A New Nonlinear Mixture Response Surface Paradigm for the Study of Synergism: A Three Drug Example

Abstract

A flexible approach to response surface modeling for the study of the joint action of three active anticancer agents is used to model a complex pattern of synergism, additivity and antagonism in an in vitro cell growth assay. The method for determining a useful nonlinear response surface model depends upon a series of steps using appropriate scaling of drug concentrations and effects, raw data modeling, and hierarchical parameter modeling. The method is applied to a very large in vitro study of the combined effect of Trimetrexate (TMQ), LY309887 (LY), and Tomudex (TDX) on inhibition of cancer cell growth. The base model employed for modeling dose-response effect is the four parameter Hill equation [1]. In the hierarchical aspect of the final model, the base Hill model is treated as a function of the total amount of the three drug mixture and the Hill parameters, background B, dose for 50% effect D50, and slope m, are understood as functions of the three drug fractions. The parameters are modeled using the canonical mixture polynomials from the mixture experiment methodologies introduced by Scheff [2]. We label the model generated a Nonlinear Mixture Amount model with control observations, or zero amounts, an "NLMAZ" model. This modeling paradigm provides for the first time an effective statistical approach to modeling complex patterns of local synergism, additivity, and antagonism in the same data set, the possibility of including additional experimental components beyond those in the mixture, and the capability of modeling three or more drugs.

Concentration-Dependent Synergy and Antagonism within a Triple Antifungal Drug Combination againstAspergillusSpecies: Analysis by a New Response Surface Model

“…Overall, the synergistic interactions of the voriconazole-caspofungin and amphotericin B-caspofungin double combinations decreased, whereas the antagonistic interactions increased, with increasing concentrations of amphotericin B and voriconazole, respectively. The complexity of these interactions was well described by an E max -based mixture-amount response surface model using full cubic canonical mixture polynomials (40). The sigmoid E max model has been used extensively to successfully describe the concentration-effect relationships of amphotericin B, caspofungin, and azoles (1,22,26,32,41).…”

“…A new nonlinear mixture-amount response surface model for analyzing three-drug combinations was recently described (40). By contrast with other fully parametric response surface models, the new model is flexible enough to describe complicated response surfaces and to determine complex patterns of synergy and antagonism.…”

“…If all coefficients in equation 2 equal 0, then U 50 ϭ 1, indicating Loewe additivity. The U 50 value is equivalent to the classical combination index of Loewe additivity (9,40). This derives from Loewe additivity theory.…”

Tumor models for efficacy determination

“…Response surface modeling provides an effective statistical approach to modeling complex patterns of synergy, additivity, and antagonism in the same data set (39,88). Tan et al (39) developed a class of repeated-measures models to analyze the dose-response relationship in xenograft experiments while accounting for incomplete (missing at random and informative censoring) and variable constraints.…”

Quantifying Subpopulation Synergy for Antibiotic Combinations via Mechanism-Based Modeling and a Sequential Dosing Design

“…However, quantitative approaches for antibiotic combinations that characterize the time course of synergistic killing or of resistance prevention are scarce (7)(8)(9). Existing methods to describe synergy are limited to outcomes at a single time point (often 24 h) and do not implement the mechanism(s) of synergy (10,11).…”