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Quantifying Combination Drug Synergy

John J. PetersonResearch Statistics Unit

GlaxoSmithKline Pharmaceuticals

Midwest Biopharmaceutical Statistics Workshop25 May 2010

1

OUTLINE

• “Excess over highest single agent” – a useful first hurdle

• Loewe Synergy

• Loewe Dosewise additivity

• Nonlinear Blending

• Dose Reduction Ratios

• Summary

2

3

A Simple (bottom-line) Criterion for Combination Drug Assessment

• “Excess over highest single agent”response at (A1,B1) > max {‘response at A1 alone’, ‘response at B1 alone’}

• It is also an FDA criterion for (21 CRF 300.50) for combination drug approval.

(A1,B1) combination

A1

B1

Drug A

Drug B

4

Using “Excess over Highest Single Agent” as a Screening Criterion

• “Excess over Highest Single Agent” provides a low-level efficacy hurdle for combination drug screening.

• “Excess over Highest Single Agent” provides the largest window for screening drugs for “dose reduction potential”.

(A1,B1) combination

A1

B1

Drug A

80% isobologram contour lineDrug B • Note that even with mild antagonism

“excess over highest single agent”guarantees some room for “dose reduction potential”, i.e. we can lower both doses of drugs A and B whilestill maintaining the same response.

• This is useful when one or both of each drugalone has different dose-related side effects.

5

Testing for Excess over Highest Single Agent

(A1,B1) combination

A1

B1

Drug A

Drug B

We can test H0 vs. H1 using the ‘min’ test.This is done by performing two one-sided tests:

Testing for EOHSA with one fixed dose combination.

0 : AB A AB BH orµ µ µ µ≤ ≤

1 : AB A AB BH andµ µ µ µ> >

( ) ( )0 1: . :A A

AB A AB AH vs Hµ µ µ µ≤ >

( ) ( )0 1: . :B B

AB B AB BH vs Hµ µ µ µ≤ >

If we reject both and then the min test rejects H0 .( )0

AH ( )0 ,BH

6

Compound B Dose Levels 8 S C C C C C C C C

7 S C C C C C C C C

6 S C C C C C C C C

5 S C C C C C C C C

4 S C C C C C C C C

3 S C C C C C C C C

2 S C C C C C C C C

1 S C C C C C C C C

0 V S S S S S S S S

0 1 2 3 4 5 6 7 8

Compound A Dose Levels

Simultaneous testing for EOHSA across compound combinations

C = “combination”S = “single compound”V = “vehicle (control)”

• Suppose there is a trend forcompound A (for dose levels 0 to 5)at dose level 4 of compound B.

• Suppose also there is a trend forcompound B (for dose levels 0 to 4)at dose level 5 of compound A.

• It follows then that the compoundcombination (5,4) has EOHSA.

7

Compound B Dose Levels 8 S C C C C C C C C

7 S C C C C C C C C

6 S C C C C C C C C

5 S C C C C C C C C

4 S C C C C C C C C

3 S C C C C C C C C

2 S C C C C C C C C

1 S C C C C C C C C

0 V S S S S S S S S

0 1 2 3 4 5 6 7 8

Compound A Dose Levels

Simultaneous testing for EOHSA across compound combinations

C = “combination”S = “single compound”V = “vehicle (control)”

• So intersecting trends can be used test for EOHSA.

• Since there are k dose levels of eachcompound there are 2(k-1) simultaneoustrend tests involving exactly l dose levels.

• At each dose level, l, we can do 2(k-1)Bonferroni-adjusted trend tests.

• The Tukey step-down trend test can be used as we step from level l=(k-1) to l=1.

• The Tukey step-down test requires noadjustment of the (Bonferroni-adjusted) α-level.

See: Peterson, J. J. (2006), “Testing for Excess Over Highest Single Agent with an Application to High-throughputScreening of Pairs of Compounds”, GSK BDS Technical Report 2006-04, available at: http://biometrics.com/?p=128

88

Loewe SynergyIsobolograms and Combination Index

(A,B) combination50% contour line

(isobologram)

( ) ( )d d I

ED ED+ = <1 2

1 250 50

1

( ),d d1 2

( )ED 250

( ) ( )d d

ED ED+ =1 2

1 250 50

1• The red contour line is a 50% isobologram, i.e. the locusof (d1,d2) combination points producing a 50% response.

• Since the contour bows inwardwe say that we have Loewe synergy.

• Here, the combination index, I, is less than1 for any (d1,d2) combination point on the red dotted line.

• The point (d1,d2) has Loewe synergy since I is less than 1.

Drug 1

Drug 2

( )150EDd1

d2

9

What the Combination Index is not Telling You.

(i) Is there enough synergy to overcome the “dilution effect”?

(ii) Will your “synergistic combination” have a better IC50than you most potent single agent?

(iii) If you have “synergy”, which drug (in combination) provides the best dose-reduction ratio of itself to its (dose-response equivalent) single agent?

( ) ( )d d I

ED ED+ = <1 2

1 250 50

1

10

Loewe SynergyIsobolograms and Combination Index

(A,B) combination50% contour line

(isobologram)

( ) ( )d d I

ED ED+ = <1 2

1 250 50

1

( ),d d1 2

( )ED 250

Drug 1

Drug 2

( )150ED d1

d2

1 2d d A+ =

A

A

Line of

Line of 45:55dose ratio

• The blue line is the locus of points such that the total dose equals A.

• The ED50 for the drug productformed by the 45:55 dose ratio isA since at the green point.

• Here the ED50 for the combinationdrug product is greater than that fordrug 1 alone despite having Loewe synergy!

1 2d d A+ =

11

Ray of constant dose ratio.

50% contour line(isobologram)

d d A+ =1 2

dose d1

dose d2

*d1

*d2

A00

( )ED 150



dose d1

dose d2

*d1

*d2

A

A

A

( )ED 150

( )ED 250

( )ED 250

00



d d A+ =1 2

dose d1

dose d2

*d1

*d2

A00

( )ED 150





dose d1

dose d2

*d1

*d2

A

A

A

( )ED 150

( )ED 250

( )ED 250

00

• If the ED50’s of two compounds are sufficiently different, then the interaction index can produce a troubling result.

• In figures 1 and 2 below both interaction indices are less than one for the combination (green point) on the isobologram.

• However, the ED50 for the combination in Fig. 1 is greater than that for Drug A alone despite a synergistic interaction index!

Fig. 1 Fig. 2

Is there a synergy potency?

Here I<1Here I<1

12

If you have “synergy”, which drug (in combination) provides the best dose-reduction ratio of itself to its a single agent?

• Recall that the Loewe combination index is

• I refer to the individual ratios, as dose-reduction ratios.

• So if I < 1 then are both less than 1, I=0.7, say,

we could have

( ) ( ) .d dIED ED

= +1 21 2

50 50

( ) ( )andd d

ED ED1 21 2

50 50

( ) ( ). and .d d

ED ED= =1 2

1 250 50

0 35 0 35

or, we could have ( ) ( ). and .d d

ED ED= =1 2

1 250 50

0 05 0 65

( ) ( )andd d

ED ED1 21 2

50 50

• The combination index confounds information about the dose ratios!

13

Plotting Dose Reduction Ratios for Various Drug Combinations

Dose Reduction Ratios for different combinations

1

0 1

( )1

1 150

drED

=

( )2

2 250

drED

=

Note that the blue dots correspond to combinations with the sameinteraction index value.

Dru g 1

Drug 2

14log( Conc )

Res

pons

e

0

20

40

60

80

100

-4 -2 0 2

Monotherapy 1

-4 -2 0 2

Monotherapy 2

-4 -2 0 2

50:50 Ratio

Another Problem with the Interaction Index

Cannot always compute the interaction index!

Monotherapies do not achieveY = 50%

Yet, excellent synergy existsAt a 50:50 ratio!

Acknowledgements to Steve Novick

1515

Problems with Isobologram Approaches to Combination Drug Synergy

• Isobolograms attempt to model the locus of compound-combination points corresponding to a fixed response value, e.g. the 50% (inhibition) response.

• If one (or both) compounds alone do not produce the expected response (e.g. 50% inhibition), then we have may a comparison problem with the combination.

• Furthermore, with isobolograms alone it can be difficult to tell how much they should bow inwards in order to have a synergy of potency (i.e. in order to overcome the “dilution effect” if one of the compounds is not as potent as the other.)

But…suppose we looked at synergy on the response scale, instead if theconcentration scale.

16

Nonlinear BlendingMixture-Amount Experiments

An alternative approach to analyzing dose-response for combination drugs

• The methodology of “Mixture-amount experiments” has beensuccessfully applied to substance blending experiments in theareas of: fertilizer crop yield studies, pharmaceutical tablet optimization, and detergent formulation.

• A key conceptual difference: - Mixture-amount experiments quantify the blending properties ofsubstances for fixed total dose amounts.- Isobolograms, on the other hand, attempt to model the locus ofcompound-combination points corresponding to a fixed response value.

• The isobologram approach can be awkward for situations where themaximum dose-response asymptotes are different or where one or bothdrugs have no dose response when applied alone (e.g. Augmentin,especially when applied to bacteria that are resistant to amoxicillin.)

17

The Concept of Nonlinear Blending

• Mixture-amount experiments quantify the blending properties ofsubstances for fixed total dose amounts.

• Consider a fixed total dose, D, of two drug substances (A and B) at varying proportions of drug A

weak nonlinear blending

0% drug A 100% drug A

response strong nonlinearblending


response

total dose=D total dose=D

18

Nonlinear Blending and IsobologramsOne can show mathematically thatstrong nonlinear blending synergy of potencyfor strictly increasing dose-response surfaces

XA=DxA+xB=D

Synergy on the ray of constant dose ratio. ( p% drug A and (1-p)% drug B)

(A’,B’) combination50% contour line (isobologram)

XB=D

strong nonlinearblending


response

total dose=D

A “slice” throughthe response surface for totaldose fixed at D

AX 50

BX 50

compound A

compound B

19

Two examples: Virology and Cancer Chemotherapy• Virology Example: Drugs AZT and FLG

• The isobologram contour plot for the combination-drug dose response surface.

• There appears to be Loewe synergy here are the interaction index is lessthan 1 for the 50% (and some other) isobolograms.

FLG

0

1

2

3

4

5

6

7

8

9

10

AZT

0. 00 0. 01 0. 02 0. 03 0. 04 0. 05 0. 06

FLG

0

1

2

3

4

5

6

7

8

9

10

AZT

0. 00 0. 01 0. 02 0. 03 0. 04 0. 05 0. 06 AZT

FLG

• However, the ED50’s for these two drugs are quitedifferent!

20

Virology Example: Drugs AZT and FLG

Percent AZT

Res

pons

e Pe

rcen

t

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.000.02

0.040.06

0.080.10

0.120.14

0.160.18

0.000.01

0.020.03

0.040.05

0.060.07

0.080.09

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0000.005

0.0100.015

0.0200.025

0.0300.035

0.0400.045

0.0000.002

0.0040.006

0.0080.010

0.0120.014

0.0160.018

0.0200.022

Total dose=0.035

Total dose=0.07

Total dose=0.14

Total dose=0.28 Total dose=0.56 Total dose=1.13

Total dose=2.25 Total dose=4.5 Total dose=9.0

• Below, the nonlinear blending plots show that despite a “synergistic”Interaction index, blending in FLG with AZT does not help!

Proportion of AZT

Resp

onse

Pro

port

ion

21

In-vitro Cancer chemotherapy example

• Two drugs: 5’FU and a B-Raf inhibitor

• Below, it appears that there is little or no synergy for isobolograms below a60% response level. (For 5’FU we do not even have an ED70, ED80, etc.)

TX

1

12501

25001

37500

50000

5FU

1 12501 25001 37500 50000

22

In-vitro Cancer chemotherapy example

Percent 5FU

Res

pons

e Pe

rcen

t

0102030405060708090

100

0102030405060708090

100

0.0 0.2 0.4 0.6 0.8 1.00

102030405060708090

100

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Total dose=5555 (nM)Total dose=7312(nM)

Total dose=9623(nM)

Total dose=12664(nM) Total dose=16667(nM) Total dose=21935(nM)

Total dose=28868(nM) Total dose=37992(nM) Total dose=50000(nM)

• Below we can see that for some total dose levels, we have strong nonlinear blending, despite little or no evidence of Loewe synergy.

Proportion 5FU

23

Strong Nonlinear Blending Can Also Be Assessed by ComparingSeveral Dose Response Profiles for Fixed-Dose Ratios

Drug A alone

Drug B alone

50:50 drug ratio

Same total dose

50:50 drug ratio

80:20 drug ratio

20:80 drug ratio

Drug A dose

Drug B dose

Line of same total dose

This perspective is sometimeseasier for clients to understand.

24

Constrained Mixture-Amount Experiments

drug 1 dose

drug

2 d

ose

drug 1 dose response

Line of same total dose

25

An Example of Two Drugs with Combination Indices Much Less Than 1Yet Show No Synergy of Potency

• Drug combination experiment of TMQ (trimetrexate) and AG2034in the presence of high folic acid.

• Minto-White (parametric) response surface model fitted to data

• Contour plot of dose response surface:

0.00 0.01 0.02 0.03 0.04TMQ

0.00

0.02

0.04

0.06

0.08

AG20

34

0.2 0.3 0.4 0.5

0.6 0.7 0.8

0.9TMQ

AG

2034

Isobologram contours

26


• Drug combination experiment of TMQ (trimetrexate) and AG2034

• Minto-White (parametric) response surface model fitted to data

• Contour plot of dose response surface:

0.00 0.01 0.02 0.03 0.04TMQ

0.00

0.02

0.04

0.06

0.08

AG20

34

0.2 0.3 0.4 0.5

0.6 0.7 0.8

0.9TMQ

AG

2034

27


Key: Black line = AG2034 alone, Gray Line = TMQ alone, Red line= combination. Note that

for TMQ proportions of 0.7, 0.8, and 0.9 the red and gray lines virtually overlap.

Proportion of TMQ

Nor

mal

ized

Mea

n R

espo

nse

Total dose = 0.005 µM

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.0

28log Total Dose

Nor

mal

ized

Mea

n R

espo

nse

0.2

0.4

0.6

0.8

1.0

-3 -2 -1 0 1

prop. of TMQ=0.1 prop. of TMQ=0.2

-3 -2 -1 0 1

prop. of TMQ=0.3


0.2

0.4

0.6

0.8

1.0prop. of TMQ=0.6

0.2

0.4

0.6

0.8

1.0prop. of TMQ=0.7

-3 -2 -1 0 1



Key: Black line = AG2034 alone, Gray Line = TMQ alone, Red line= combination. Note that

for TMQ proportions of 0.7, 0.8, and 0.9 the red and gray lines virtually overlap.

29

Synergy vs. “Dose Reduction Potential”• Intuitively, when we think of combination drug synergy we think of two (or more) drugs combining to produce a response greater thanwhat we would expect.

• On the other hand, no such response synergy may exist between two drugsbut they may have “dose reduction potential”.

• In other words, we do not expect better responses by combining the drugs but we can nonetheless achieve the same response with smaller amounts of each drug than if we have to use either one alone.

- This is helpful if two drugs have side effects that are different. Or, if one drug has a serious side effect that can be greatly lessened by the addition of a second drug, while still maintaining the same level of efficacy.

30

Dose Reduction Profile Plots

• Suppose instead we are interested in examining the potential of twodrugs to reduce the amount used over each one alone while retaining thesame desired level of efficacy.

50% isobologramcontour line

Drug 1 dose

Drug 2 dose

( )150D

( )250D

1d

2d

Here, it is easy to see qualitatively that ( ) ( )1 2

1 250 50andd D d D .< <

But it is difficult to determine from theIsobologram the precise values of

( ) ( )1 21 2

50 50

andd d .D D

31

Dose Reduction Profile Plots• Suppose instead we are interested in examining the potential of twodrugs to reduce the amount used over each one alone while retaining thesame desired level of efficacy.

Here, it is easy to see qualitatively that ( ) ( )1 2

1 250 50andd D d D .< <

But it is difficult to determine from theIsobologram the precise values of

( ) ( )1 21 2

50 50

andd d .D D50% isobologram

contour line

Drug 1 dose

Drug 2 dose

( )150D

( )250D

1d

2d

• Chou & Chou (1988) first introduced the notion of looking at the ratiosindividually.

( ) ( )1 21 2

50 50

andd d .D D

32

Dose Reduction Profile Plots• Suppose instead we are interested in examining the potential of twodrugs to reduce the amount used over each one alone while retaining thesame desired level of efficacy.

• It is possible to sweep acrossthe isobologram contour line and record each dose ratio

• Each red arrow is uniquely associated with a specific proportion of drug 1.

• So we should be able to plotfor each isobologram contour, where p1 is the

proportion of drug 1.

( ) ( )1 2

1 21 250 50

andd dr rD D

= =

50% isobologramcontour line

Drug 1 dose

Drug 2 dose

( )150D

( )250D

1d

2d

1 1 2 1vs and vsr . p r . p

33

Nonlinear Blending Plots

log Total Dose

Nor

mal

ized

Mea

n R

espo

nse

0.2

0.4

0.6

0.8

1.0

-3 -2 -1 0 1


-3 -2 -1 0 1

prop. of TMQ=0.3


0.2

0.4

0.6

0.8

1.0prop. of TMQ=0.6

0.2

0.4

0.6

0.8

1.0prop. of TMQ=0.7

-3 -2 -1 0 1


Key: Black line = AG2034 alone, Gray Line = TMQ alone, Red line= combination. Note that forTMQ proportions of 0.7, 0.8, and 0.9 the red and gray lines virtually overlap.

Combinations of TMQ and AG2034 in the presence of high folic acid

Proportion of TMQ

Nor

mal

ized

Mea

n R

espo

nse


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0


0.0 0.2 0.4 0.6 0.8 1.0


0.0 0.2 0.4 0.6 0.8 1.0

34

Dose Reduction Profile Plots

Proportion of TMQ

Rat

io o

f Com

bina

tion

Dos

e to

Sin

gle-

Dru

g D

ose

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Isobol value=0.2 Isobol value=0.3

0.0 0.2 0.4 0.6 0.8 1.0

Isobol value=0.4

Isobol value=0.5 Isobol value=0.6

0.0

0.2

0.4

0.6

0.8

1.0Isobol value=0.7

0.0

0.2

0.4

0.6

0.8

1.0Isobol value=0.8

0.0 0.2 0.4 0.6 0.8 1.0

Isobol value=0.9

.

Dose reduction profile plots (high folic acid experiment).The gray line (TMQ) is the ratio

while the black line (AG2034) is the ratio TMQ TMQ TMQ/ ,r d D=

AG2034 2034 2034/ .AG AGr d D=

Combinations of TMQ and AG2034 in the presence of high folic acid

• Chou & Chou (1988) and Chou (2006)propose a “dose reduction index” equal to and

• But they only plot or vs. the corresponding isobol values fora fixed proportion of one of the drugs.

• Peterson, J. J. (2010) "A Review of SynergyConcepts of Nonlinear Blending and Dose-ReductionProfiles“, Frontiers of Bioscience. S2, 483-503

TMQ TMQ/ 1 / TMQD d r=

AG2034 AG2034 2034/ 1 / AGD d r=

1TMQr− 1

2034AGr−

35

Some Take-Home Messages for Combination Drug Synergy• “Excess over highest single agent” (EOHSA) can be a good first hurdle for combination drug screening.

• Nonlinear blending can be applied no matter what the shape of the doseresponse surface. Issues involving partial inhibitors, potentiation, or coalism pose no problem. “Potentiation” = one compound has little or no efficacy by itself.“Coalism” = both compounds have little or no efficacy by themselves.(e.g. Augmentin applied to amoxicillin resistant bacteria)

• Strong nonlinear blending implies:(i) A synergy of potency (ii) A response synergy (iii) Loewe synergy (if it exists)

• The combination index may be less than 1 but the drugs may notachieve a synergy of potency. This may happen if the compoundshave very different relative potencies.

• The combination index confounds information about the ‘dose reduction ratios’. As such, it may be better to examine these ratios individually.

36

Some ReferencesChou, J. H. and Chou, T. C. (1988) Computerized simulation of dose reduction index (DRI) in synergistic drug combinations. Pharmacologist 30:A231.

Loewe, S. (1953), "The Problem of Synergism and Antagonism of Combined Drugs", Arzeim. Forsch. 3, 285-290.

Peterson, J. J. (2006), “Testing for Excess Over Highest Single Agent with an Application to High-throughput Screening of Pairs of Compounds”, GSK BDS Technical Report 2006-04, available at: http://biometrics.com/?p=128

Peterson, J. and Novick, S. (2007), “Nonlinear Blending: A Useful General Concept for the Assessment of Combination Drug Synergy”, Journal of Receptors and Signal Transduction, 27,125-146.

Peterson, J. J. (2010) "A Review of Synergy Concepts of Nonlinear Blending and Dose-Reduction Profiles“, Frontiers of Bioscience. S2, 483-503, January issue. (Acknowledgements to Bill Greco for his editorial work on this FBS special issue.)

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