Prediction of multidimensional drug dose responses based on measurements of drug pairs

Abstract

Finding potent multidrug combinations against cancer and infections is a pressing therapeutic challenge; however, screening all combinations is difficult because the number of experiments grows exponentially with the number of drugs and doses. To address this, we present a mathematical model that predicts the effects of three or more antibiotics or anticancer drugs at all doses based only on measurements of drug pairs at a few doses, without need for mechanistic information. The model provides accurate predictions on available data for antibiotic combinations, and on experiments presented here on the response matrix of three cancer drugs at eight doses per drug. This approach offers a way to search for effective multidrug combinations using a small number of experiments.

Keywords: cancer treatment; drug cocktails; drug combinations; mechanism-free formula; predictive formula.

Fig. 1.

Response matrix of three chemotherapy drugs at eight doses shows nonmonotonic behavior that is well captured by the dose model but not by other models. (A) Survival of A549 lung cancer cells when treated with the three pairs of drugs (taxol−doxorubicin, cisplatin−doxorubicin, cisplatin−taxol) at eight doses each: (Left) the measured response, (Middle) the predicted response using the present dose model, and (Right) the predicted response by Bliss independence. (B) Slices of the three-drug dose–response matrix. The first column is the measured response, followed by the prediction of the three drugs’ interactions using different models. Note that the Isserlis and regression models apply only to triplets and above, not to pairs.

Fig. 2.

Response of three cancer drug mixtures at multiple doses is best captured by the present dose model. Experimental measurements of the mixture of the three anticancer drugs taxol, doxorubicin, and cisplatin at all dose combinations compared with (A) Bliss independence, (B) Isserlis predictions, (C) regression model, and (D) present dose model predictions. R² = 1 − ∑(Model − Experiment)/var(Experiment) can yield negative values when the mean model prediction is different from the mean experimental measurement, indicating a very poor fit.

Fig. 3.

The present dose model accurately describes response surfaces of antibiotic and chemotherapy drug pairs. Examples of response surfaces for antibiotic pairs that are (A) antagonistic (B) hyperantagonistic, and (C) synergistic. (D) Interactions of anticancer drug pairs also well described by the model. (E) R² values of the 20 drug pairs considered here for Bliss independence (o), and the present dose model of Eqs. 3 and 4 (x). Experimental data from Wood et al. (14). Cip, ciprofloxacin; Cm, chloramphenicol; Dox, doxycycline; Ery, erythromycin; Linc, lincomycin; Ofl, ofloxacin; Sal, salicylate; Tet, tetracycline; Tmp, trimethoprim (see also Fig. S1A).

Fig. S1.

The errors of the dose model seem to result mainly from experimental noise. (A) RMSE of the dose model for 20 drug pairs (o) as a function of the maximal RMSE of the Hill model for the single drugs. The correlation between the two shows that much of the error in the dose model errors can be attributed to noise. Similar results for three anticancer drugs are also presented (□). (B) RMSE of the dose model for three- and four-drug combinations as a function of the maximal RMSE of the dose model for the pairs of drugs in the combination (o). The correlation between the two suggests that much of the dose model error can be attributed to noise. Similar results for three anticancer drugs are also presented (□).

Fig. S2.

Accuracy analysis suggests that a single interaction parameter suffices for the pair model in most cases, and that only about 10 measurements suffice to specify the pair parameters. (A) The loss of accuracy (ΔRMSE) when using only one parameter (a₁₂ or a₂₁) to describe pair interactions. The best results (lower RMSE) out of the two options are presented. In most cases, the loss of accuracy is small (<0.5%). In the case of superantagonism (Tmp-Ofl), the loss of accuracy is large, because both interaction terms are needed to describe the effect. (B) The loss of accuracy as a function of the number of pairs of doses measured to estimate the parameters of a given pair of drugs. Accuracy is relative to the measurement of all dose pairs in the dataset (∼100). For 10 measurements, the loss of accuracy is about 0.5%.

Fig. 4.

A hierarchy between antibiotics. For most of the drug pairs, one of the two interaction parameters a₁₂ or a₂₁ can be set to zero without significant loss of fit quality, resulting in a single parameter for the interaction. In these cases, we can say that drug 1 changes the effective dose of drug 2 but not vice versa (1→2). We find that these relations are transitive: If drug 1 affects drug 2 (1→2) and drug 2 affects drug 3 (2→3), then drug 1 will affect drug 3 (1→3) but not vice versa. The direction of the interaction is transitive, but the absolute strength of the interaction is not always transitive. To quantify the strength of the hierarchy we use the goodness of fit (RMSE) difference in the case of a₁₂ = 0 and a₂₁ = 0. This quantity is indicated by the arrow thickness. The strength of the interaction is indicated by the color of the arrow (red, strong antagonistic; blue, strong synergistic). We plotted only interaction with nonnegligible hierarchy (∆RMSE > 0.005%). Note that 8 out of the 28 possible interactions were not present in the published dataset.

Fig. S3.

A mechanistic motivation for the two-drug interaction term based on drug efflux pump induction. We use the well-understood drug interaction mechanism in the mar system. In this system, salicylate and other drugs interact through the induction of the mar efflux system by salicylate. The efflux system exports the other drug and effectively reduces its dose (14). The figure shows data from ref. on the promoter activity of the mar system as a function of salicylate dose. We find that this induction curve is fit very well by the present Michaelis–Menten-like interaction term between the two drugs salicylate and tetracycline. Additional mechanisms are probably at play in other cases.

Fig. 5.

Combinations of three or four drugs are well described by the present dose model (x). Bliss independence (o) results are displayed for comparison. The three-drug combination Cm-Ofl-Sal shows strong antagonism (B) The three-drug combination Cm-Ofl-Tmp show synergism. (C) The three-drug combination Cm-Ery-Tmp shows synergism. (D) The three-drug combination Sal-Ery-Cm shows mild antagonism (E) The four-drug combination Linc-Cm-Ofl-Tmp shows complex interactions captured by the model. (F) The four-drug combination Dox-Ery-Linc-Sal shows antagonism. (G) R² values for three- and four-drug combinations: Bliss model (o) and the present dose model (x). Antibiotic data are from Wood et al. (32); results for the present experiments on three anticancer drugs on A549 cells (Figs. 1 and 2) are also presented. Cm/C, chloramphenicol; Dox/D, doxycycline; Ery/E, erythromycin; Linc/L, lincomycin; Ofl/O, ofloxacin; Sal/S, salicylate; Tmp/T, trimethoprim. Isserlis model for quadruplets is g₁₂₃₄ = g₁₂g₃₄ + g₁₃g₂₄ + g₁₄g₂₃ − 2g₁g₂g₃g₄ (see also Fig. S1B).

Fig. 6.

Search for a mixture with high efficacy at low doses. We seek a combination of three antibiotics with a given large effect (g = 0.1). The doses that satisfy this condition form a surface in the 3D dose space for chloramphenicol (Cm), erythromycin (Ery), and trimethoprim (Tmp). On this surface, we seek the combination with minimal side effects. Assuming that side effects increase with drug doses, we can seek to minimize, for example, the maximal dose in the combination M = max(D_i/D_0i). The colors indicate the decrease in M relative to the lowest D_i/D_0i of the individual drugs. The optimal drug combination indicated by an arrow on the g = 0.1 surface has a fourfold-lower M than the best individual drug (see also Fig. S4).

Fig. S4.

Search for high efficacy at low dose combinations. We seek a combination with a given large effect (g = 0.1). The doses that satisfy this condition form a surface in the 3D dose space, for example, the g = 0.1 surface for the three antibiotics: Cm, Ery, and Tmp. On this surface, we seek the combination with minimal side effects. Assuming that side effects increase with drug doses, we can seek to minimize, for example, the sum of normalized doses S = ${\displaystyle \sum }$ D_i/D_0i. The color indicates the decrease in the side effects relative to the side effect of the best individual drug. The optimal drug combination, indicated by an arrow on the g = 0.1 surface, has an approximate twofold-lower dose than the best individual drug. Another possibility is to minimize the maximal dose in the combination M = max(D_i/D_0i) (Fig. 6).

Fig. S5.

The Isserlis model agrees with the present model in the limit of weak interactions and low noise. (A) We generated synthetic data using the present model and fit the data using the Isserlis model. We used doses D/D₀ = 0 to Dmax, where g(Dmax) = 0.2, Hill coefficients for the two simulated drugs were in the range n = 1–7, and interaction parameters a_ij in the range −1 to 6 (note that the smallest possible value for a_ij is −1). The models agree well when the interactions are weak (n < 2, |a_ij| < 3) (R² > 0.9). Agreement breaks down when the interactions are stronger or Hill coefficients for at least one drug exceed 2. (B) The agreement breaks down even for weak interactions in the presence of noise. Here sigma is the SD of Gaussian noise added to the simulated log effect data. Hill coefficient for the two drugs in this example was n = 2. (C and D) We also tested a smoothed version of the Isserlis model on the present cancer drug dose–response matrix, in which the single and pair drug inputs to the Isserlis model are taken from the present model. Because the present model uses multiple dose measurements, this input has reduced noise. This smoothed Isserlis model fits the present cancer drug data better than the unsmoothed Isserlis model [R² = 0.7 (C) vs. R² = 0.6 (D)] but not as well as the present model (R² = 0.82) (Fig. 2D). This indicates that part of the improvement in R² is due to noise insensitivity (∼0.1) and part of it is systematic (∼0.1).