The Search for Synergism A Data Analytic Approach

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

Overview § 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 2

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

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

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 5

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: 331 -385 6

Sarriselkä agreement (1992) 7

Loewe 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 8

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

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) 10

Isobologram Antagonism Synergy 11

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 12

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 13

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 14

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 15

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) 16

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 17

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 18

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: D r u g A Drug B 19

Ray Design Composition of Mixtures § Tested concentration Ci of mixture is composed of: D r u g A § Proportion of constituents in mixture: Drug B 20

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 21

Layout of Screening Experiments in Oncology § Ray design reference compound A, tested compound B f = 0, 0. 125, 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 22

Experimental Data 23

Percentages 24

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: 25

Fit of 2 Parameter Logistic Ignoring Plate 26

Individual Fits of 2 Parameter Logistic per Plate 27

Studentized Residuals versus Fitted Values after Individual Model Fitting 28

Normal Quantile Plot of Pooled Residuals after Individual Model Fits 29

Individual Estimates per Plate-Factor 30

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 ? 31

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 32

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

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 34

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

Conclusions § 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 36