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 {\displaystyle d_{1}} and d 2 {\displaystyle d_{2}} be doses of compounds 1 and 2 producing in combination an effect e {\displaystyle e} . We denote by D e 1 {\displaystyle D_{e1}} and D e 2 {\displaystyle D_{e2}} the doses of compounds 1 and 2 required to produce effect e {\displaystyle e} alone (assuming this conditions uniquely define them, i.e. that the individual dose-response functions are bijective).
D e 1 / D e 2 {\displaystyle D_{e1}/D_{e2}} quantifies the potency of compound 1 relatively to that of compound 2.
d 2 D e 1 / D e 2 {\displaystyle d_{2}D_{e1}/D_{e2}} can be interpreted as the dose d 2 {\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 d 1 + d 2 D e 1 / D e 2 = D e 1 {\displaystyle d_{1}+d_{2}D_{e1}/D_{e2}=D_{e1}} or
d 1 / D e 1 + d 2 / D e 2 = 1 {\displaystyle d_{1}/D_{e1}+d_{2}/D_{e2}=1} . Geometrically, Loewe additivity is the situation where isoboles are segments joining the points ( D e 1 , 0 ) {\displaystyle (D_{e1},0)} and ( 0 , D e 2 ) {\displaystyle (0,D_{e2})} in the domain ( d 1 , d 2 ) {\displaystyle (d_{1},d_{2})} . If we denote by f 1 ( d 1 ) {\displaystyle f_{1}(d_{1})} , f 2 ( d 2 ) {\displaystyle f_{2}(d_{2})} and f 12 ( d 1 , d 2 ) {\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
d 1 f 1 − 1 ( f 12 ( d 1 , d 2 ) ) + d 2 f 2 − 1 ( f 12 ( d 1 , d 2 ) ) = 1 {\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}
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