1 Your Combination Drug Study – What the Loewe Interaction Index is Not Telling You John Peterson [email protected]Research Statistics Unit November 17th, 2009 Ray of constant dose ratio. 50% contour line (isobologram) d d A + = 1 2 dose d 1 dose d 2 * d 1 * d 2 A 0 0 ( 29 ED 1 50 Ray of constant dose ratio. 50% contour line (isobologram) dose d 1 dose d 2 * d 1 * d 2 A A A ( 29 ED 1 50 ( 29 ED 2 50 ( 29 ED 2 50 0 0 Ray of constant dose ratio. 50% contour line (isobologram) d d A + = 1 2 dose d 1 dose d 2 * d 1 * d 2 A 0 0 ( 29 ED 1 50 Ray of constant dose ratio. 50% contour line (isobologram) Ray of constant dose ratio. 50% contour line (isobologram) dose d 1 dose d 2 * d 1 * d 2 A A A ( 29 ED 1 50 ( 29 ED 2 50 ( 29 ED 2 50 0 0 1 0 1 ( 29 1 1 1 50 d r ED = ( 29 2 2 2 50 d r ED = Proportion of TMQ Normalized Mean Response Total dose = 0.005 μ M 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 = 0.01 μ M 0.0 0.2 0.4 0.6 0.8 1.0 Total dose = 0.015 μ M 0.0 0.2 0.4 0.6 0.8 1.0 Total dose = 0.02 μ M 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 = 0.025 μ M 0.0 0.2 0.4 0.6 0.8 1.0 Total dose = 0.03 μ M 0.0 0.2 0.4 0.6 0.8 1.0 Total dose = 0.035 μ M 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 = 0.04 μ M 0.0 0.2 0.4 0.6 0.8 1.0 Total dose = 0.045 μ M 0.0 0.2 0.4 0.6 0.8 1.0
Transcript
1
Your Combination Drug Study – What the Loewe Interaction Index is Not Telling You
1. Why is Combination Drug Product DevelopmentPotentially Useful?
2. Brief synergy overview
3. What the Combination Index is not Telling You.
4. Nonlinear Blending
5. Dose Reduction Profiles
6. References
3
Why is Combination Drug Product DevelopmentPotentially Useful?
• There is growing interest in the pharmaceutical industry in the discovery and development of combination drug products.
• This is due to the flexibility a combination drug product offers in developingstrategies to treat a disease.
• For example - A combination drug product (with low doses of each drug) may achieve a desired level of efficacy with a low side effects profile if each compound is associated with biologically different and independent side effects.
- A disease may have two biological pathways which each of which can be blocked by a different drug compound (Keith et al, 2005, Nature Reviews - Drug Discovery).
- Improved kill rates for infectious agents such as bacteria and viruses. - Improved kill rates for cancer cells.
- Treating multiple aspects of a disease (e.g. bronchoconstriction and inflammation in asthma)
4
What do we mean by “synergy”?
On-line Medical dictionary:Synergy: The interaction of two or more treatments such that their combined effect is greater than the sum of the individual effects observed when each treatment is administered alone.
AIDS research Web site:Synergy: the action of two or more drugs working together to produce an effect greater than the expected combined effect of the same agents used separately.
Random House Dictionary:Synergy: the joint action of agents, as drugs, that when taken together increase each other's effectiveness
One more defintion…
5
What do we mean by “synergy”?
On-line Medical dictionary:Synergy: The interaction of two or more treatments such that their combined effect is greater than the sum of the individual effects observed when each treatment is administered alone.
AIDS research Web site:Synergy: the action of two or more drugs working together to produce an effect greater than the expected combined effect of the same agents used separately.
Random House Dictionary:Synergy: the joint action of agents, as drugs, that when taken together increase each other's effectiveness
Dunkin’ Donuts definition:
6
Some Basic Synergy Quantification Methods
1. Bliss Independence (Bliss, 1939)
2. Loewe synergy (Loewe, 1928)
3. Strong nonlinear blending (Peterson and Novick, 2007)
7
Bliss Independence model
Both drugs act independently against different targets.
Let f1 = fraction of possible response from compound 1.Let f2 = fraction of possible response from compound 2.Let f12= fraction of possible response from combination.
Bliss model quantifications:f12 = f1 + f2 - f1×f2. Bliss independence
f12 > f1 + f2 - f1×f2. Bliss synergy
f12 < f1 + f2 - f1×f2. Bliss antagonism
Bliss (1939) Annals of Applied Biology
8
Bliss independence model limitations
• Has problems when viewed from dose-response continuum perspective
• Example: Consider amount ½A for drug 1 and amount ½A for drug 2.• Suppose that
– f1(½A) = f2(½A) = .1
– f12(½A, ½A) = .4 > .1 + .1 - .1 × .1 = .19.
• The combination (½A, ½A) exhibits Bliss synergy.
• What if f1(A) = f2(A) = .6 > .4 = f12(½A, ½A) ?
• In this case one has to admit that at total amount, A, either drug 1 or drug 2 alone produces a better response than the combination amount, A, of drugs 1 and 2 together, despite Bliss synergy.
9
Loewe SynergyIsobolograms and Combination Index
(A,B) combination50% contour line (isobologram)
( ) ( )d d
IED ED
+ = <1 21 2
50 50
1
( ),d d1 2
( )ED2
50
( ) ( )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
( )150ED d1
d2
10
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 IC50 than 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?
11
Loewe SynergyIsobolograms and Combination Index
(A,B) combination50% contour line (isobologram)
( ) ( )d d
IED ED
+ = <1 21 2
50 50
1
( ),d d1 2
( )ED2
50
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+ =
12
Ray of constant dose ratio.
50% contour line(isobologram)
d d A+ =1 2
dose d1
dose d2
*d1
*d2
A00
( )ED1
50
Ray of constant dose ratio.
50% contour line(isobologram)
dose d1
dose d2
*d1
*d2
A
A
A
( )ED1
50
( )ED2
50( )ED2
50
00
Ray of constant dose ratio.
50% contour line(isobologram)
d d A+ =1 2
dose d1
dose d2
*d1
*d2
A00
( )ED1
50
Ray of constant dose ratio.
50% contour line(isobologram)
Ray of constant dose ratio.
50% contour line(isobologram)
dose d1
dose d2
*d1
*d2
A
A
A
( )ED1
50
( )ED2
50( )ED2
50
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 enough synergy to overcome the “dilution effect”?
Here I<1 Here I<1
13
Ray of constant dose ratio.
50% contour line(isobologram)
d d A+ =1 2
dose d1
dose d2
*d1
*d2
A00
( )ED1
50
Ray of constant dose ratio.
50% contour line(isobologram)
dose d1
dose d2
*d1
*d2
A
A
A
( )ED1
50
( )ED2
50( )ED2
50
00
Ray of constant dose ratio.
50% contour line(isobologram)
d d A+ =1 2
dose d1
dose d2
*d1
*d2
A00
( )ED1
50
Ray of constant dose ratio.
50% contour line(isobologram)
Ray of constant dose ratio.
50% contour line(isobologram)
dose d1
dose d2
*d1
*d2
A
A
A
( )ED1
50
( )ED2
50( )ED2
50
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
14
• Suppose that– ED501=1 and ED502=50
– (d1=.1, d2=9.9) Y = 50% response
• The interaction index I = 0.298 (Strong “Synergy”)
• A (fixed-dose ratio) combination with a synergistic interaction index can have larger ED50 (total dose) than the more potent monotherapy.
I==+ 298.50
9.9
0.1
1.0
Loewe Combination Index vs. Combination Total Dose
15
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 d
IED ED
= +1 21 2
50 50
( ) ( )andd d
ED ED
1 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 ED
1 21 2
50 50
• The combination index confounds information about the dose-reduction ratios!
16
Plotting Dose Reduction Ratios for Various Drug Combinations
Dose Reduction Ratios for different combinations
1
0 1
( )1
1 150
dr
ED=
( )2
2 250
dr
ED=
Note that the blue dots correspond to combinations with the same interaction index value.
17log( Conc )
Re
spo
nse
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!
18
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 of substances for fixed total dose amounts. - Isobolograms, on the other hand, attempt to model the locus of compound-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.)
19
The Concept of Nonlinear Blending
• Mixture-amount experiments quantify the blending properties of substances 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 nonlinear blending
0% drug A 100% drug A
response
total dose=D total dose=D
20
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 nonlinear blending
0% drug A 100% drug A
response
total dose=D
A “slice” throughthe response surface for totaldose fixed at D
AX50
BX50
compound A
compound B
21
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!
22
Virology Example: Drugs AZT and FLG
Total dose=0.035
Total dose=0.07
Total dose=0.14
Total dose=0.28Total dose=0.56 Total dose=1.13
Total dose=2.25Total 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
Res
pons
e P
ropo
rtio
n
23
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 a 60% 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
24
In-vitro Cancer chemotherapy example
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
25
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 sometimes easier for scientists to understand.
26
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 0.04
TMQ
0.00
0.02
0.04
0.06
0.08
AG
203
4
0.2 0.3 0.4 0.5
0.6 0.7 0.8
0.9TMQ
AG
2034
Isobologram contours
27
An Example of Two Drugs with Combination Indices Much Less Than 1Yet Show No Synergy of Potency
• 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 0.04
TMQ
0.00
0.02
0.04
0.06
0.08
AG
203
4
0.2 0.3 0.4 0.5
0.6 0.7 0.8
0.9TMQ
AG
2034
28
An Example of Two Drugs with Combination Indices Much Less Than 1Yet Show No Synergy of Potency
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 µ M
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Total dose = 0.01 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.015 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.02 µ M
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Total dose = 0.025 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.03 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.035 µ M
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Total dose = 0.04 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.045 µ M
0.0 0.2 0.4 0.6 0.8 1.0
29log Total Dose
No
rma
lize
d M
ea
n R
esp
on
se
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
prop. of TMQ=0.4 prop. of TMQ=0.5
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
prop. of TMQ=0.8 prop. of TMQ=0.9
An Example of Two Drugs with Combination Indices Much Less Than 1Yet Show No Synergy of Potency
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.
30
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”. - Such a “response” could be a true response (e.g. blood pressure reduction) - Or, it may be more indirect (e.g. an increase in potency).
• On the other hand, no real synergy (on the response scale) may exist between two drugs but 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 and unrelated. 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.
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.
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 21 250 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.
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 21 250 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 21 250 50
andd d
.D D
33
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 plot
for each isobologram contour, where p1 is the
proportion of drug 1.
( ) ( )1 2
1 21 250 50
andd d
r rD D
= =
50% isobologramcontour line
Drug 1 dose
Drug 2 dose
( )150D
( )250D
1d
2d
1 1 2 1vs and vsr . p r . p
34
Nonlinear Blending Plots
log Total Dose
No
rma
lize
d M
ea
n R
esp
on
se
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
prop. of TMQ=0.4 prop. of TMQ=0.5
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
prop. of TMQ=0.8 prop. of TMQ=0.9
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.
Combinations of TMQ and AG2034 in the presence of high folic acid
Proportion of TMQ
Nor
mal
ized
Mea
n R
espo
nse
Total dose = 0.005 µ M
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Total dose = 0.01 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.015 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.02 µ M
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Total dose = 0.025 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.03 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.035 µ M
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Total dose = 0.04 µ M
0.0 0.2 0.4 0.6 0.8 1.0
Total dose = 0.045 µ M
0.0 0.2 0.4 0.6 0.8 1.0
35
Dose Reduction Profile Plots
Proportion of TMQ
Ra
tio o
f Co
mb
ina
tion
Do
se 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
1.0
Isobol value=0.7 0.0
0.2
0.4
0.6
0.8
1.0
Isobol 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.
TMQ TMQ/ 1 / TMQD d r=
AG2034 AG2034 2034/ 1 / AGD d r=
1TMQr− 1
2034AGr−
36
Some Take-Home Messages for Combination Drug Synergy
• 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
• The combination index may be less than 1 but the drugs may not achieve a synergy of potency. This may happen if the compounds have different relative potencies.
• The combination index confounds information about the ‘dose reduction ratios’. As such, it may be better to examine these ratios individually.
37
Some ReferencesBliss CI. The toxicity of poisons combined jointly. Ann Appl Biol 1939, 26, 585–615.
Chou, 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. 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. "A Review of Synergy Concepts of Nonlinear Blending and Dose-Reduction Profiles". (2010) Frontiers of Bioscience. (to appear)
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