# Loewe additivity

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

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

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

## References

• William R Greco, Gregory Bravo, and John Parsons "The Search for Synergy: A Critical Review from a Response Surface Perspective", Pharmacological Reviews 47(2) 331-385 (1995)
• Loewe and Muischnek. "Effect of combinations: mathematical basis of the problem" Arch. Exp. Pathol. Pharmakol. 114: 313-326, 1926
• Pamela J. Yeh, Matthew J. Hegreness, Aviva Presser Aiden & Roy Kishony "Nature Reviews Microbiology" 7, 460-466 (June 2009)
• Tang, J., K. Wennerberg, and T. Aittokallio. "What Is Synergy? The Saariselkä Agreement Revisited." Frontiers in Pharmacology 6. Frontiers: 181. (2015)