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In toxicodynamics and pharmacodynamics, Loewe additivity (or dose additivity) is one of several common reference models used for measuring the effects of drug combinations.[1][2][3]
Let [math]\displaystyle{ d_1 }[/math] and [math]\displaystyle{ d_2 }[/math] be doses of compounds 1 and 2 producing in combination an effect [math]\displaystyle{ e }[/math]. We denote by [math]\displaystyle{ D_{e1} }[/math] and [math]\displaystyle{ D_{e2} }[/math] the doses of compounds 1 and 2 required to produce effect [math]\displaystyle{ e }[/math] alone (assuming this conditions uniquely define them, i.e. that the individual dose-response functions are bijective). [math]\displaystyle{ D_{e1}/D_{e2} }[/math] quantifies the potency of compound 1 relatively to that of compound 2.
[math]\displaystyle{ d_2 D_{e1}/D_{e2} }[/math] can be interpreted as the dose [math]\displaystyle{ d_2 }[/math] 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 [math]\displaystyle{ d_1 + d_2 D_{e1}/D_{e2} = D_{e1} }[/math] or [math]\displaystyle{ d_1 / D_{e1} + d_2/D_{e2} = 1 }[/math].
Geometrically, Loewe additivity is the situation where isoboles are segments joining the points [math]\displaystyle{ (D_{e1},0) }[/math] and [math]\displaystyle{ (0,D_{e2}) }[/math] in the domain [math]\displaystyle{ (d_1,d_2) }[/math].
If we denote by [math]\displaystyle{ f_1(d_1) }[/math], [math]\displaystyle{ f_2(d_2) }[/math] and [math]\displaystyle{ f_{12}(d_1,d_2) }[/math] the dose-response functions of compound 1, compound 2 and of the mixture respectively, then dose additivity holds when
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][clarification needed]
Original source: https://en.wikipedia.org/wiki/Loewe additivity.
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