A foundation for reference models for drug combinations with an application to Loewe's reference model

Treating patients with combinations of drugs that have synergistic effects has become widespread practice in the clinic. Drugs work synergistically when the observed effect of a drug combination is larger than the effect predicted by the reference model. The reference model is a theoretical null model that returns the combined effect of given doses of drugs under the assumption that these drugs do not interact. There is ongoing debate on what it means for drugs to not interact. The controversy transcends mathematical punctuality, as different non-interaction principles result in different reference models. A famous reference model that has been in existence for already a long time is Loewe's reference model. Loewe's vision on non-interaction was purely intuitive: two drugs do not interact if all combinations of doses that result in a certain given effect lie on a straight line.

We show that Loewe's reference model can be obtained from much more fundamental principles. First, we introduce the new notion of complementary dose. Secondly, we reformulate the existing concept of equivalent dose, whereby our formulation is more general than existing ones. Finally, a very general non-interaction principle is put forward. The proposed non-interaction principle represents a certain interplay between complementary and equivalent doses: drugs are non-interacting if complementarity is preserved under equivalence. It is then shown that Loewe's reference model naturally follows from these principles by an appropriate choice of complementarity.

The presented work increases insight into Loewe's reference model for drug combinations, which is realized by the introduction of a very general non-interaction principle that does not refer to any specific dose-response curve, nor to any property of applicable dose-response curves.