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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
7
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
9
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)
11
IsobologramIsobologram
Synergy
Antagonism
BxBx
AxAxAx
Bx
Bx
Ax
Ax CIC
ICICC
IC
C
IC
C,
,
,,,
,
,
,
, 1
AxC ,
AxIC ,
BxC ,BxIC ,
Bx
Ax
IC
IC
,
,
12
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
13
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
14
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
15
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
16
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)
17
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
18
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
19
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
20
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
21
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
22
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
23
Experimental DataExperimental Data
24
PercentagesPercentages
25
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
26
Fit of 2 Parameter Logistic Fit of 2 Parameter Logistic Ignoring PlateIgnoring Plate
27
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
31
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
33
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
34
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
35
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
36
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