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Edited by: Thomas J. Anastasio, University of Illinois at Urbana-Champaign, USA

Reviewed by: Jiang Liu, University of Southern California, USA; Murat Cokol, Sabanci University, Turkey

*Correspondence: Jing Tang, Krister Wennerberg and Tero Aittokallio, Institute for Molecular Medicine Finland (FIMM), University of Helsinki, Tukholmankatu 8, FI-00290 Helsinki, Finland

This article was submitted to Experimental Pharmacology and Drug Discovery, a section of the journal Frontiers in Pharmacology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Many biological or chemical agents when combined interact with each other and produce a synergistic response that cannot be predicted based on the single agent responses alone. However, depending on the postulated null hypothesis of non-interaction, one may end up in different interpretations of synergy. Two popular reference models for null hypothesis include the Bliss independence model and the Loewe additivity model, each of which is formulated from different perspectives. During the last century, there has been an intensive debate on the suitability of these synergy models, both of which are theoretically justified and also in practice supported by different schools of scientists. More than 20 years ago, there was a community effort to make a consensus on the terminology one should use when claiming synergy. The agreement was formulated at a conference held in Saariselkä, Finland in 1992, stating that one should use the terms Bliss synergy or Loewe synergy to avoid ambiguity in the underlying models. We review the theoretical relationships between these models and argue that one should combine the advantages of both models to provide a more consistent definition of synergy and antagonism.

Evaluation of interaction effects between biologically active agents has become an important topic in many disciplines, including pharmacology (Cokol et al.,

For the rest of the review, we will use drug combination as an example of interaction data. A drug's effect _{1} produces a response _{1} and drug 2 at dose _{2} produces a response _{2}. Next, we combine the two drugs at the dose pair (_{1}, _{2}) and observe a combination response _{c}. To quantify the degree of drug interaction, we need to formulate a reference model to answer the following question: if there is no interaction between the drugs, what would be the expected combination response _{EXP} at (_{1}, _{2})? So far, two major reference model classes have been proposed, the Bliss independence model (Bliss,

The Bliss independence model adopts a probabilistic perspective by treating a drug combination under non-interaction as a joint action of independent, yet competing perturbations by the individual drugs. Such a probabilistic independence allows the expected combination response to be computed as the product of the individual drug responses:
_{BLISS} can be interpreted as a departure from the probabilistic independence, which thus implies an interaction between the two drugs. The Loewe additivity model, on the other hand, requires additional information about the dose-response relationships of the individual drugs. Namely, let _{1}(_{2}(_{LOEWE} can be represented as, ^{−1} is an inverse function which maps the response _{LOEWE} must satisfy:
_{1}(_{2}(_{LOEWE} = _{1} + _{2}) for the sham experiment, reflecting the intuition that combining two drugs of the same type should induce neither synergy nor antagonism.

The assumptions and performance of the two reference models have been compared and discussed in many review articles (e.g., Berenbaum,

After reaching the common understanding on the model assumptions, the Saariselkä agreement allowed the researchers for the flexibility to choose a preferred reference model to evaluate interactions of multiple agents, with the only precondition that the names of the specific models need to be explicitly reported. Namely, depending on which model is used, a combination response greater or less than _{EXP} will be termed as Loewe synergy, Loewe antagonism, Bliss synergy or Bliss antagonism, respectively. Following these recommendations, the controversy over the definitions of synergy and antagonism seemed subsided. More recently, a variety of interaction assessment methods have been further developed and applied to a wide range of biological research fields. Notably, most of these methods can be traced back to the two basic model classes. For example, variants of the Loewe additive model include combination index (Lee et al.,

Paradoxically, even with the clear distinction that has been made between the reference models, we feel that the fundamental question still remains unanswered, if not becoming even more serious: What is synergy after all? Since the expected combination responses _{BLISS} and _{LOEWE} most often are not identical (Berenbaum,

To ease the model selection burden, we propose here the use of new terminology that incorporates both of the two reference models, together with the single drug responses, to distinguish non-interaction, synergy and antagonism. With simple algebra, one can show that max(_{1}, _{2}) ≤ _{BLISS}. For the Loewe additivity model with a monotonically increasing dose-response relation, one can also show that max(_{1}, _{2}) ≤ _{LOEWE}. We note that, max(_{1}, _{2}) is also the expected response from a popular reference model, called highest single agent (HSA) model (Berenbaum, _{c} is lower than max(_{1}, _{2}), then one would intuitively infer antagonism. Therefore, we may use max(_{1}, _{2}), to distinguish antagonism from non-interaction. Similarly, one can use the response of the less effective single drug, that is min(_{1}, _{2}), to further distinguish between weak and strong antagonisms. For distinguishing synergy from non-interaction the answer is less obvious, as it depends on the comparison between _{BLISS} and _{LOEWE}. There has been considerable interest in the mathematical relationships between the Bliss independence and the Loewe additivity models to understand how much difference in the characterization of drug interaction one can expect when choosing one model over another (see e.g., Goldoni and Johansson, _{LOEWE} > _{BLISS} is generally observed for very steep dose-response curves, while _{LOEWE} < _{BLISS} when the curves become more flat. Since, _{BLISS} and _{LOEWE} differ in a complex way depending on the parameterization of the dose-response functions, we propose two cut-offs, min(_{BLISS}, _{LOEWE}) and max(_{BLISS}, _{LOEWE}), for characterizing synergistic combinations. We reason that the consistency between the Bliss independence and the Loewe additivity models should be indicative of the degree of synergy: If both the Bliss model and the Loewe model classify a drug combination as synergistic, that is, _{c} > max(_{BLISS}, _{LOEWE}), then we call it a strong synergy; If the combination is classified as synergistic according to one model only, that is, min(_{BLISS}, _{LOEWE}) < _{c} < max(_{BLISS}, _{LOEWE}), then it is called weak synergy. Finally, non-interacting drugs have max(_{1}, _{2}) < _{c} < min(_{BLISS}, _{LOEWE}), reflecting our view that non-interaction should also be defined similarly as a range, rather than a single point as in the individual reference models. Given such a classification, one may continue to develop statistical testing methods for evaluation of its significance for replicate data. To facilitate better understanding of these definitions, we designed an interaction barometer that enables a systematic comparison of these proposed interaction terms along an axis of drug combination response _{c} (Figure

_{c} is lower than the maximum single drug effect max(_{1}, _{2}) but higher than the minimum effect min(_{1}, _{2}), then the combination is called weak antagonism; if _{c} < min(_{1}, _{2}) it is called strong antagonism. To classify synergy, we consider the Bliss and Loewe models, with the expected effects denoted as _{BLISS} and _{LOEWE}, respectively. If max(_{1}, _{2}) < _{c} < min(_{BLISS}, _{LOEWE}), then we call the combination as non-interaction; if _{HSA} < _{c} < min(_{BLISS}, _{LOEWE}) it is called weak synergy, and for _{c} > max(_{BLISS}, _{LOEWE}) strong synergy.

The benefits of adopting the proposed terminology for the degree of interaction are two-fold. First, the definitions of synergy and antagonism are based on a simultaneous evaluation of the two reference models, as well as the individual drug responses. Such a data-driven approach avoids any pre-defined preference either for the Bliss independence or Loewe additivity when characterizing drug interactions, and thus it minimizes the biases toward either of the models. This is consistent with the idea that any synergy model should be treated as an exploratory ranking statistic for prioritization of the most potent combinations for further evaluation, rather than a “true model” for explaining synergy or antagonism mechanisms. Further, this terminology enables a more intuitive definition of non-interaction, under which the combination response may be higher than the single drug response, but not as high as the expected responses from the Bliss independence and the Loewe additivity models. Note that a drug combination falling into such an interval would be classified as antagonistic according to both of the two models, but since it produces a higher response than the single drugs, one would rather characterize it as an additive effect or non-interaction. For the sake of clarity, we would call it non-interaction, and in fact discourage the terms additive or additivity since these may be confused with the additivity implicated by the Loewe additivity model. The interval of non-interaction shown in Figure

Secondly, the different interaction terms are positioned along the common response axis (e.g., measured as the percentage inhibitions of cell growth), which makes it easier to relate the degree of drug interaction with its outcome in the drug response. With the proposed interaction barometer (Figure _{c} and the responses of individual drugs (_{1}, _{2}), as well as the expected combination responses (_{BLISS}, _{LOEWE}) based on the two reference models. The clear correspondence between the degree of synergy and the combination response is in many ways superior to the use of an interaction index, such as combination index or other similar approaches (see e.g., Lee et al.,

So far, we have merely discussed about the assessment of drug interactions using the difference between the observed combination response and its theoretical expectation, i.e., _{c} − _{EXP}, to classify a drug combination as synergistic or antagonistic. However, a drug combination can be also classified as either effective or ineffective based solely on its actual combination response _{c}. It is important not to confuse these two concepts, synergy and efficacy, as the nomenclatures are related but should not be treated the same. Synergy is a measure of the degree of drug interaction, while efficacy is a measure of phenotypic response of a drug combination. It is possible that a drug combination is highly synergistic, while its actual response may be insufficient to reach therapeutic efficacy. On the other hand, a drug combination that exhibits strong response does not necessarily imply a synergistic interaction. For instance, only one of its component drugs may produce the response, while the other one is simply lowering the adverse effect of the first drug without affecting its on-target activity. In preclinical testing, a drug combination with strong synergy and efficacy should be prioritized for further mechanistic investigation, with an additional requirement of tolerable toxicity profile (Fitzgerald et al.,

The definition of synergistic interaction is still under debate. After a careful investigation of the Bliss independence model and the Loewe additivity model, we argue that, without jeopardizing the validity of both models, a more consistent terminology for classifying synergy and antagonism can be made. By comparing the observed combination response with the expected combination responses from the two models, as well as the single drug responses, one can classify the drug combination into five categories including strong antagonism, weak antagonism, non-interaction, weak synergy, and strong synergy. We propose the use of the interaction barometer to visualize the degree of interaction on the common axis of drug response, which has been shown to facilitate the interpretation and comparison between different combinations. We view our efforts as a continuation to what the Saariselkä agreement started more than 20 years ago but has not yet concluded: a consensus on concepts and terminology for interaction assessment. We acknowledge that our proposal is not yet solving the practical issues for analyzing real data which typically contain combination responses tested at different dose ranges. How to maximize the benefits of the interaction barometer to summarize the interaction patterns of a drug combination would be a source of future research initiatives. We hope that such a classification scheme will raise more discussions about the standardization of the interaction assessment, toward finally reaching a consensus not only on the definition itself, but also on the other important issues, such as the experimental design of combination experiments, their quality control and the statistical evaluation of synergy and antagonism.

This work was supported by the Academy of Finland (grants 272437, 269862, 279163, and 292611 for TA, 277293 for KW); Cancer Society of Finland (JT, TA, and KW).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.