The most widely used combination reference is Loewe additivity1,2, or "dose additivity" which describes the trade-off in potency between two agents when both sides of a dose matrix contain the same compound. For example, if 50% inhibition is achieved separately by 1 uM of drug A or 1 uM of drug B, a combination of 0.5 uM of A and 0.5 uM of B should also inhibit by 50%. Synergy over this level is especially important when justifying the clinical use of proposed combination therapies, as it defines the point at which the combination can provide additional benefit over simply increasing the dose of either agent. Synergy relative to Loewe additivity can be graphically illustrated using an "isobologram"3, where experimental dose matrix data are used to draw a contour, as a function of the component concentrations X and Y, at which the combination achieves a chosen inhibition level I. Loewe additive combinations produce contours that are straight lines connecting the single agent effective concentrations XI and YI, and synergy occurs where the data contour falls between the additivity line and the origin. Mathematically, synergy can be measured in terms of a Combination Index4, CI = X/XI + Y/YI, or the total ratio of drug required in combination to achieve I over the corresponding single agent concentrations, where CI<1 for="" synergistic="">

Model Response Surface

Single agent responses at varying concentrations X and Y are shown along the bottom and left edges of the dose matrix, using colors that run from black (no effect) through the rainbow to pink/white (total effect). Here, the X-agent reaches ~60% and the Y-agent ~40% effect.

The response follows the closest single agent until concentrations where both are active, at which point the two agents help each other reach higher effects in the same way that adding more of one component helps itself. When plotted on a linear scale, equal-effect contours of a dose-additive combination are straight lines.


In Chalice, at each combined concentration (X,Y), an iterative approach5 is used to find the inhibition ILoewe that satisfies (X/XI) + (Y/YI) = 1, where XI and YI are the single agent effective concentrations for the observed combination effect I. Starting with a guess that I = IHSA, the single agent curves are interpolated to find XI,YI that produce I, the corresponding combination index is calculated, and bisection6 is used to converge on a value of ILoewe with combination index CI = 1. The single agent curve interpolations use ither: (a) a linear interpolation between single agent values, returning blank if data do not exist both for higher and lower non-zero single agent concentrations; or (b) the fitted sigmoidal curve to the single agent data (see SigmoidCurve description). By default, the fitted curves (option b) are used for the ADD model's calculation.

Limitations and Constraints

This reference model is appropriate for all effect measures, including raw values, inhibition, and fold increases. However, the ADD calculation as implemented in Chalice only works well when both agents have effects that are positive and monotonically increasing at all concentrations (excluding noise). Because the iterative solution to the dose additivity equation requires many calculations, the ADD model is computationally intensive, especially compared to the much faster HSA or Bliss models.

  1. Loewe (1928), Ergebn. Physiol. 27:47-187.
  2. Chou, Talalay (1981), Eur J Biochem 115:207-16.
  3. Greco, Bravo, Parsons (1995), Pharmacol Rev 47:331-85.
  4. Chou, Talalay (1984), Adv Enzyme Regul 22:27-55.
  5. Berenbaum (1985), J Theor Biol 114:413-31.
  6. Press et al. (1997), Numerical Recipes in C: the Art of Scientific Computing, 2nd Ed. Cambridge:CUPpp.