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.drug combinations | drug cocktails | cancer treatment | mechanism-free formula | predictive formula T o kill cancer cells or bacteria, combination therapy can be more effective than individual drugs (1-6). Combination therapy is thought to allow increased efficacy at low doses, thus reducing side effects and toxicity; it is also believed to minimize the chances of resistance (7-9), a pressing problem in treating cancer and infectious diseases.Much work has been devoted to classifying how pairs of drugs interact (10)(11)(12)(13)(14). Across systems, a good first approximation is the Bliss independence model (15,16), in which the pair effect is the product of the individual drug effects: If the effect of the drugs are g 1 and g 2 , the effect of the combination is g 12 = g 1 · g 2 . The Bliss model ignores interactions in which drugs enhance each other effectssynergism-or inhibit each other's effects-antagonism. Some drugs even inhibit each other so much that the combined effect is lower than either drug alone, an effect called hyperantagonism (17)(18)(19)(20)(21).Going beyond drug pairs has been difficult. Experimentally testing high-order combinations beyond pairs in a systematic way is challenging because it requires an exponentially large number of experiments (22-25): For N drugs at D doses, one needs D N experiments. For N = 10 drugs and D = 8 doses, this means ∼10 9 measurements. The combinatorial explosion makes exhaustive testing of drug dose combinations unfeasible. This problem is especially acute in cases where material is scarce, such as testing of patient-derived samples (26)(27)(28)(29). Hence, models for predicting high-order effects are essential.Apart from detailed simulations of particular systems (22,30,31), there has been little study of general mechanism-independent models for multiple drugs. An exception is the elegant study by Wood et al. (32) that showed that combinations of antibiotics can be predicted by an Iserliss-like formula that uses pair effects to predict the effects of higher-order mixtures. For example, the effect of a drug triplet is modeled as g 123 = g 12 g 3 + g 13 g 2 + g 23 g 1 − 2g 1 g 2 g 3 . This formula has not been tested on cancer drug mixtures, to the best of our knowledge.Another line of research u...

Uncovering emergent interactions in three-way combinations of stressors

“…Dose-dependent interactions with three or more drugs [37,[52][53][54] have been studied by extending the Loewe additivity measure that classifies two-drug interactions based on lines of equivalent growth rates (isobolograms) across a range of concentrations of the combined drugs. In this regard, a study by Jonker et al [55] uses Loewe additivity and provides a model that can test whether the interaction type is independent of the absolute concentrations of the combined drugs or dependent only upon the dose ratios of the drugs.…”

“…As part of this study, they discovered a simple and highly informative algebraic metric related to the one for emergent interactions we derive below. More recently, Zimmer et al [37] used a framework that incorporates interaction coefficients as part of a model based on Hill functions to show how to increase predictive power for three-way interactions based on limited information about pairwise interactions across a range of drug concentrations.…”

Pairwise interactions and the battle against combinatorics in multidrug therapies

“…Unfortunately, the inherent flexibility of combination therapy also presents a considerable practical hurdle: the number of possible drug combinations grows exponentially with the number of drugs, making exhaustive screening with even a modest number of drugs intractable. In PNAS, Zimmer et al (6) develop a robust method for predicting the effects of multidrug combinations for microbial infections and cancer, potentially sidestepping the combinatorial explosion that limits systematic design of combination therapies.…”

“…The study by Zimmer et al (6) provides several innovative and fundamental advances over previous work, potentially opening the door to widespread practical application of pairwise approximations to multidrug treatments. ; D 2 + D 3 ) measurements at these concentrations but also potentially measurements at other doses.…”

“…Using their estimate of ∼10 measurements per drug pair, our hypothetical 10-drug screen would now be reduced to several hundred measurements-a task easily achievable even for modest-sized academic laboratories. As an elegant proof of principle, they optimize a combination of three antibiotics to achieve growth inhibition comparable to single-drug therapy but with a fourfold reduction in drug concentration (6).…”