Loewe additivity

Let ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$ be doses of compounds 1 and 2 producing in combination an effect ${\displaystyle e}$. We denote by ${\displaystyle D_{e1}}$ and ${\displaystyle D_{e2}}$ the doses of compounds 1 and 2 required to produce effect ${\displaystyle e}$ alone (assuming this conditions uniquely define them, i.e. that the individual dose-response functions are bijective). ${\displaystyle D_{e1}/D_{e2}}$ quantifies the potency of compound 1 relatively to that of compound 2.

${\displaystyle d_{2}D_{e1}/D_{e2}}$ can be interpreted as the dose ${\displaystyle d_{2}}$ 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 ${\displaystyle d_{1}+d_{2}D_{e1}/D_{e2}=D_{e1}}$ or ${\displaystyle d_{1}/D_{e1}+d_{2}/D_{e2}=1}$.

Geometrically, Loewe additivity is the situation where isoboles are segments joining the points ${\displaystyle (D_{e1},0)}$ and ${\displaystyle (0,D_{e2})}$ in the domain ${\displaystyle (d_{1},d_{2})}$.

If we denote by ${\displaystyle f_{1}(d_{1})}$, ${\displaystyle f_{2}(d_{2})}$ and ${\displaystyle f_{12}(d_{1},d_{2})}$ the dose-response functions of compound 1, compound 2 and of the mixture respectively, then dose additivity holds when

${\displaystyle {\frac {d_{1}}{f_{1}^{-1}(f_{12}(d_{1},d_{2}))}}+{\frac {d_{2}}{f_{2}^{-1}(f_{12}(d_{1},d_{2}))}}=1}$

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).[4]

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.[5]