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CombiTool - Terminology and Definitions
Criteria for calculating zero interaction combination effects
Two widely used methods for calculating the expected combination effect for the case of no interaction from single-agent effects are dose-additivity and independence. Both models are also recommended by the so-called Saariselkä agreement (Greco et al., 1992). The terminology proposed by these authors is Loewe additivity for dose-additivity and Bliss independence for the independence criterion. The cases for which the combination effects deviate from the zero interaction effect are then called Loewe/Bliss synergism/antagonism. This terminology avoids the difficulties which usually arise when the terms synergism or antagonism are used without stating explicitly according to which criterion the evaluation of the combination experiment was performed.
A variety of particular methods can be traced back to these basic approaches. To give only one example, the median-effect approach developed by Chou and Talalay is identical to Loewe additivity for the case of mutually exclusive agents and bears resemblance to Bliss independence for mutually non-exclusive agents (Chou and Talalay, 1984).
So far, there is no generally accepted agreement which of the two models is more appropriate. In addition to personal views the application of one or the other methods seems to be dependent on the particular field of biomedical research. For example, in radiation research the independence criterion is widely used (Suzuki, 1994). In other fields, Loewe additivity has been called the ‘gold standard’ (Gebhart, 1992).
CombiTool adopts both criteria for calculating the expected effect of a combination for the case of no (zero) interaction.
Bliss independence
The independence criterion (Bliss independence) was originally derived from probability theory. In this case the expected effect for a combination of two or three agents can be calculated from the single-agent effects EA(dA), EB(dB) by
EBIAB = EA + EB - EA EB
EBIAB = EA + EB + EC - EA EB - EA EC - EB EC + EA EB EC
where the dependence of the effects on the doses dA and dB of agents A and B is not written out for the sake of brevity. Note, that this criterion only be applied to fractional effects 0<E<1. In this case the effect can simply be replaced by the fractional survival S using E=1-S.
Loewe additivity
The dose-additivity criterion (Loewe additivity) defines zero interaction by
(dA / DA ) + (dB / DB ) + (dC / DC ) + .... = I
I > 1 indicates Loewe antagonism and I < 1 Loewe synergism. This is only correct, however, for monotonic dose-response curves. For non-monotonic relations the interpretation is more involved (Sühnel, 1992a). The equation can also applied to cases where one agent does not exhibit an effect when used alone. In this case DA or DB is set to infinity.
It is obvious from these equations that the Bliss independence criterion is defined in terms of effects but the Loewe additivity criterion in terms of doses. This difference has hampered a thorough comparison of the two criteria. We have, therefore, developed an approach which allows for the treatment of both criteria on an equal footing. In the equations decribing Bliss independence single-agent dose-response relations can easily be inserted which leads finally to an expression for the zero-interaction response surface EBI(dA,dB). The same procedure was not possible for Loewe additivity. One can, however, combine response-surface modeling techniques with the Loewe additivity criterion. This leads then also to mathematical expressions for the Loewe additivity zero interaction response surface ELA(dA,dB) (Sühnel, 1990, 1992a,b, 1993, 1997 in press).
Response surface approach
Recently, it has been pointed out that the response surface perspective offers a good possibility to compare various approaches for combined-action assessment (Greco et al, 1995). Response surfaces have really a variety of conceptual and practical advantages. For example, widely used experimental designs like the variation of the dose of one agent in the presence of a fixed amount of a second agent or the simultaneous variation of two doses keeping the dose ratio fixed, however, represent nothing more than cross sections through the response surface and are thus particular cases of a more general treatment. However, the reliable fitting of response surfaces to combination data is often not possible because of an inappropriate experimental design or simply because of too few data. On the other hand, the calculation of zero interaction response surfaces requires only the knowledge of single-agent dose-response relations. Any statistically significant deviation from the surface indicates an interaction. Note, however, that zero interaction response surfaces cannot predict interactive combination effects. Zero interaction response surfaces are also useful for a comparison with mechanistic models.
Calculation of zero interaction response surfaces from single-agent dose-response relations
For Bliss independence the expected zero interaction combination effect for two or three agents can be easily calculated from the single-agent effects EA(dA), EB(dB), EC(dC) from the equations defining Bliss independence. The usual procedure of applying the Loewe additivity criterion is either to calculate the index of interaction I according to equation given above or to plot an isobologram. An isobologram is a two-dimensional graph with the doses of agents A and B as coordinate axes, in which one or several lines, the isoboles, are shown connecting different dose combinations which all produce the same magnitude of effect. This means that the Loewe additivity isoboles (I=1) are straight lines in an isobologram with linear dose scales touching the dose axes at the single-agent doses DA and DB. To the best of our knowledge an isobologram, and thus indirectly the Loewe additivity definition equation , was first used in a paper by Fraser (1870/1871). Later this method was especially advocated by Loewe (1953). A detailed derivation of the definition equation starting out from empirical monotonic dose-response relations was given by Berenbaum (1989). We have recently reinforced his arguments and shown that the Loewe additivity method can be applied to non-monotonic dose-response relations as well (Sühnel, 1993). For the non-monotonic case, however, the usual interpretation of I<1 as synergism and I>1 as antagonism has to be reversed in passing from the increasing to the decreasing part of the non-monotonic dose-response relation.
Power function dose-response relation
We have combined the Loewe additivity definition equation with various single agent dose-response relations and this leads to equations which describe the dose-dependence of the effect under the assumption that the Loewe additivity criterion has to be fulfilled. The approach is to be illustrated for the power function relation. In the following a, b, s, µ indicate parameters of dose-response relations. We do not mention the mechanistic implication of these parameters but simply use them as empirical quantities.
E(d) = (d/a)µ
This equation can be recast after d. With the the assumption
E(dA,dB) = E(DA) = E(DB)
this leads to
{dA/[aA(ELAAB)1/µA]} + {dB/[aB(ELAAB)1/µB]} = 1
This is an implicit equation which has to solved by iteration. For µA=µB=µ the equation simplifies to
ELAAB = [(dA/aA) + (dB/aB)]µ = [(EA)1/µ + (EB)1/µ]µ (µA=µB=µ)
A further simplification is obtained for µA=µB=1 (linear relation). In this case
ELAAB = (dA/aA) + (dB/aB) = EA + EB (µA=µB=1)
is obtained. For a variety of dose-response relations the equations for the Loewe additivity zero interaction response surfaces are given in the following:
Median-effect dose-response relation
E(d) = dµ/(aµ + dµ)
dA/{aA[ELAAB/(1 - ELAAB)]1/µA } + dB/{aB[ELAAB /(1 - ELAAB)]1/µB} = 1
ELAAB = [(dA/aA) + (dB/aB)]µ / {1 + [(dA/aA) + (dB/aB)]µ} (µA=µB=µ)
Weibull dose-response relation
E(d) = 1 - exp[-(a d)µ]; S(d) = exp[-(a d)µ]
S is the survival fraction, which is related to the effect E by S=1-E. Note that S is always restricted to the range 0<S<1.
a AdA/(- ln SLAAB)1/µA + aBdB/(- ln SLAAB)1/µB = 1
SLAAB = exp[-(aAdA + aBdB)µ] (µA=µB=µ)
Logistic dose-response relation
E(d) = 1 - {1/[1 + (d/s)µ]}
{(dA /sA) / [ELAAB/(1 - ELAAB)]1/µA } +
{(dB /sB) / [ELAAB/(1 - ELAAB)]1/µB } = 1 [16]
ELAAB = 1 / {1 + [(dA/sA) + (dB/sB)]µ} (µA=µB=µ)
Linear-quadratic survival function
E(d) = 1 - exp( -ad - bd2); S(d) = exp( -ad - bd2)
2bAdA/[(a2A - 4bAlnSLAAB)1/2 - aA] +
2bBdB/[(a2B - 4bBlnSLAAB)1/2 - aB] = 1
SLAAB = exp[ -a (dA + dB) - b ( dA + dB)2 (aA = aB; bA = bB)
References
Berenbaum, M.C. What is synergy? Pharmacol. Rev. 41, 93 (1989). [PubMed] [Journal] [PDF]
Gebhart, G. F. Comments on the isobole method for analysis of drug interactions. Pain 51, 381 (1992). [Pubmed] [Journal] [PDF]
Greco, W., Unkelbach, H.-D., Pöch, G., Sühnel, J., Kundi, M. and Bödeker, W. Consensus on concepts and terminology for combined-action assessment: The Saariselkä Agreement. Arch. Complex Environmen. Studies 4, 65 (1992a). [PDF]
Comment:
The preceding paper is the original citation for the so-called Saariselkä Agreement including the Loewe Additivity and Bliss Indepence terminology and should thus be used when referring to the corresponding terminology or concepts. Unfortunately, the journal in which this paper was published is no longer in existence. So, understandably in many papers on combined action of biologically active agents the following reference is given. This fails, however, to give appropriate credit to all the authors of the Agreement paper.
Check out the number of citations according to Google Scholar.
Greco, W. R., Bravo, O. and Parsons, J.C. The search for synergy: A critical review from a response surface perspective. Pharmacol. Rev. 47, 331 (1995). [PubMed] [Journal] [PDF]
Chou, T.-C. and P. Talalay. Quantative analysis of dose-effect relationships: The combined effects of multiple drugs or enzyme inhibitors. Adv. Enzyme Regul. 22, 27 (1984). [PubMed] [Journal] [PDF]
Sühnel, J. Evaluation of synergism or antagonism for the combined action of antiviral agents. Antiviral Res. 13, 23 (1990). [PubMed] [Journal] [PDF]
Sühnel, J. Zero interaction response surfaces, interaction functions and difference response surfaces for combinations of biologically active agents. Arzneimittelforschung 42, 1251 (1992). [PubMed] [PDF]
Sühnel, J. Assessment of interaction of biologically active agents by means of the isobole approach: Fundamental assumptions and recent developments. Arch. Complex Environmen. Studies 4, 35 (1992b). [PDF]
Sühnel, J. Evaluation of interaction in olfactory and taste mixtures. Chem. Senses 18, 131 (1993). [Journal] [PDF]
Sühnel, J. Zero-interaction response surfaces for combined-action assessment. Food Chemical Toxicol. 34, 1151 (1997). [PubMed] [Journal] [PDF]
Sühnel, J. Parallel dose-response curves in combination experiments. Bull. Math. Biol. 60, 197 (1998) [Journal] [PDF]
Fraser, T. R. An experimental research on the antagonism between the actions of physostigma and atropia. Proc. Roy. Soc. Edinburgh 7, 506 (1870-1871) [PDF]
Loewe, S. The problem of synergism and antagonism of combined drugs. Arzneimittelforsch. 3, 285 (1953). [PDF]
See also:
Groten, J., Feron, V. and Sühnel, J. Toxicology of simple and complex mixtures. Trends Pharmacol. Sci. 22, 316-22 (2001). [Pubmed] [Journal] [PDF]