Emax model and interaction index for assessing drug interaction in combination studies

Abstract

Applying the Emax model in a Lowe additivity model context, we analyze data from a combination study of trimetrexate (TMQ) and AG2034 (AG) in media of low and high concentrations of folic acid (FA). The Emax model provides a sufficient fit to the data. TMQ is more potent than AG in both low and high FA media. At low TMQ:AG ratios, when a smaller amount of the more potent drug (TMQ) is added to a larger amount of the less potent drug (AG), synergy results. When the TMQ:AG ratio reaches 0.4 or larger in low FA medium, or when the TMQ:AG ratio reaches 1 or larger in high FA medium, synergy is weakened and drug interaction becomes additive. In general, synergistic effect in a dilution series is stronger at higher doses that produce stronger effects (closer to 1-Emax) than at lower dose levels that produce weaker effects (closer to 1). The two drugs are more potent in the low compared to the high FA medium. Drug synergy, however, is stronger in the high FA medium.

Figure 1

Dose-response curves under the E_max model by varying the parameters E_max, ED₅₀, and m.

Figure 2

Variable width percentile plot for the observed effect in experiments with low and high folic acid media. Each vertical bar indicates a five percent increment. The middle 20% of the data are shaded in a light orange color.

Figure 3

Experimental design showing the logarithmically transformed AG2034 (AG) dose versus the logarithmically transformed trimetrexate (TMQ) dose in the fixed ratio experiments. 16 curves are shown. Curves A and B are controls; no drugs applied. Curves C and D are single-drug studies for TMQ and AG, respectively. Curves E through P are the combination drug studies. Each curve has 11 dilutions shown in circles. Panel A: low folic acid medium. Panel B: high folic acid medium.

Figure 4

Distribution of the effect versus dose level for curves A through P for the experiment in a low folic acid medium.

Figure 5

Distribution of the effect versus dose level for curves A through P for the experiment in a high folic acid medium.

Figure 6

Perspective plots (A, B, C), contour plots (D, E), and image plot (F) for the effect versus logarithm transformed doses of trimetrexate and AG2034 for the experiment in a low folic acid medium.

Figure 7

Perspective plots (A, B, C), contour plots (D, E), and image plot (F) for the effect versus logarithmically transformed doses of trimetrexate and AG2034 for the experiment in a high folic acid medium.

Figure 8

Effect versus logarithmically transformed dose plot for the combination study of trimetrexate and AG2034 in a low folic acid medium. Raw data are shown in open circles. Blue dashed line and red dotted line indicate the fitted marginal dose-response curves for trimetrexate and AG2034, respectively. Black solid line indicates the fitted dose-response curve for the combination of trimetrexate and AG2034.

Figure 9

Trellis plot of the estimated interaction index (solid line) and its point-wise 95% confidence interval (red solid lines) and the 95% simultaneous confidence band (dashed lines) for the low folic acid experiment. Estimates at the design points where experiments were conducted are in red. The interaction index is plotted on the logarithmically transformed scale but labeled on the original scale.

Figure 10

Effect versus logarithmically transformed dose plot for the combination study of trimetrexate and AG2034 in a high folic acid medium. Raw data are shown in open circles. Blue dashed line and red dotted line indicate the fitted marginal dose-response curves for trimetrexate and AG2034, respectively. Black solid line indicates the fitted dose-response curve for the combination of trimetrexate and AG2034.

Figure 11

Trellis plot of the estimated interaction index (solid line) and its point-wise 95% confidence interval (red solid lines) and the 95% simultaneous confidence band (dashed lines) for the high folic acid experiment. Estimates at the design points where experiments were conducted are in red. The interaction index is plotted on the logarithmically transformed scale but labeled on the original scale.