BRAID: A Unifying Paradigm for the Analysis of Combined Drug Action

Abstract

With combination therapies becoming increasingly vital to understanding and combatting disease, a reliable method for analyzing combined dose response is essential. The importance of combination studies both in basic and translational research necessitates a method that can be applied to a wide range of experimental and analytical conditions. However, despite increasing demand, no such unified method has materialized. Here we introduce the Bivariate Response to Additive Interacting Doses (BRAID) model, a response surface model that combines the simplicity and intuitiveness needed for basic interaction classifications with the versatility and depth needed to analyze a combined response in the context of pharmacological and toxicological constraints. We evaluate the model in a series of simulated combination experiments, a public combination dataset, and several experiments on Ewing's Sarcoma. The resulting interaction classifications are more consistent than those produced by traditional index methods, and show a strong relationship between compound mechanisms and nature of interaction. Furthermore, analysis of fitted response surfaces in the context of pharmacological constraints yields a more concrete prediction of combination efficacy that better agrees with in vivo evaluations.

Figure 1. Example BRAID response surfaces.

Unless otherwise specified, in all surfaces, ID_M,A = 1.5 μM, ID_M,B = 1.5 μM, n_a = n_b = 3, E₀ = 0, E_f,A = E_f,B = 100, and κ = 0. Isoboles in all plots represent a change in effect equal to 10. (a) A basic additive response surface. (b) Lower Hill slopes (n_a = n_b = 1). (c) Differing Hill slopes (n_a = 1, n_b = 3). (d) Mild synergy (κ = 0.75). (e) Strong synergy (κ = 2.5). (f) Differing final effects (E_f,A = 100, E_f,B = 55). (g) Mild antagonism (κ = −0.3). (h) Severe antagonism (κ = −1.2). (i) A more complex surface, with n_a = 2.7, n_b = 3.2, E_f,A = 100, E_f,B = 75, and κ = −0.5. Note that the isoboles, particularly at higher effect levels, show regions of convexity and concavity; traditionally, surfaces like this would necessitate varying domains of synergy and antagonism. However, by allowing the maximal effects of the two drugs to differ, such surfaces can be modeled with a single interaction parameter.

Figure 2. Comparison of CI and BRAID using simulations.

(a) Estimated CI at the 99% effect level and best fit estimate of BRAID κ for 50 simulated additive combinations. Vertical bars indicate bootstrapped confidence intervals. In each trial, measurement noise level was 5%, and the Hill slopes of both drugs’ dose response curves were 1. Vertical dashed lines indicate the additivity range used for CI classification. (b) Results from simulations in which the measurement noise level was 7.5% and the Hill slope of the two drugs were 0.8 and 2.4. CI variability has substantially increased and is biased higher (due to incorrect assumptions about the shape of constant ratio combination dose response), while BRAID κ has remained centered around zero with a slight increase in confidence intervals.

Figure 3. Results of BRAID analysis on a large combination dataset.

(a) Plot of the interaction parameter α developed by Cokol et al. versus BRAID κ in all 200 combinations tested. Vertical bars indicate bootstrapped confidence intervals on κ. Points in red were designated antagonistic by BRAID analysis, while points in green were designated synergistic. Vertical dashed lines indicate the ‘window of additivity’ on α used by Cokol et al. to designate significant synergy or antagonism. (b) Graph of sign and magnitude of interaction between all pairs of 13 compounds, as represented by α. (c) Graph of sign and magnitude of interaction between all pairs of 13 compounds, as represented by κ. (d) Plot of all 13 compounds tested in a two-dimensional space extracted by applying multidimensional scaling to similarity of interaction as determined by κ.

Figure 4. Illustration of the BRAID analysis approach.

The combination of BMN-673 and SCH-900776 in cell line ES8 (a,c,e,g,i) and the combination of olaparib and SN-38 in cell line ES1 (b,d,f,h,j). (a,b) Cell survival (modeled by Cell-Titer Glo luminescence) is calculated relative to negative controls on all plates, and the average logarithm of each dose-pair is plotted as a response surface. (c,d) Smoothing data using a simple Gaussian interpolation allows the viewer to quickly evaluate the shape of the response surface. (e,f) The additive BRAID surface estimated using the best-fit individual dose-response parameters of the two drugs. (g,h) The best fit BRAID surface in which the interaction parameter κ is allowed to vary freely. The best fit BRAID surface for BMN-673 vs. SCH-900776 is very close to the estimated additive surface, while for olaparib vs. SN-38 it is visibly different from the additive surface. This is due to strong synergy between olaparib and SN-38. (i,j) Comparing the best fit BRAID surface with the measured unsmoothed data allows the viewer to quickly gauge whether the surface contains non-uniform errors, suggesting artifacts or a poor fit.

Figure 5. Comparison of CI-based methods and BRAID in EWS.

(a) Plot of CI values for all dose pairs tested in the ES8 cell line. Dark blue corresponds to strong synergy (CI < 0.5); light blue to mild or moderate synergy (0.5 < CI < 0.85); white to near-additivity (0.85 < CI < 1.2); light red to mild or moderate antagonism (1.2 < CI < 2); and dark red to strong antagonism (CI > 2). (b) FIC plots for all six PARP_i/SOC combinations in ES8 at the 99% effect level. Dark shaded grey regions indicate the areas of near-additivity, while light grey shaded regions indicate areas of mild or moderate synergy or antagonism. Points are colored according to whether they lie above, within, or below the near-additive region. (c) Network graphs depicting the magnitude and sign of the best fit BRAID κ for all tested combinations. For reference, “strong synergy” corresponds to κ = 2.5, “mild synergy” corresponds to κ = 1, “mild antagonism” corresponds to κ = −0.66, and “strong antagonism” corresponds to κ = −1.

Figure 6. Modeling potentiation.

Calculation of potentiation with the BRAID model for the combinations of veliparib and SN-38 in ES1 (a–c), olaparib and temozolomide in ES8 (d–f), and temozolomide and SN-38 in ES1 (g–i). (a,d,g) Best fit BRAID surface for each combination with estimated value of κ and confidence interval. (b,e,h) Estimated effect of the second drug in each combination in the presence of various levels of the first drug. Curve shifts to the left indicate potentiation. (c,f,i) EC₉₉ values of the second drug in each combination in the presence of various levels of the first drug. Note that though the combinations of veliparib vs SN-38 and olaparib vs. temozolomide exhibit similar κ values, olaparib produces considerably more potentiation of temozolomide than veliparib does of SN-38. This is due both to the increased potency of olaparib and the lower Hill slope of temozolomide.

Figure 7. Predictions of the index of achievable efficacy (IAE).

(a) Illustration of the calculation of IAE. The area bounded by the maximum achievable concentrations of both compounds and the isobole of the desired efficacy is the “area of achievable efficacy”. The square root of the ratio of the area of all achievable dose pairs (C_max,A∙ C_max,B in this case) to the area of achievable efficacy is the IAE. (b) Measured values of IAE₉₉ in cell lines ES1 and ES8 and IAE₉₀ in cell line EW8 for all PARPi and SOC agent combinations. Black bars indicate bootstrapped confidence intervals on IAE values. (c,d) Survival of mice orthotopically implanted with ES8 cells treated with combinations of all three PARP_i with temozolomide (c) and irinotecan (d). Black survival curves indicate the SOC only treatment of irinotecan and temozolomide.