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Review Article |
Preclinical Pharmacology Core Laboratory, Molecular Pharmacology and Chemistry Program, Memorial Sloan-Kettering Cancer Center, New York, New York
Abstract
Abstract I. Introduction A. Why Drug Combination? B. Pitfalls in Drug Combination Studies 1. Synergism versus Enhancement or Potentiation. 2. The Most Common Errors. C. Truth or Fallacy and Its Consequences D. An Approach for Extinguishing Controversies II. Theoretical Basis for Dose-Effect Analysis A. An Approach of Merging the Mass-Action Law with Mathematical Induction and Deduction 1. The Power of Mathematical Induction and Deduction. 2. Nature's Law. 3. Dealing with Diversified Biological and Pharmacological Systems. B. The Derivation of Equations and Theorems Based on the Mass-Action Law 1. The Median-Effect Equation. 2. The Unified Theory. C. Extension of Mass-Action Law to Multiple Drug-Effect Systems 1. The Multiple Drug-Effect Equation. 2. The Combination Index Theorem and Plot. 3. The General Equation for Combination of n Drugs. 4. Algorithms for Determining Synergism and Antagonism. 5. Main Features of the General Equation. 6. The Fa-Combination Index Plot and Isobologram Are Two Sides of the Same Coin. 7. How Much Synergism Is Synergy? D. The Dose-Reduction Index Equation and Plot E. The Polygonogram III. Experimental Design for Drug Combinations A. The Prerequisite and Theoretical Minimum Requirements for Drug Combination Studies B. Constant Ratio Drug Combinations, Dose Range, Dose Density, and Experimental Scheme C. The Nonconstant Ratios of Drug Combinations D. The Optimal Combination Ratio for Maximal Synergy E. Combination Designs for Three or More Drugs F. Drug Combination in Vitro, in Vivo, and in Clinics G. Schedule Dependence H. Condition-Dependent Synergism or Antagonism and Combination of Drugs with Different Modalities, Different Units, and Mechanisms IV. Computerized Automation, Graphic Simulation, and Informatics A. Computer Software B. The Median-Effect Plot and the Simulation of Dose-Effect Curve C. Simulation of the Fa-Combination Index Plot D. Construction of the Classic and Normalized Isobologram E. Simulation of the Fa-Drug-Reduction Index Plot F. Step-by-Step Use of CompuSyn Software for Single Drug and for Drug Combination Studies G. Statistical Considerations V. Selected Examples of Cited Applications A. Cited Methods and Evaluation of Single Drug and Drug Discovery 1. Exploration of Potency, Toxicity, Parameters, and Structure-Activity Relations for New Compounds. 2. Low-Dose Risk Assessment for Carcinogens and Radiation: 3. Calculation of Ki from the IC50 Value: 4. Exclusive and Nonexclusive Inhibitors and Topology of Binding Sites: 5. Drug Resistance Evaluation and Other Applications: 6. Cellular Pharmacological Studies: 7. Tissue Pharmacological Studies: 8. Cardiovascular Pharmacological Studies: 9. Pharmacological Studies on Animals: 10. Behavioral Studies: 11. Cancer Prevention Agents: B. Examples of Cited Applications in Drug Combinations 1. Anticancer Agent Combinations. 2. Antiviral Agent Combinations. 3. Immunosuppressant Combinations for Organ Transplantations. 4. Schedule Dependence of Combinations. 5. Drug Combinations That Highlight Antagonism. 6. Topological Analysis of Multiligand Bindings. 7. Selectivity of Synergism. 8. Gene Therapy or Molecular Biology by Combinations. 9. Combinations of Other Anti-Infectious Disease Agents. 10. Cardiovascular Drug Combinations: 11. Combination for Animal Growth: 12. Anesthetic Combinations: 13. Radiation and Drug Combinations: 14. Antiparasitic Combination: 15. Segmental Reviews for Median-Effect Principle and Combination Index Methods. VI. Illustrations of Real Data Analysis with Mass-Action Law-Based Computer Software A. Single-Drug, Two-Drug, and Three-Drug Combination Analysis with Computer Software 1. Single-Drug Analysis and Two-Drug Combinations. 2. Topological Analysis for the Multiple Ligand Sites in the Steady-State System. 3. Two- and Three-Drug Combinations against Cancer Cell Growth and the Construction of Polygonograms. B. Other Applications of the Median-Effect Principle of the Mass-Action Law 1. Estimating Low-Dose Risk of Carcinogens. 2. Risk Assessment for Radiation. 3. Therapeutic Index and Safety Margin 4. Age-Specific Cancer Incidence Rate Analysis. ;_Chou_and_Miller,_1980 ):">5. Epidemiological Applications (Chou, 1978
6. Calculation of Ki from IC50. C. Sample Analysis of Drug Combination Data with Computerized Summaries 1. Synergism of Two Insecticides on Houseflies. 2. Antagonism between Methotrexate and Arabinosylcytosine. 3. Seven-Drug Combination against Human Immunodeficiency Virus and Their Polygonograms. a. Introduction. b. Summaries of results. c. Conclusions. D. Approaches for the Conservation of Laboratory Animals 1. The Median-Effect Principle. 2. Experimental Design. 3. Serial Deletion Analysis. 4. Polygonogram. Appendix I: Derivation of the Multiple Drug-Effect Equation A. Summation of the Effects B. Alternative Equations for Multiple Inhibitors in First-Order Systems C. Inhibition of Higher-Order Kinetic Systems by a Single Inhibitor D. Inhibition of the Higher-Order Kinetic Systems by Mutually Exclusive Inhibitors E. Multiple Inhibitions by Mutually Nonexclusive Inhibitors 1. First Order. a. Case 1. b. Case 2. 2. Multiple Inhibitions by Inhibitors with Different Kinetic Orders. 3. Higher-Order Multiple Mutually Nonexclusive Inhibitors. Glossary
The median-effect equation derived from the mass-action law principle at equilibrium-steady state via mathematical induction and deduction for different reaction sequences and mechanisms and different types of inhibition has been shown to be the unified theory for the Michaelis-Menten equation, Hill equation, Henderson-Hasselbalch equation, and Scatchard equation. It is shown that dose and effect are interchangeable via defined parameters. This general equation for the single drug effect has been extended to the multiple drug effect equation for n drugs. These equations provide the theoretical basis for the combination index (CI)-isobologram equation that allows quantitative determination of drug interactions, where CI < 1, = 1, and > 1 indicate synergism, additive effect, and antagonism, respectively. Based on these algorithms, computer software has been developed to allow automated simulation of synergism and antagonism at all dose or effect levels. It displays the dose-effect curve, median-effect plot, combination index plot, isobologram, dose-reduction index plot, and polygonogram for in vitro or in vivo studies. This theoretical development, experimental design, and computerized data analysis have facilitated dose-effect analysis for single drug evaluation or carcinogen and radiation risk assessment, as well as for drug or other entity combinations in a vast field of disciplines of biomedical sciences. In this review, selected examples of applications are given, and step-by-step examples of experimental designs and real data analysis are also illustrated. The merging of the mass-action law principle with mathematical induction-deduction has been proven to be a unique and effective scientific method for general theory development. The median-effect principle and its mass-action law based computer software are gaining increased applications in biomedical sciences, from how to effectively evaluate a single compound or entity to how to beneficially use multiple drugs or modalities in combination therapies.
Ever since the earliest days of recorded history, drug combinations have been used for treating diseases and reducing suffering. The traditional Chinese medicines, especially herbal medicines, are vivid examples. As the science of isolation technology and chemical synthetic capability advance, drug combinations have been more defined and sophisticated and their scope continues to broaden. Attempts have been made during the past century to quantitatively measure the dose-effect relationships of each drug alone and its combinations and to determine whether or not a given drug combination would gain a synergistic effect. Because biological systems as well as dose-effect models are exceedingly complex, there have been numerous models, approaches, hypotheses, and theories as well as controversies on drug combination analysis during the past century, as elaborated in many review articles, such as those by Fraser (1872), Loewe (1928, 1957), Le Pelley and Sullivan (1936), Plackett and Hewlett (1948), Finney (1952, 1971), Elion et al. (1954), Veldstra (1956), Goldin and Mantel (1957), Lacey (1958), Ariens and Simonis (1961), Venditti and Goldin (1964), Goldin et al. (1968), Skipper (1974), Schabel (1975), Grindey et al. (1975), Chou and Talalay (1977, 1981, 1983, 1984), Steel and Peckham (1979), Ashford (1981), Berenbaum (1981, 1989), Copenhaver et al. (1987), Carter et al. (1988), Greco et al. (1990), Poch et al. (1990), Prichard and Shipman (1990), Suhnel (1990), Chou (1991), Schinazi (1991), Jackson (1991), Lam (1991), Tallarida (1992), and Greco et al. (1995). Among them, a recent review by Berenbaum (1989) has listed >560 references and another review by Greco et al. (1995) has categorized 13 different approaches and methods for the determination of synergism and antagonism. The main difference of the present review from the earlier reviews is that the rather fruitless and confusing debates of the past will not be repeated. Instead, in this review article the focus will be on the drug combination analyses that have physical-chemical bearings and have mathematically verifiable equations and theories. After continued and persistent devotion on this single subject for >35 years, I propose a seemingly simple way to hopefully end all the controversies on how to determine synergism or antagonism and introduce an explicit mass-action law-based method that allows automated computerized simulation of synergism and antagonism. Here, the general theory of dose and effect will be presented, the experimental design will be illustrated, the algorithms for computer simulation will be given, and examples of applications in various fields of biomedical sciences and on real data sample sets will be demonstrated.
The use of multiple drugs may target multiple targets, multiple subpopulations, or multiple diseases simultaneously. The use of multiple drugs with different mechanisms or modes of action may also direct the effect against single target or a disease and treat it more effectively. The possible favorable outcomes for synergism include 1) increasing the efficacy of the therapeutic effect, 2) decreasing the dosage but increasing or maintaining the same efficacy to avoid toxicity, 3) minimizing or slowing down the development of drug resistance, and 4) providing selective synergism against target (or efficacy synergism) versus host (or toxicity antagonism). For these therapeutic benefits, drug combinations have been widely used and became the leading choice for treating the most dreadful diseases, such as cancer and infectious diseases, including AIDS.
B. Pitfalls in Drug Combination Studies
1. Synergism versus Enhancement or Potentiation. Let us consider the simplest situation in which two drugs, A and B, are combined. If drug A has an effect and drug B has no effect and if in combination they have an effect that is greater than that of drug A, then it is enhancement or potentiation. We can describe the effect simply as percent enhancement or -fold of potentiation. If A and B alone each has an effect, then in combination they may produce a synergistic, an additive, or an antagonistic effect. By definition, synergism is an effect that is more than additive, whereas the definition for antagonism is an effect that is less than additive. Clearly, defining what is an "additive effect" is the most crucial criterion for defining synergism and antagonism. I spent more than 10 years (1972-1983) in attempting to define the additive effect by deriving and publishing several hundred specific equations and several general equations.
2. The Most Common Errors. In most cases, medical researchers or clinical practitioners perform drug combinations for perusing synergism. However, there are many common errors associated with these claims:
), this type of problem can be solved by (1 - 0.6)(1 - 0.6) = 0.16, 1 - 0.16 = 0.84. Chou and Talalay (1984) called it the fractional product method. This method will never lead to a combination effect exceeding 100% inhibition. Chou and Talalay (1984), however, have also proved that this method has limited validity because it takes into account the potency (e.g., fractional inhibition) but ignores the shape of the dose-effect curve (e.g., hyperbolic or sigmoidal). The importance of the "shape" in a dose-effect analysis is shown in Fig. 1. Chou and Talalay (1984) indicated that Webb's method is valid only when both drugs have hyperbolic curves (i.e., in simple Michaelis-Menten kinetics when dose-effect curves are hyperbolic, i.e., m = 1 in the median-effect plot) and is not valid when m 
1, such as sigmoidal (m > 1) or flat sigmoidal (m < 1) curves. Furthermore, Webb's method is valid when the effects of two drugs are mutually nonexclusive (e.g., totally independent) and is not valid for mutually exclusive (e.g., similar mechanisms or modes of actions, as assumed for the classic isobologram, see below).
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On an Internet Web search, the term "drug combination" had 43,722 hits and 6,350,000 hits by PubMed and Google, respectively, and the term "synergistic effect" has been cited 14,296, 7186, and 963,000 times by PubMed, ISI, and Google, respectively. However, it is to be noted that in one review article by Goldin and Mantel (1957) alone, seven different definitions for synergism were given, and none of them supported the others. In a more recent review by Greco et al. (1995), 13 different methods for determining synergism were listed. Again, none of them supported the others. Thus, it is hard to find any other field in biomedical science that has more controversy and more confusion than drug combinations. The meaning of synergism has become an individual's preference, agenda, or wishes. The seriousness of faulty or unsubstantiated or erroneous claims of synergy is clearly obvious, since it is frequently referred to as therapy for treating patients.
It is not likely that the different definitions of synergism are all correct or the different methods for determining synergism are all valid. In the presence of so much ambiguity, doubt, bias, and confusion, science has been under siege and challenged. The longing for fact and truth in this field of discipline of research is ever strengthening.
D. An Approach for Extinguishing Controversies
For each hypothesis, approach, and theory of drug combination, we need to demand theoretical thoroughness and rigorous derivations. A mathematical formula does not really constitute a proven method, if it is empirical without actual derivation (Chou, 1977b). Often, a formula with obscure origin and lack of sound theoretical basis emerges and dominates the drug combination field for decades until a new one replaces it (Table 1). The complexity in biology and pharmacology apparently underlies this imperfectness. Therefore, the issues that have been raised are 1) How was the formula obtained? 2) Are all the parameters or constants defined and do they have any chemicophysical bearings? and 3) Are the formulae for a single drug expandable to multidrug systems? During the past seven decades, evidence indicated that dose-effect analysis per se was a physicochemical problem rather than a statistical problem. In other words, it was deterministic rather than probabilistic. At this time, I recommend a set of criteria for the credibility of a theory or a method, whether it existed or it will be newly proposed. These criteria include 1) Are there any derived equations to be based upon? If so, how, when, and where were they derived? 2) Are there any algorithms? If not, how can the procedure be executed or how can a computer program be established? and 3) Are the conclusions or claims quantitatively indicated or merely descriptive? We demand a quantitative conclusion. It is proposed that the right or wrong of a method for drug combination data analysis can be illustrated by the following fictional narrative, which can serve as a simple litmus test:
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Once upon a time, there was a Master who held two bottles of antitumor ingredients. The red bottle contained drug A, and the blue bottle contained drug B. He then gave the two bottles to his disciples, John and Paul. The Master asked them to conduct drug combination studies in different proportions and to determine whether they were synergistic or antagonistic by using any of their best choice of assay method and by using the best choice of theory for their dose-effect data analysis. Weeks later, John said they were synergistic, whereas Paul said they were antagonistic. However, the Master, without hesitation, said "No!" to both of them. Why? It was because drug A and drug B were the same, and therefore, it could only be an "additive effect"! The ingredient in the bottles was panaxytriol isolated from red ginseng that yielded highly sigmoidal dose-effect curves in a variety of assays. In fact, the additive effect conclusion should always be valid no matter what assay method was used, and it was not relevant whether or not the shape of a dose-effect curve was hyperbolic or sigmoidal and whether the drug interaction was determined at ED30, ED50, ED70, ED95, or ED99 levels. It should yield an additive effect under all circumstances. Using this approach, one should be able to determine whether any hypothetical method for determining synergism or antagonism is valid or faulty. The sigmoidicity of a dose-effect curve (e.g., for panaxytriol) greatly magnifies the differences among the different methods or theories. Thus, the main controversies in drug combination analysis in the past century can be readily resolved.
II. Theoretical Basis for Dose-Effect Analysis
A general equation of dose and effect and its theorem of combination index have been developed by using the approach of merging the physicochemical principle of the mass-action law with the mathematical principle of induction and deduction. After deriving hundreds of equations and three and one-half decades of progression, Chou presented an overview of this systematic approach to complex biosystems that leads to the genesis of some of the fundamental rules in nature. Remarkably, the derived general theory of dose and effect has been proven to be the unified theory of the four basic equations in biomedical sciences pioneered by Henderson-Hasselbalch, Michaelis-Menten, Hill, and Scatchard. Furthermore, the present theory not only leads to the derivation of the combination index theorem but also leads to the derivation of the isobologram equation, the dose-reduction index equation, and the generation of polygonograms. Their informatics has been explored on theoretical grounds. Their algorithms have allowed for the creation of computer software to facilitate their applications into a broad discipline in biomedical sciences, especially in the field of parameter determination, and have allowed for the simulation of synergism or antagonism in drug combinations at all dose and effect levels. Based on an ISI Web of Science search (Institute for Scientific Information 1976-2006; http://portal.isiknowledge.com/portal.cgi?DestApp=WOS&Func=Frame), one article alone on the median-effect principle (Chou and Talalay, 1984) has been cited in >1294 scientific papers in hundreds of biomedical journals.
A. An Approach of Merging the Mass-Action Law with Mathematical Induction and Deduction
1. The Power of Mathematical Induction and Deduction. Whether the proposition 1 + 2 + 3 + 4 + 5 +... + 6789 = 23,048,655 is right or wrong can be determined in several ways. One may actually count and add, step by step to prove it, whereas another may create an iterative program using a computer for a virtually error-free calculation. But for an individual, familiar with mathematical induction and deduction, it can be proven in less than 30 s with the aid of a pocket calculator or even by hand, given approximately 3 minutes, by using a pencil and a piece of paper.
Therefore, there is a seemingly magical power in mathematical induction and deduction. Since the 1970s and early 1980s, Chou has attempted to use this approach for biological systems by using the basic rules of physics and chemistry. Three and one-half decades later, we now have the second-degree Pascal's triangle (Chou, 1970, 1972), the median-effect equation (Chou, 1974, 1976, 1977; Chou and Talalay, 1977), the combination index equation (Chou and Talalay, 1981, 1983, 1984; Chou, 1991; Chou et al., 1994), the dose-reduction index equation (Chou and Talalay, 1984; Chou, 1987; Chou, 1991, 1994), the general equation for the isobologram (Chou and Talalay, 1984, 1987; Chou, 1991; Chou et al., 1991), and the creation of the polygonogram (Chou et al., 1991; Chou and Martin, 2005), along with their computer software (Chou JH et al., 1983; Chou JH and Chou, 1985; Chou and Hayball, 1997; Chou and Martin, 2005). Remarkably, the median-effect equation, which has been independently derived mathematically, is, in fact, the "unified theory" for the Michaelis-Menten equation of enzyme kinetics, the Hill equation for higher-order ligand binding saturation, the Henderson-Hasselbalch equation for pH ionization, and the Scatchard equation for receptor binding (Chou, 1977, 1991).
2. Nature's Law.
In the physical world of nature, there is the mass-action law (C. M. Guldberg and P. Waage, 1864), the equilibrium law (A. F. Horstmann, J. W. Gibbs, and J. H. Van't Hoff, 1873-1886), and the absolute reaction rate theory (M. Polanyi and H. Eyring, 1935) [For references, see Bothamley (2002).] Using the equilibrium and steady-state approach at a constant temperature and pressure, enzyme kinetics and receptor theory have flourished. These developments have provided a golden opportunity to merge these physicochemical approaches with mathematical induction and deduction for biological systems. A general theory and their theorems with broad applicability for diversified biological applications have thus been created. Consequently, the algorithms for computerized simulation and analysis on these theorems have also been developed.
3. Dealing with Diversified Biological and Pharmacological Systems.
Using the well-developed field of enzyme kinetics as a model, Chou in the 1960s, as a pharmacology Ph.D. graduate student at Yale University, learned how to derive the Michaelis-Menten equation, which described a single substrate-single product reaction at a steady-equilibrium state (Chou 1970, 1974). Intrigued by the intricacy of this process, Chou gradually and systematically extended the derivation to various multiple substrate-multiple product reactions. This extension and pattern analysis used various notations introduced by Cleland (1963), such as the sequential ordered mechanism, the ping-pong mechanism, and the random mechanism, with different numbers of complexes, enzyme species, and stable enzyme forms. A systematic approach, similar to mathematical induction, with substrate and product for n = 1, 2, 3,... allows the derivation of many specific equations (Chou, 1972, 1974). Through the combination and permutation of different numbers of substrate and product reactants in conjunction with the above mechanisms, hundreds of specific equations have been derived (Chou, 1974, 1976, 1977; Chou and Talalay, 1977, 1981, 1983, 1984).
Most drugs, as described in pharmacology, are inhibitors that suppress enzymes, receptors, or pathways. The enzymatic mechanisms, indicated above, can be considered the mini-pathways. Introduction of a competitive, noncompetitive, or uncompetitive inhibitor to the above enzyme kinetic derivations (e.g., I, competitive with substrate A, noncompetitive with substrate B, uncompetitive with substrate C, etc.) again, allows for hundreds of specific equations to be derived (Chou, 1974). The mathematical deduction of these specific equations is greatly facilitated by taking the ratio of reaction rate equations in the presence (
i) and absence (0) of an inhibitor. This ratio (i/0) is the fraction that is unaffected or uninhibited (fu). Taking the ratio of fi/fu or (1 - fu)/fu is equal to [(fu)-1 - 1] or [(1 - fu)-1 - 1]-1, where 1 - fu = fa (the fraction that is affected or inhibited). By the system and pattern analysis and by mathematical induction and deduction, it is shown that the ratio of (1 - fu)/fu, in turn, is always equal to the ratio of (I/IC50)m,1 where m is the kinetic order. More importantly, it is shown that Ki/I50 = Ex/Et, where Et is the total amount of enzyme and Ex is the fractional availability of the enzyme species with which the inhibitor may combine to (Chou, 1974; Chou and Talalay, 1981). This relationship holds irrespective of the number of substrates or products or their reaction mechanisms and it is also irrespective of the mechanism type of inhibition of the inhibitor. Therefore, this approach allows kinetic constants such as Km and Ki as well as Vmax, to be canceled out, leaving only the dose-effect relationship of the inhibitor.
Pattern analysis on numerous specific equations by using an approach similar to the mathematical deduction created the median-effect equation in 1976 (Chou, 1976, 1977), which is called the general theory of dose and effect. It is shown that the dose-generated effects are not random variables. There is a fundamental rule, i.e., the massaction law, that underlies and governs them. They are not governed by empirical curves. Most importantly, it is shown that "dose" and "effect" are interchangeable.
Similarly, the introduction of multiple inhibitors, such as I1 competitive with substrate A and noncompetitive with substrate B, and I2 competitive with substrate B and noncompetitive with substrate C, etc., again, allows for the derivation of hundreds of specific equations (Chou and Talalay, 1981). Using pattern analysis and the ratio of equations for mathematical deduction, the multiple drug-effect equation was derived and introduced by Chou and Talalay (1977, 1981), presented as the combination index equation (Chou and Talalay, 1983, 1984), and is also called the combination index theorem (Chou and Martin, 2005).
The logical steps for the derivation of the above theory and theorems are given in Fig. 2a and the flow chart of the method of derivation using the median-effect as the common link is given in Fig. 2b, along with the relevant references. More details of derivations are given in Appendix I. The fundamental equation of dose and effect, as well as the general theorems indicated above, should hold regardless of the number of reactants (substrates, products, and inhibitors), of the reaction mechanism (ordered sequential, ping-pong, or random), or the type of inhibition (competitive, noncompetitive, or uncompetitive) and, therefore, can be generally applied to diverse fields of biology, including biochemistry, pharmacology, and medicine.
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B. The Derivation of Equations and Theorems Based on the Mass-Action Law
1. The Median-Effect Equation.
A systematic analysis, with the classic kinetic models of enzyme-substrate-inhibitor interactions with different number of substrates, different reaction mechanisms, and different types or mechanisms of inhibition has been carried out (Chou, 1974). It has been concluded that for all cases, fractional velocity (fv) and fractional inhibition (fi) in the presence of an inhibitor (I) can be expressed by
 | (1) |
 | (2) |
; Chou, 1974). By definition, fv = vi/v and fi = (v - vi)/v, where vi and v are the reaction velocities in the presence and absence of an inhibitor.
Another general relation was induced from the analysis (Chou, 1974) which gives
 | (3) |
Therefore, Ki will never be greater than I50, and the ratio of Ki and I50 expresses the fractional distribution of the enzyme species in an enzyme reaction under specified experimental conditions. Thus, the ratio provides a simple experimental basis for determining the availability of the enzyme species for inhibitor binding.
Substitution of eq. 3 into eqs. 1 and 2 gives
 | (4) |
 | (5) |
Dividing eq. 5 by eq. 4 gives another form of describing the median-effect principle:
 | (6) |
Further analysis with the pharmacological receptor system, yielded a similar conclusion, which led to a general median-effect equation. The median-effect equation (Chou, 1976, 1977) describes dose-effect relationships in the simplest possible term, which is given by
 | (7) |
, 1977). Equation 7 is believed to be the simplest possible form for relating the dose (right side) and the effect (left side). Rearranging it yields
 | (8) |
 | (9) |
).
Plotting x = log(D) versus y = log(fa/fu) based on the logarithm form of eq. 7, as defined by Chou, is called the median-effect plot (Chou, 1976), where
 | (10) |
 | (11) |
 | (12) |
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; Armitage and Doll, 1954), the logit law (Reed and Berkson, 1929), or the probit law (Finney, 1952) can linealize dose-effect curves reasonably well. However, their coefficients or parameters and their slopes and intercepts in the plots have no physical or chemical bearings. These empirical approaches or statistical approaches render enormous difficulties when dealing with more complicated situations, such as drug combinations.
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2. The Unified Theory.
It should be noted that the parameters of the median-effect equation (eq. 7) have physical bearings related to the mass-action law. Thus, Dm signifies potency, m signifies the shape of the dose-effect curve (m = 1, hyperbolic; m > 1, sigmoidal; or m < 1, flat sigmoidal), and r signifies the conformity of the data to the mass-action law (Chou, 1976). Computer software has been developed to facilitate the simulation and the automated calculation of these parameters from the dose-effect data (Chou and Chou, 1985; Chou and Hayball, 1997; Chou and Martin, 2005). It should also be noted that both sides of the generalized median-effect equation (eq. 7) are ratios and, thus, are dimensionless quantities in equality.
As shown in Fig. 4, rearrangement of the median-effect equation and/or taking its logarithmic form gives rise to the four major equations in biomedical sciences, i.e., the Michaelis-Menten equation (Michaelis and Menten, 1913) for first-order enzyme kinetics (m = 1); the Hill (1910, 1913) equation for primary ligand occupancy at high order of biological receptors, such as oxygen-hemoglobin interaction (m = n); the Scatchard (1949) equation for ligand-receptor binding and dissociation; and the Henderson-Hasselbalch equation for pH ionization (Clark, 1928; Goldstein et al., 1968). Thus, equations that share the same mathematical form may have different physicochemical meanings. A comparison of distinctions between the median-effect equation (Chou, 1976) and the Hill (1913) equation are given in Table 2. When the endpoint of the measurement of the fraction affected (fa) (e.g., the fractional inhibition, fi, or percent inhibition/100) in the median-effect equation is changed to fractional saturation, fractional occupancy, fractional binding, and fractional ionization, respectively, the correspondence among the above-mentioned equations becomes clear. Retrospectively, it is not surprising that half-affected (Dm) is corresponding to half-saturated (Km), half-occupied (K), half-bound and half-free (KD), and half-ionization, the antilog of (pKa), where pH = -log [H+] (Chou, 1977, 1991; Chou and Chou, 1990b).
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Thus, the four major equations pioneered by Michaelis-Menten, Hill, Henderson-Hasselbalch, and Scatchard with obviously different appearances for different purposes in biomedical sciences can all be derived from the median-effect equation of the mass-action law principle. Thus, the median-effect equation is called the general theory of dose and effect. The normalization of the doseeffect curves based on eq. 4 is illustrated in Fig. 5. As indicated in the this subsection, the general equation of median and effect even allows drawing a dose-effect curve for only two data points.
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1. The Multiple Drug-Effect Equation.
The median-effect equation for a single drug can be extended to multiple drugs. Thus, for a two-drug combination, in a first-order system (when m = 1), we get (Chou and Talalay, 1981, 1984; Chou et al., 1983)
 | (13) |
1, then
 | (14) |
More detailed derivations for the multiple drug-effect equation are given in Fig. 2; see also Appendix I.
Equations 13 and 14 are based on the generalized assumption that two drugs share similar modes of action (i.e., effects are mutually exclusive), which is in complete agreement with the assumption of the classic isobologram. If one assumes that two drugs have totally different modes of actions (i.e., effects are purely mutually nonexclusive), then the resulting equation should, in theory, have a third term, which is the product of the first two terms (Chou and Talalay, 1984), thus,
 | (15) |
; Chou JH, 1991).
When two drugs are combined and subjected to serial dilutions, the combined mixture of the two drugs behaves as the third drug for the dose-effect relationship. Thus, y = log [(fa)1,2/(fu)1,2] versus x = log [(D)1 + (D)2] will give m1,2, (Dm)1,2, and r1,2 values (Chou, 1991).
2. The Combination Index Theorem and Plot.
Based on eqs. 13 and 14, Chou and Talalay in 1983 introduced the term combination index (CI) for quantification of synergism or antagonism for two drugs (Chou and Talalay, 1983, 1984; Chou, 1991):
 | (16) |
A plot of CI on the y-axis as a function of effect levels (fa) on the x-axis is called Fa-CI plot or in brief, CI plot (Chou and Talalay, 1981, 1984). It should be noted that the extreme end of the CI values for synergism is 0 to 1 and for antagonism is 1 to infinity. Fa-log(CI) plot, not only reduces the out-of-scale points in the Fa-CI plot, but also makes the presentation symmetrical with the additive effect axis (CI = 1) locating at zero [i.e., log(CI) = log(1) = 0]. Therefore, in the Fa-log(CI) plot, synergism is indicated by a negative value [i.e., log(CI) <0], and antagonism is indicated by a positive value [i.e., log(CI) > 0]. Therefore, at a special case of eq. 16 when CI = 1, the classic ED50 isobologram for two drugs at ED50 is described as (Chou and Talalay, 1981, 1984)
 | (17) |
 | (18) |
Figure 6a illustrates the conventional "classic isobologram" and Fig. 6b illustrates the dose-normalized isobologram. The typical isobolograms and their interpretations are also illustrated in Fig. 8b and 8c.
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), m = 3 (
), and m = 5(
). b, median-effect plots of a. c, dose-effect curves when m = 1 and when Dm = 0.5 µM(
), 6 µM(
), 8 µM (x), and 16 µM(+). d, median-effect plots of c. Similar graphics can be generated by using other m and Dm values in 
