In toxicodynamics and pharmacodynamics, Loewe additivity (or dose additivity) is one of several common reference models used for measuring the effects of drug combinations.

Definition

Let d_1 and d_2 be doses of compounds 1 and 2 producing in combination an effect e. We denote by D_{e1} and D_{e2} the doses of compounds 1 and 2 required to produce effect e alone (assuming this conditions uniquely define them, i.e. that the individual dose-response functions are bijective). quantifies the potency of compound 1 relatively to that of compound 2. can be interpreted as the dose d_2 of compound 2 converted into the corresponding dose of compound 1 after accounting for difference in potency. Loewe additivity is defined as the situation where or. Geometrically, Loewe additivity is the situation where isoboles are segments joining the points (D_{e1},0) and (0,D_{e2}) in the domain (d_1,d_2). If we denote by f_1(d_1), f_2(d_2) and the dose-response functions of compound 1, compound 2 and of the mixture respectively, then dose additivity holds when

Testing

The Loewe additivity equation provides a prediction of the dose combination eliciting a given effect. Departure from Loewe additivity can be assessed informally by comparing this prediction to observations. This approach is known in toxicology as the model deviation ratio (MDR). This approach can be rooted in a more formal statistical method with the derivation of approximate p-values with Monte Carlo simulation, as implemented in the R package MDR.