Mathematical definitions of "additive effect of two (or more) drugs" and their synergism and/or antagonism based on mass-action law (MAL) algorithms for pharmacodynamics (PD), biodynamics (BD) and bioinformatics (BI) simulations.

During the past century, there are over 15 different definitions of "synergism" or methods of its measurement, but none supported the others. These generate tremendous confusions and controversies in biomedical science and biomedical R&D, despite the fact that treatments of the most dreadful diseases such as cancer and AIDS, are mainly using drug combinations or multi-modality therapies (e.g., drug and radiation). In the absence of consensus of "exact definition of synergism", the researchers, peer reviewers and governmental regulatory agencies (such as FDA, NIH, Patent Office, etc.) have no clear standard to judge or to regulate the "Synergy Claims". This paper indicates that the combination index equation (CIE), mathematically derived from MAL-PD/BD, provides mathematical exact definition of synergism (CI<1), additive effect (CI=1) and antagonism (CI>1), quantitatively/digitally by automated computer simulation (e.g., using CompuSyn software, available for free download from www.combosyn.com, upon registration). This capability is enabled by the doctrine of the median, i.e., the median-effect equation (MEE) which is the unified equation of Michaelis-Menten (enzyme kinetics), Henderson-Hasselbalch (pH ionization), Hill (ligand occupancy), Scatchard (receptor binding) and Langmuir (adsorption isotherm). In addition, the median-effect plot (MEP) with x= log Dose vs y= log [Fa/(1-Fa)] (Where Fa is the fraction affected by a drug or effector) linearized all PD dose-effect curves (SEC), with x-intercept yields the median-effect dose (Dm) signifying potency, and the slope (m) yields dynamic order, signifying quantitatively the shape of DEC, where m=1, >1, and <1 indicates hyperbolic, sigmoidal, and flat sigmoidal, respectively. Furthermore, the linearization of DECs by the MEP leads to the discovery the Minimum Two-Dose-Data-Points Theory that allows default 3 point as Dose-zero, and the 4 points as Dm, the universal reference-point and dynamic common-link, for cost-effect studies. This purely MAL approach (non-statistical non-empirical approach), creates a new paradigm-shift since all terms of MEE and CIE are relativity ratio, therefore, the theory is valid regardless of nature of effectors (e.g., chemical drug, natural product, radiation, UV, temperature, pH); units (e.g., mg, nM, IU, rad, multiple of infection, mg, mg/kg, or mg/M ), mechanisms or mode of action (e.g., competitive, noncompetitive, uncompetitive, sequential, ordered, ping-pong, or random); and regardless of in vitro (molecular, cellular, tissue), in animal or in clinical trial protocol design and data analysis/ simulation for BI diagnostic plots (MEP, DEC, Fa-CI Plot, classic Isobologram, Non-constant-ratio-combo normalized Isobologram, Dose-Reduction Index (DRI), and Polygonogram for cocktail design for 3 or more drugs. The original article introducing the CIE theorem/algorithm and computer software (T.C. Chou and P. Talalay, Adv. Enz. Regul 22: 27-55, 1984), as of Nov.29, 2021, received 7,401 citations in 1,420 biomedical journals internationally, indicating a record broad applications.