A reference model for the combination of an arbitrary number of drugs: A generalization of the Bliss independence model

Abstract It is commonplace to determine the effectiveness of the combination of drugs by comparing the observed effects to a reference model that describes the combined effect under the assumption that the drugs do not interact. Depending on what is to be understood by non-interacting behavior, several reference models have been developed in the literature. One of them is the celebrated Bliss independence model, which assimilates non-interaction with statistical independence. Intuitively, this requires the dose-response curves to have zero as minimal effect and one as maximal effect, a restriction that was indeed adopted by Bliss. However, we show how non-interaction can be interpreted in terms of statistical independence, while nevertheless allowing arbitrary values for the minimal and the maximal effect. Furthermore, our reference model allows the maximal effects of the dose-response curves to be different. In a first step, we construct a basic reference model for the case of two drugs and where the maximal effects of the two individual dose-response curves are assumed to be equal. By relying on the notion of non-interaction in terms of statistical independence, and by introducing two consistency principles, we show how a unique reference model can be derived. In a second step, a more general reference model, allowing the maximal effects to be different while still restricting to two drugs, is then easily constructed from the basic reference model. Finally, an induction step is applied to generalize the reference model to the case of an arbitrary number of drugs, allowing each dose-response curve to have a possibly different maximal effect. Although the minimal effect of the dose-response curves are restricted to be equal, which we show to be a necessary consequence of consistency rules, its value is arbitrary. Author summary The Bliss independence model is a very popular reference model for drug combinations, meaning that it predicts the combined effect of doses of given drugs under the assumption of non-interaction between these drugs. However, because Bliss described non-interaction as statistical independence, he thought that he had to assume that the minimal effect of all dose-response curves are zero, while the maximal effect of all dose-response curves are one. While it is acceptable that all dose-response curves have minimal effect zero, because this amounts to having a common reference state (i.e. the response when no drug at all is given), it is a severe restriction to force all dose-response curves to have maximal effect one. On the other hand, the Bliss independence model has the advantage that it relies on sound statistical theory, and the assimilation of non-interaction with statistical independence is rather intuitive. We have extended the Bliss independence model to allow the involved dose-response curves to have different maximal effects. This has been done in a rigorous way, where the statistical underlying theory that was used by Bliss remains essentially intact.