Loewe additivity

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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]

## Definition

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

[math]\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 }[/math]

## 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).[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]

## References

1. ↑ Greco, W.R.; Bravo, G.; Parsons, J. (1995). "The Search for Synergy: A Critical Review from a Response Surface Perspective". Pharmacol. Rev. 47 (2): 331–385. PMID 7568331.
2. ↑ Loewe, S. (1926). "Effect of combinations: mathematical basis of problem". Arch. Exp. Pathol. Pharmakol. 114: 313–326. doi:10.1007/BF01952257.
3. ↑ Tang, J.; Wennerberg, J.K.; Aittokallio, T. (2015). "What Is Synergy? The Saariselkä Agreement Revisited.". Frontiers in Pharmacology 6: 181. doi:10.3389/fphar.2015.00181. PMID 26388771.
4. ↑ Belden, J. B.; Gilliom, R.; Lydy, M.J. (2007). "How well can we predict the toxicity of pesticide mixtures to aquatic life?". Integr. Environ. Assess. Manag. 3 (3): 364–72. doi:10.1002/ieam.5630030307. PMID 17695109.
5. ↑ "Github development repository for the R package MDR". 2020-01-20. https://github.com/gilles-guillot/MDR.

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