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The Search for Synergism A Data Analytic Approach L. Wouters, J. Van Dun, L. Bijnens May 2003 Three Country Corner Royal Statistical Society

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2Overview Combined action of drugs Screening for synergism Experimental Design Fitting concentration response curves, estimation of IC 50 Graphical analysis of combined action –isobolograms –fraction plots –combination index

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3 Drug Combinations Additive Sub-additive: antagonism fight against one another Super-additive: synergism work together

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4 Drug Combinations: Antagonism - Synergism Major therapeutic areas: –Oncology –Infectious disease Ideal combination: –Synergistic for therapeutic activity –Antagonistic for toxicity

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5 Non-additivity and Statistical Interaction 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

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6 Problems with Synergism - Antagonism Synergism is controversial issue Literature large but confusing Different definitions Different methods and experimental designs Pharmacological - biostatistical approaches Greco (1995) Pharmacol Rev 47:

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7 Sarriselkä agreement (1992)

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8 Loewe Additivity IC x,A, IC x,B concentrations required for each drug A, B individually to obtain a certain effect x (x % inhibition) Let C x,A, C x,B doses of drug A and drug B in the combination that jointly yield same effect x Drug A has lower potency IC x,A > IC x,B Relative potency of A: IC x,A / IC x,B

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9 Loewe Additivity (cont.) Assume constant relative potency and additivity Combination can be expressed as equivalent concentrations of either drug :

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10 Methods 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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11Isobologram Synergy Antagonism

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12 Bliss Independence i 1, i 2, i 12 inhibition as a fraction [0; 1] by drug 1, drug 2, and their combination from a probabilistic point of view, when fraction i 1 is inhibited by drug 1, only (1 - i 1 ) is available to respond to drug 2. Assuming independence: can be reformulated in terms of u. = 1 - i., the fraction remaining unaffected

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13 Bliss Independence Counter-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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14 Screening for Synergism in Oncology 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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15 Screening Experiment Requirements –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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16 Plate Location Effects in 96-well Plates 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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17 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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18 How to Eliminate Bias & 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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19 Experimental Design Ray Design Mixtures are composed based on preliminary estimates of IC 50 of constituents Assuming additivity: Construct concentration response curve for different mixture factors: DrugADrugA Drug B

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20 Ray Design Composition of Mixtures Tested concentration C i of mixture is composed of: Proportion of constituents in mixture: DrugADrugA Drug B

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21 Advantages 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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22 Layout of Screening Experiments in Oncology 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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23 Experimental Data

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24Percentages

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25 Lessons 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:

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26 Fit of 2 Parameter Logistic Ignoring Plate

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27 Individual Fits of 2 Parameter Logistic per Plate

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28 Studentized Residuals versus Fitted Values after Individual Model Fitting

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29 Normal Quantile Plot of Pooled Residuals after Individual Model Fits

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30 Individual Estimates per Plate-Factor

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31 Lessons from EDA for Functional Model 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 IC 50 and standard errors ?

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32 Nonlinear 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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33Isobologram Synergism Antagonism Decompose IC 50,M of mixture into IC 50 of constituents C 50,A and C 50,B : Plot of drug B versus drug A and line of additivity

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34 Fraction Plot Based upon refined estimates of IC 50 of Drug A and B recalculate the correct fraction f : Plot of IC 50 of mixture versus recalculated fraction

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35 Combination Index Chou and Talalay (1984) 95% Confidence intervals by parametric bootstrap (n = 10000) based upon estimates and standard errors from nlme fit

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36Conclusions 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

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