testing approach of Laska (1994)

Response surface methodology of Greco (1990), Machado (1994)

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IsobologramIsobologram

Synergy

Antagonism

BxBx

AxAxAx

Bx

Bx

Ax

Ax CIC

ICICC

IC

C

IC

C,

,

,,,

,

,

,

, 1

AxC ,

AxIC ,

BxC ,BxIC ,

Bx

Ax

IC

IC

,

,

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Bliss IndependenceBliss Independence

i1, i2, i12 inhibition as a fraction [0; 1] by drug 1, drug 2,

and their combination

from a probabilistic point of view, when fraction i1 is

inhibited by drug 1, only (1 - i1) is available to respond to

drug 2. Assuming independence:

can be reformulated in terms of u. = 1 - i., the fraction

remaining unaffected

212121112 )1( iiiiiiii

21

21211212 111111

uu

uuuuiu

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Bliss IndependenceBliss IndependenceCounter-argumentCounter-argument

A drug can be synergistic with itself

75 % of control at 0.9 mg/kg

Assume a dose of 0.9 mg/kg of the drug is combined with 0.9 mg/kg of the same drug

Total dose = 1.8 mg/kg

Under Bliss independence:0.75 x 0.75 = 0.56 = 56 % for combination

1.8 mg/kg yields 15.7 % of control

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Screening for Synergism in Screening for Synergism in OncologyOncology

Screening experiment

– as simple as possible with limited resources

– carried out on a routine basis

– analysis must be automated

Screening experiments on tumor cells grown in 96-well microtiter plates

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Screening ExperimentScreening ExperimentRequirementsRequirements

– Unbiased estimates of responses

– Avoidance of confounding of random error and drug effects

– Elimination of plate effects and plate location effects in 96-well plates

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Plate Location Effects in 96-well Plate Location Effects in 96-well PlatesPlates

Microtiter plates contain a substantial amount of unexplainable systematic error along their rows & columns (Faessel, et al. 1999)

Importance of standardization experiment (low, middle, and high response)

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Standardization Experiment (n = 3)Standardization Experiment (n = 3)

Standardization experiment at high level of response, n=3

Within assay presence of systematic differences of important magnitude (up to 50 %) in untreated microtiter plates after edge removal

Not repeatable between different runs of assay

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How to Eliminate Bias & How to Eliminate Bias & Confounding ?Confounding ?

Randomization assures:

– Equal probability to attain a specific response for each well

– Independence of results

– Absence of confounding

– Proper estimation of random error

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Experimental DesignExperimental DesignRay DesignRay Design

Mixtures are composed based on preliminary estimates of IC50 of constituents

Assuming additivity:

Construct concentration response curve for different mixture factors:

Drug

A

Drug B

factor mixture :

1 ,50,50,50

f

ICffICIC BAMixture

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Ray DesignRay DesignComposition of MixturesComposition of Mixtures

Tested concentration Ci of mixture is composed of:

Proportion of constituents in mixture:

factordilution :

factor mixture :

1 ,50,50

k

f

ICffICkCC BAi

C

ICf

C

fIC BB

AA

,50,50 1

Drug

A

Drug B

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Advantages of strategyAdvantages of strategy

Simplified analysis:

– Consider mixture as new drug

– Fit concentration response curve to different dilutions of mixture

Easy to carry out in laboratory

Limited number of samples

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Layout of Screening Experiments in Layout of Screening Experiments in OncologyOncology

Ray design reference compound A, tested compound B

f = 0, 0.125, 0.25, 0.5, 0.75, 1

Experiments carried out in 3 independent 96-well plates

Dilutions (k): 10/1, 10/2, 10/3, 10/4, 1/1, 1/2, 1/4, 1/10

All dilutions tested within single plate

Wells for background and maximum effect

Allocation of different treatment is randomized within

plate by robot

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Experimental DataExperimental Data

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PercentagesPercentages

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Lessons from EDALessons from EDA

Asymptotes of sigmoidal curve not reached always

Some part of sigmoidal curve is still present

Computing percentages makes sense (common system maximum)

Proposed functional model:

ii CIC

yloglogexp1

100

50

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Fit of 2 Parameter Logistic Fit of 2 Parameter Logistic Ignoring PlateIgnoring Plate

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Individual Fits of 2 Parameter Individual Fits of 2 Parameter Logistic per PlateLogistic per Plate

28

Studentized Residuals versus Fitted Studentized Residuals versus Fitted Values after Individual Model FittingValues after Individual Model Fitting

29

Normal Quantile Plot of Pooled Normal Quantile Plot of Pooled Residuals after Individual Model FitsResiduals after Individual Model Fits

30

Individual Estimates per Plate-FactorIndividual Estimates per Plate-Factor

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Lessons from EDA for Functional Lessons from EDA for Functional Model FittingModel Fitting

Sigmoidal shape as described by 2-parameter logistic model

Importance of plate effect even after correcting for background, etc. by calculating percentages

How to obtain reliable estimate of IC50 and standard errors ?

32

Nonlinear Mixed EffectsNonlinear Mixed Effects

Nonlinear Mixed Effects Model (Pinheiro, Bates) allows to model individual response curves within plates and provides reliable estimate of standard error

Result = estimates and standard errors of model parameters as fixed effects

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IsobologramIsobologram

Synergism

Antagonism

• Decompose IC50,M of mixture into IC50 of constituents C50,A

and C50,B :

• Plot of drug B versus drug A and line of additivity

1,50

,50

,50

,50 B

B

A

A

IC

C

IC

C

BBB

MAA

ICC

ICC

,50,50

,50,50

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Fraction PlotFraction Plot

Based upon refined estimates of IC50 of Drug A and B recalculate the correct fraction f :

Plot of IC50 of mixture versus recalculated fraction

BAAA

BA

ICICIC

ICf

,50,50,50

,50

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Combination IndexCombination IndexChou and Talalay (1984)Chou and Talalay (1984)

95% Confidence intervals by parametric bootstrap (n = 10000) based upon estimates and standard errors from nlme fit

antagonism

additivity

synergism

:1

:1

:1

,50

,50

,50

,50

B

B

A

A

IC

C

IC

CCI

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ConclusionsConclusions

Present graphical approach appealing to scientists

Still a lot to be done

– T. O’Brien’s approach (TOB)

– Incorporating design issues in TOB

– Alternative distributions (e.g. gamma)

– Optimal design