Abstract
The medianeffect equation derived from the massaction law principle at equilibriumsteady 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 MichaelisMenten equation, Hill equation, HendersonHasselbalch 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 doseeffect curve, medianeffect plot, combination index plot, isobologram, dosereduction index plot, and polygonogram for in vitro or in vivo studies. This theoretical development, experimental design, and computerized data analysis have facilitated doseeffect 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 stepbystep examples of experimental designs and real data analysis are also illustrated. The merging of the massaction law principle with mathematical inductiondeduction has been proven to be a unique and effective scientific method for general theory development. The medianeffect principle and its massaction 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.
I. Introduction
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 doseeffect 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 doseeffect 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 physicalchemical 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 massaction lawbased 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.
A. Why Drug Combination?
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 (19721983) 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:

A + B > A or A + B > B says nothing about synergism. This is a simple arithmetic fact that needs neither proof nor requires an elaborated statistical task, such as determining p values.

Additive effect is not a simple arithmetic sum of two (or more) drugs. If A and B each inhibits 30%, then the additive effect is not 60% because if A and B each inhibits 60%, the combined additive effect cannot be 120%.

If A and B each inhibits 60%, then it is oversimplification to say that the additive effect is 84% inhibition. Based on the reasoning by Webb (1963), 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 doseeffect curve (e.g., hyperbolic or sigmoidal). The importance of the “shape” in a doseeffect 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 MichaelisMenten kinetics when doseeffect curves are hyperbolic, i.e., m = 1 in the medianeffect 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).
C. Truth or Fallacy and Its Consequences
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 doseeffect 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:
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 doseeffect 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 doseeffect 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 doseeffect curve was hyperbolic or sigmoidal and whether the drug interaction was determined at ED_{30}, ED_{50}, ED_{70}, ED_{95}, or ED_{99} 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 doseeffect 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 DoseEffect 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 massaction law with the mathematical principle of induction and deduction. After deriving hundreds of equations and three and onehalf 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 HendersonHasselbalch, MichaelisMenten, 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 dosereduction 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 19762006; http://portal.isiknowledge.com/portal.cgi?DestApp=WOS&Func=Frame), one article alone on the medianeffect principle (Chou and Talalay, 1984) has been cited in >1294 scientific papers in hundreds of biomedical journals.
A. An Approach of Merging the MassAction 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 errorfree 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 onehalf decades later, we now have the seconddegree Pascal's triangle (Chou, 1970, 1972), the medianeffect 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 dosereduction 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 medianeffect equation, which has been independently derived mathematically, is, in fact, the “unified theory” for the MichaelisMenten equation of enzyme kinetics, the Hill equation for higherorder ligand binding saturation, the HendersonHasselbalch 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 massaction law (C. M. Guldberg and P. Waage, 1864), the equilibrium law (A. F. Horstmann, J. W. Gibbs, and J. H. Van't Hoff, 18731886), and the absolute reaction rate theory (M. Polanyi and H. Eyring, 1935) [For references, see Bothamley (2002).] Using the equilibrium and steadystate 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 welldeveloped 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 MichaelisMenten equation, which described a single substratesingle product reaction at a steadyequilibrium state (Chou 1970, 1974). Intrigued by the intricacy of this process, Chou gradually and systematically extended the derivation to various multiple substratemultiple product reactions. This extension and pattern analysis used various notations introduced by Cleland (1963), such as the sequential ordered mechanism, the pingpong 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 minipathways. 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 (ν_{i}) and absence (ν_{0}) of an inhibitor. This ratio (ν_{i}/ν_{0}) is the fraction that is unaffected or uninhibited (f_{u}). Taking the ratio of f_{i}/f_{u} or (1  f_{u})/f_{u} is equal to [(f_{u})^{1}  1] or [(1  f_{u})^{1}  1]^{1}, where 1  f_{u} = f_{a} (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  f_{u})/f_{u}, in turn, is always equal to the ratio of (I/IC_{50})^{m},^{1} where m is the kinetic order. More importantly, it is shown that K_{i}/I_{50} = E_{x}/E_{t}, where E_{t} is the total amount of enzyme and E_{x} 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 K_{m} and K_{i} as well as V_{max}, to be canceled out, leaving only the doseeffect relationship of the inhibitor.
Pattern analysis on numerous specific equations by using an approach similar to the mathematical deduction created the medianeffect equation in 1976 (Chou, 1976, 1977), which is called the general theory of dose and effect. It is shown that the dosegenerated 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 I_{1} competitive with substrate A and noncompetitive with substrate B, and I_{2} 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 drugeffect 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 medianeffect 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, pingpong, 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.
B. The Derivation of Equations and Theorems Based on the MassAction Law
1. The MedianEffect Equation.
A systematic analysis, with the classic kinetic models of enzymesubstrateinhibitor 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 (f_{v}) and fractional inhibition (f_{i}) in the presence of an inhibitor (I) can be expressed by and respectively, where K_{i} is the enzymeinhibitor dissociation constant, E_{t} is the total amount of enzyme, and E_{x} is the amount of the enzyme species with which the inhibitor may combine. The ratio E_{x}/E_{t} can be quantitatively expressed by the distribution equation for each reaction mechanism of the enzyme and for each inhibition mechanism of the inhibitor (Cleland, 1963; Chou, 1974). By definition, f_{v} = v_{i}/v and f_{i} = (v  v_{i})/v, where v_{i} 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 where I_{50} is the concentration of I required for 50% inhibition.
Therefore, K_{i} will never be greater than I_{50}, and the ratio of K_{i} and I_{50} 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 and Therefore, if I_{50} is known, the degree of inhibition at any other concentrations of the inhibitor can be calculated.
Dividing eq. 5 by eq. 4 gives another form of describing the medianeffect principle:
Further analysis with the pharmacological receptor system, yielded a similar conclusion, which led to a general medianeffect equation. The medianeffect equation (Chou, 1976, 1977) describes doseeffect relationships in the simplest possible term, which is given by where D is the dose (or concentration) of a drug, f_{a} is the fraction affected by D (i.e., percentage inhibition/100), and f_{u} is the fraction unaffected (i.e., f_{u} = 1  f_{a}). D_{m} is the medianeffect dose (e.g., IC_{50}, ED_{50}, or LD_{50}) that inhibits the system under study by 50%, and m is the coefficient signifying the shape of the doseeffect relationship, where m = 1, > 1, and < 1 indicate hyperbolic, sigmoidal, and flat sigmoidal doseeffect curves, respectively (Chou, 1976, 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 and Therefore, the dose and the effect are interchangeable since the dose (D) for any given degree of effect (f_{a})in eq. 8 can be determined if the values for D_{m} and m are known. Likewise, in eq. 9, the effect (f_{a}) for any given dose (D) can be determined if the values for D_{m} and m are known (Chou, 1975).
Plotting x = log(D) versus y = log(f_{a}/f_{u}) based on the logarithm form of eq. 7, as defined by Chou, is called the medianeffect plot (Chou, 1976), where linealizes all the hyperbolic and sigmoidal doseeffect curves. Note that eq. 10 has the form of a classic straight line equation: In the medianeffect plot, m is the slope and (D_{m}) is the antilog of the xintercept, which can be easily determined. As indicated above, m = 1, > 1, and < 1 signify hyperbolic, sigmoidal, and flat sigmoidal doseeffect curves, respectively. On the basis of eqs. 10 and 11, the D_{m} value can be calculated easily by the following: An example for transforming the doseeffect curves into a linear form by the medianeffect plot is illustrated in Fig. 3.
The conformity of the data to the medianeffect plot of the massaction law can be readily manifested by the linear correlation coefficient (r) of the medianeffect plot in which r = 1 indicates perfect conformity. The use of the medianeffect principle for a doseeffect analysis is a distinct departure from the conventional statistical approach in which an empirical curve is drawn to fit the scattering data points. However, in Chou's approach, the scattering data points are used to fit the medianeffect principle of the massaction law. Using a statistical approach, it is not possible to draw a “curve” for the accurately determined “two data points,” whereas this can be accomplished with ease using the medianeffect principle (see analysis for Tables 7, 8, 9). Many empirical formulae in biomedical sciences, such as the power law (Nordling, 1953; Armitage and Doll, 1954), the logit law (Reed and Berkson, 1929), or the probit law (Finney, 1952) can linealize doseeffect 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.
2. The Unified Theory.
It should be noted that the parameters of the medianeffect equation (eq. 7) have physical bearings related to the massaction law. Thus, D_{m} signifies potency, m signifies the shape of the doseeffect curve (m = 1, hyperbolic; m > 1, sigmoidal; or m < 1, flat sigmoidal), and r signifies the conformity of the data to the massaction law (Chou, 1976). Computer software has been developed to facilitate the simulation and the automated calculation of these parameters from the doseeffect data (Chou and Chou, 1985; Chou and Hayball, 1997; Chou and Martin, 2005). It should also be noted that both sides of the generalized medianeffect equation (eq. 7) are ratios and, thus, are dimensionless quantities in equality.
As shown in Fig. 4, rearrangement of the medianeffect equation and/or taking its logarithmic form gives rise to the four major equations in biomedical sciences, i.e., the MichaelisMenten equation (Michaelis and Menten, 1913) for firstorder enzyme kinetics (m = 1); the Hill (1910, 1913) equation for primary ligand occupancy at high order of biological receptors, such as oxygenhemoglobin interaction (m = n); the Scatchard (1949) equation for ligandreceptor binding and dissociation; and the HendersonHasselbalch 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 medianeffect equation (Chou, 1976) and the Hill (1913) equation are given in Table 2. When the endpoint of the measurement of the fraction affected (f_{a}) (e.g., the fractional inhibition, f_{i}, or percent inhibition/100) in the medianeffect equation is changed to fractional saturation, fractional occupancy, fractional binding, and fractional ionization, respectively, the correspondence among the abovementioned equations becomes clear. Retrospectively, it is not surprising that halfaffected (D_{m}) is corresponding to halfsaturated (K_{m}), halfoccupied (K), halfbound and halffree (K_{D}), and halfionization, the antilog of (pK_{a}), where pH = log [H^{+}] (Chou, 1977, 1991; Chou and Chou, 1990b).
Thus, the four major equations pioneered by MichaelisMenten, Hill, HendersonHasselbalch, and Scatchard with obviously different appearances for different purposes in biomedical sciences can all be derived from the medianeffect equation of the massaction law principle. Thus, the medianeffect 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 doseeffect curve for only two data points.
C. Extension of MassAction Law to Multiple DrugEffect Systems
1. The Multiple DrugEffect Equation.
The medianeffect equation for a single drug can be extended to multiple drugs. Thus, for a twodrug combination, in a firstorder system (when m = 1), we get (Chou and Talalay, 1981, 1984; Chou et al., 1983) and when m ≠ 1, then
More detailed derivations for the multiple drugeffect 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, Because partially exclusive or partially nonexclusive (i.e., nonpure) cases may exist and eq. 15 may underestimate synergistic drug interactions, it is concluded that eq. 14 should be used as the “base equation” and that any mutually nonexclusive condition, if it exists, should be considered as a contributing factor for the intrinsic synergistic effect under the assumption of eq. 14 (Chou, 1991; 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 doseeffect relationship. Thus, y = log [(f_{a})_{1,2}/(f_{u})_{1,2}] versus x = log [(D)_{1} + (D)_{2}] will give m_{1,2}, (D_{m})_{1,2}, and r_{1,2} 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): where CI < 1, = 1, and > 1 indicate synergism, additive effect, and antagonism, respectively. In the denominator, (D_{x}) is for D_{1}“alone” that inhibits a system x%, and (D_{x})_{2} is for D_{2}“alone” that inhibits a system x%. The (D_{x})_{1} and (D_{x})_{2} values can be calculated from eq. 8. In the numerators, (D)_{1} + (D)_{2}“in combination” also inhibit x%. If the sum of these two fractional terms in eq. 16 is equal to 1, additive is indicated. If the CI value is smaller than 1, synergism is indicated, and if the CI value is greater than 1, antagonism is indicated.
A plot of CI on the yaxis as a function of effect levels (f_{a}) on the xaxis is called F_{a}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. F_{a}log(CI) plot, not only reduces the outofscale points in the F_{a}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 F_{a}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 ED_{50} isobologram for two drugs at ED_{50} is described as (Chou and Talalay, 1981, 1984) and the ED_{x} isobologram for two drugs for x% inhibition is described as
Figure 6a illustrates the conventional “classic isobologram” and Fig. 6b illustrates the dosenormalized isobologram. The typical isobolograms and their interpretations are also illustrated in Fig. 8b and 8c.
3. The General Equation for Combination of n Drugs.
The detailed description for the derivation of the multiple drugeffect equation based on the medianeffect principle (Chou, 1976) has been given in Chou and Talalay (1977, 1981, 1984). This process, including the flow chart, has been outlined in Fig. 2 of this article. The early steps of derivations are summarized in Appendix I, which lead to the combination index equation.
Similar to eq. 16, the general equation of a fivedrug combination at x% inhibition is The general equation for ndrug combination at x% inhibition becomes where ^{n}(CI)_{x} is the combination index for n drugs at x% inhibition, (D_{x})_{1}_{n} is the sum of the dose of n drugs that exerts x% inhibition in combination, is the proportionality of the dose of each of n drugs that exerts x% inhibition in combination, and (D_{m})_{j}{(f_{a}_{x})_{j}/[1  (f_{a}_{x})_{j}]}^{1/}^{mj} is the dose of each drug alone that exerts x% inhibition, where D_{m} is the medianeffect dose (antilog of the xintercept of the medianeffect plot), f_{a} is the fractional inhibition at x% inhibition, and m is the slope of the medianeffect plot, which depicts the shape of the doseeffect curve (where m = 1, > 1, and < 1 indicate hyperbolic, sigmoidal, and flat sigmoidal curve, respectively).
4. Algorithms for Determining Synergism and Antagonism.
The combination index equation as shown in eqs. 16 and 20, in conjunction with eqs. 8 and 9, can be used as algorithms for computerized simulation for the combination index values at different effect levels (i.e., at different f_{a} values). This multiplestep calculation for simulation is shown in Fig. 7. The computer program for fully automated simulation has been developed (Chou and Chou, 1988). Chou and Talalay (1984) and Chou and Chou (1985) have named this graphic the F_{a}CI plot and later the combination index plot or the CI plot. A typical F_{a}CI plot and its interpretation is shown in Fig. 8a, where CI < 1, = 1, and > 1 indicate synergism, additive effect, and antagonism, respectively.
5. Main Features of the General Equation.
The most unique and important feature for both the general theory of dose and effect (eq. 1) and the combination index theorem (eq. 20, including eqs. 13, 14, 15, 16, 17, 18) is that each term of the equation is a ratio. So, for each drug entity, the unit is canceled out and thus becomes a dimensionless quantity. This feature is essential for their general broad applicability. It is important to note here that based on the Web search of ISI citation records, one article alone (Chou and Talalay, 1984) has been cited in >1294 scientific papers in biomedical journals. Furthermore, due to the dimensionless feature of the combination index equation, D_{1} can be in micromolar concentrations, D_{2} in micrograms per milliliter, D_{3} in international units, and D_{4} in multiples of infection and their interactions in terms of synergism, additive effect, or antagonism can still be determined, as long as there is a conformity to the massaction law principle (e.g., r > 0.95). In addition, the drugs can be unknown entities without knowing their chemical structures, molecular weights, or mechanism of action. One may also combine one mixture with another mixture and determine their interactions (e.g., for Chinese herbal medicines). The applications have been extended to the combination of a drug with radiation or a drug combination at different oxygen tensions for cytotoxicity studies (Chou and Talalay, 1984; Chou, 1991; Chou JH, 1991). The application to hyperthermia and pH remains to be explored. However, these latter applications require very accurate measurements, since very steep doseeffect relationships (i.e., high m values in the medianeffect plot) are expected.
As indicated above (see section II.B.2.), the medianeffect equation allows the demonstration of how to draw a doseeffect “curve” with a minimum of only “two data points” if the data are accurately determined (see also Fig. 11 and Supplemental Data Appendices II and III; http://pharmrev.aspetjournals.org/cgi/content/full/pr.58.3.10/DC1). Furthermore, based on the combination index theorem, synergism or antagonism can be quantitatively determined for a minimum of only a “single combination data point” if the parameters (m and D_{m}) for each drug are determined by the medianeffect plot (for examples, see the illustrations given in Supplemental Data Appendices II and III). Thus, these feature the simplest barebones situations for any doseeffect analysis. The applications of this theory and/or theorems not only provide sound rational basis for biomedical research but also allow for smaller experiments than in the past and, therefore, lead to saving time, effort, and research cost, as well as conserving the use of laboratory animals and, thus, minimizing the pain and suffering of laboratory animals in a general way (see section VI.D.).
6. The F_{a}Combination Index Plot and Isobologram Are Two Sides of the Same Coin.
An isobologram is a graph with equipotency sum of doses. The isobol concept was conceived about a century ago (Loewe, 1928, 1957; Berenbaum, 1989). However, it was not very well received then and generated some confusion and controversies. It lacked theoretical tools for rigorous treatment in the early studies. Several decades ago, researchers used graph paper for drawing isoeffective points from extrapolation or interpolation of doseeffect curves, which would take ½ h to complete. Now with the isobol equation, the medianeffect principle, and the computer program that is widely available, multiple effect level isobolograms (e.g., ED_{50}, ED_{75}, and ED_{95} isobolograms) for a pair of drugs can be constructed in a split second. For an example of an ED_{50} isobologram, see Fig. 6.
Webb's fractional product method for the inhibition of single substratesingle product MichaelisMenten system with two inhibitors (Webb, 1963) had been expanded to the multiplesubstrate and multipleproduct MichaelisMenten system (m = 1) with n inhibitors by Chou and Talalay (1977, 1981). A comparison of Webb's method and Loewe's method and the CI method for multidrug inhibition is given in Table 3.
In theory, both the isobologram and the F_{a}CI plot of Chou and Talalay should yield exactly identical conclusions in drug combination studies, since both graphics are based on the same combination index equation. They are like two sides of the same coin: the isobologram is a doseoriented graphic and the CI plot is an effectoriented graphic, but both are based on the combination index theorem of Chou and Talalay.
However, the isobologram has some practical limitations. First, an isobologram has two dimensions, which is convenient for twodrug combinations. For threedrug combinations, it is not convenient to construct a threedimensional isobologram, and even if it were constructed, it would not be easy to read (Chou and Chou, 1992). Alternatively, one can split ndrug combination into two parts and assign them to two axes (Chou, 1991). For example, where 1 ≤ m ≤ n  1 and n and m are integers. Second, if twodrug isobolograms are shown at three or fewer effect levels (e.g., at ED_{50}, ED_{70}, and ED_{90}), usually the graph will be readable by inspection. But if an isobologram were to be constructed for four or more effect levels, it would usually be too messy to read because of data scattering or data point congestion or overlapping. By contrast, the CI plot of Chou and Talalay can be shown at all f_{a} levels simultaneously with any number of drugs that are combined.
7. How Much Synergism Is Synergy?
We frequently hear that “drug A and Drug B in combination are synergistic.” However, when this claim is made, there are a string of conditions that are attached to it. Obviously, the claim is valid for the experimental conditions and experimental designs that are used (e.g., temperature, oxygen tension, cell lines, species, gender, age, race, sequence of drug addition, ratio of drug combination, schedule of drug administration, and so on), although some variations in conditions may make little or no difference in the conclusions. After all, the synergism or antagonism determinations are based on the massaction law principle.
At the extremes, synergism is CI = 0 to 1, and antagonism is CI = 1 to infinity. Therefore, log(CI) will provide symmetric presentation of the CI graphics [i.e., in the F_{a}log(CI) plot for additive effect, log(CI) = 0; for synergism CI = 0.1 and 0.01 will give log(CI) = 1 and 2; for antagonism CI = 10 and 100 will give log(CI) = 1 and 2, etc.]. It is important to note that synergism and antagonism can be different at different dose levels or different effect levels. Therefore, in many publications, the CI and dosereduction index (DRI) values are presented in the summary table for the ED_{50}, ED_{75}, ED_{90}, and ED_{95} levels (e.g., Chou et al., 1994, 2005). For some chronic or physiological conditions or diseases, synergism or antagonism at low dose or low effect levels is important. But for infectious diseases or cancer therapies, synergism at high effect levels (e.g., at ED_{90}, ED_{95}, or ED_{99}) is much more therapeutically relevant than at low effect levels (e.g., ED_{30} or ED_{50}). Therefore, synergism or antagonism at different effect levels may have different significance for different diseases. In addition, selective synergism against the target and antagonism toward the host is also of practical importance. Furthermore, whether the concentrations for synergism are achievable in the body (e.g., blood or body fluid) is also an issue to be considered.
In the past, Chou and coworkers have proposed semiquantitative methods for describing the degrees of synergism or antagonism (Chou and Talalay, 1984; Chou and Hayball, 1997). These methods are now refined and expanded as shown in Table 4. Using the log(CI) grading, synergism is subdivided into (near) additive (±), slight synergism (+), moderate synergism (++), synergism (+++), strong synergism (++++), and very strong synergism (+++++). Antagonism is divided in the same way, except using “” sign(s); thus, the corresponding symbols are ±, ,  ,   ,    , and      (Table 4). In polygonograms (Fig. 9), for visual distinction, synergism is represented by a solid line and antagonism is represented by a dash line. The degree of synergism or antagonism is represented by the thickness of the line. Synergism or antagonism in polygonograms can be represented by colors. Usually, synergism is represented by redtone colors and antagonism is represented by bluetone colors. Colors in the polygonogram are used for enhancing the visual contrast.
D. The DoseReduction Index Equation and Plot
One of the major objectives of having synergistic drug combination is to reduce the dose of the drug used, thereby reducing the toxicity while maintaining efficacy. The concept of the dosereduction index was formally introduced by Chou JH and Chou TC in 1988 and has since been used in many publications (Chou, 1991, 1994; Chou JH, 1991; Chou et al., 1991, 1994; Bertino and Chou, 1997). The DRI is a measure of how many fold the dose of each drug in a synergistic combination may be reduced at a given effect level compared with the doses of each drug alone. Simply by inverting each term of eqs. 16 and 20, the DRI value for each corresponding drug is given. Thus, for twodrug combinations and for ndrug combinations Therefore, or
The DRI is important in clinical situations, in which dose reduction leads to reduced toxicity toward the host while the therapeutic efficacy is retained. Although DRI > 1 is beneficial, it does not necessarily indicate synergism because, from the above equation, an additive effect or even slight antagonism may also lead to DRI > 1. If drug A and drug B each inhibit 50%, and if (0.5A + 0.5B) also inhibits 50% and if both drugs have no overlapping toxicity toward the host, then indeed DRI ≥1 may still be beneficial. The greater DRI value indicates a greater dose reduction for a given therapeutic effect, but does not necessarily always indicate synergism, as shown by the mathematical relationship of CI and DRI given in eq. 21.
Facilitating the use of the above, the computer program for automated simulation of DRI values at different f_{a} values for each drug in the combination (i.e., the DRI table or the DRI plot) has been developed (Chou and Martin, 2005). Selected examples for the F_{a}DRI plots and the F_{a}log(DRI) plots are illustrated in section VI. and Supplemental Data Appendices II through V (http://pharmrev.aspetjournals.org/cgi/content/full/pr.58.3.10/DC1). A typical F_{a}DRI plot and its interpretation are illustrated in Fig. 8d.
E. The Polygonogram
The concept of the polygonogram comes not from mathematical derivations but rather from practical utility (Chou et al., 1994; Chou and Martin, 2005). For example, 7 drugs may have 120 possible combinations (i.e., ). It would be difficult to memorize and to contemplate these ramifications. As shown in Fig. 9a and discussed in section VI.C.3., a heptagonal graph for 7 antiHIV agents allows a simplified visual presentation of the overall results (for the full report, see the CompuSyn printout in Supplemental Data online at http://pharmrev.aspetjournals.org/cgi/content/full/pr.58.3.10/DC1). Supplemental Data Appendix IV gives the experimental example of the polygonograms for the combination studies of five anticancer agents with different mechanism of actions. In Fig. 9b, pentagonal polygonograms based on two and threedrug combinations are shown (Chou et al., 1994) with a red solid line representing synergism and a blue broken line representing antagonism, and the thickness of the line representing strong and weak, respectively. The combination results from two to n drugs (in this case, n being five or seven drugs) can now be shown in one page graphically for easy inspection. The polygonogram in solid synergistic patterns may be used to pick out the potentially interesting combinations for further exploration. To some extent, the polygonogram also allows the projection (or prediction) of the combination outcomes in cases for which the experiments have not yet been carried out. For example, from the results of many twodrug combinations as shown in Fig. 9, one may semiqualitatively and semiquantitatively predict what would happen with three or fourdrug combinations (see also sections VI.A.3. and VI.C.3.).
III. Experimental Design for Drug Combinations
A. The Prerequisite and Theoretical Minimum Requirements for Drug Combination Studies
The prerequisite for synergism or antagonism determination is to know both the “potency” and the “shape” of the doseeffect curve for each drug. Therefore, it is necessary to know the doseeffect parameters of each drug alone [e.g., m_{1}, (D_{m})_{1}, r_{1}, m_{2}, (D_{m})_{2}, and r_{2} for twodrug combinations] and thereby determine the CI value, which can be calculated based on eq. 16 for twodrug combinations or eq. 20 for ndrug combinations. Thus, the experiment for the doseeffect curves for each drug alone needs to be carried out. The above parameters can be easily determined from the medianeffect plot (see Fig. 3) by using graph paper, but the available computer software can automatically calculate them with ease. It is highly recommended that the parameters for the combination [e.g., for two drugs, m_{1,2}, (D_{m})_{1,2}, and r_{1,2}] be acquired, but they are not absolutely necessary, because for even a single data point (from a single combination dose) of a drug combination mixture, the CI value can still be calculated by eq. 16 or eq. 20. Thus, the theoretical minimum of data points for a twodrug combination is five [i.e., two points for each drug alone and one point for the combination, with effects (f_{a}) being calculated relative to the control]. However, for the in vitro drug combination studies, in which experiments are easily carried out, each drug and its combinations usually consist of five to eight data points. By contrast, for the in vivo studies, usually the numbers of data points are somewhat reduced (e.g., three to five data points each) because of the practicality of the sample size, experimental manipulability, and costs. In drug combination studies, the experiments for a single drug and its combinations should be carried out simultaneously to ensure the same experimental conditions, such as avoiding drug decomposition due to instability, variability due to assay conditions, personnel changes, and cells or animal inconsistency.
For determining synergism or antagonism, the knowledge of “mechanisms” for each drug alone is not required. The doseeffect relationship per se does not tell the mechanism. It only tells the massaction law parameters. Furthermore, if eq. 16 or eq. 20 were mechanismdependent, it would be quite useless because of the imposed restrictions. Mechanism independence is the reason that eq. 7 is a general theory and eq. 16 or eq. 20 is a theorem.
Although it is good to know the mechanism before conducting drug combination studies, such as combining the MDRsubstrate drugs with MDRreversing agents, e.g., vinblastine + ardeemin (Chou et al., 1998a) or paclitaxel (Taxol) + ningalin (Chou et al., 2005b) and doxorubicin + M_{4}N (Chang et al., 2006), there is no way to predict synergism or antagonism when one combines cisplatin (a DNA crosslinking agent) and/or paclitaxel (a microtubule stabilization agent) and/or vincristine (a microtubule depolymerization agent) and/or topotecan (a DNA topoisomerase I inhibitor) and/or VP16 (a DNA topoisomerase II inhibitor) until one has actually performed the experiment and analyzed the data as shown by Chou et al. (1994). These examples will be illustrated in detail in section VI.A.3., including the illustrations with polygonograms; in addition, the detailed computer printout is given in Supplemental Data Appendix IV for Taxol, cisplatin, and topotecan.
When presenting a synergistic effect, a question frequently asked is “What is the mechanism of the synergism?” This is a hard question to answer. In fact, we know the mechanisms of very few drugs. We even do not know the detailed mechanisms for old drugs, such as aspirin or ether. The argument here depends on what you mean by “mechanism.” Are you talking about it at a pharmacological, biochemical, cellular, molecular, chemical, or quantum mechanics level? Even for a highly purified or crystalized enzyme such as horse alcohol dehydrogenase, when it is inhibited by a simple competitive inhibitor (ADP) in combination with a simple noncompetitive inhibitor (ophenanthrolene), Chou and Talalay (1981) demonstrated that it was difficult to predict the result quantitatively, in terms of synergism, antagonism, or additive effect, without analyzing the experimental data first (see Supplemental Data Appendices II and III).
When we combine two cytotoxic agents (or other agents) to kill a cancer cell, we really do not know how many events there are from a living cell to a dead cell. It can be several steps or it can be a hundreds of steps. Furthermore, when we talk about synergism or antagonism of two or more drugs, they are considered to be “mutual.” Although we know the dose or effect contributions in the combination, we so not know quantitatively the proportions of the mechanistic contributions of each drug that lead to the observed synergism or antagonism. Therefore, we need to be cautious when we state the causes and consequences in drug combinations, especially if one drug may have multiple mechanisms. When one of the two drugs by itself has no effect, then the enhancement (or augmentation or potentiation) or inhibition (or suppression) is easily described, simply by percent or by fold. It is not possible to determine CI because it is not possible to determine the m, D_{m}, and r values when a drug by itself has no effect.
For a simple set of drug combination studies in vitro, it will take approximately 1 to 2 weeks for the quantitative determination of synergism or antagonism. But if one wants to know how and why synergism or antagonism occurs, it may take months or years and yet the conclusions may only be tentative, suggestive, or implied. Thus, the determination of synergism or antagonism and the elucidation of how or why it occurs are separate issues.
With the current state of the art, even if we elucidate the primary, secondary, and tertiary structure of an enzyme or a receptor, it is still hard to design an inhibitor for the drug development process. To “predict” synergism or antagonism for enzymes, receptors, or biological systems is expected to be even more difficult.
B. Constant Ratio Drug Combinations, Dose Range, Dose Density, and Experimental Scheme
The diagonal constant ratio combination design proposed by Chou and Talalay (1984) and Chou (1991) can greatly reduce the number of animals needed for experiments (i.e., greatly reduce the number of data points required) and yet one can still receive the maximal amount of useful information on combinations, thus increasing the costeffectiveness of experimentation (see also section VI.D. for conservation of laboratory animals). The diagonal scheme proposed (Table 5), unlike the checkerboard or Latinsquare scheme, is an easy experiment to conduct simply by serial dilution of a mixture with a small number of data points and yet it allows the construction of F_{a}CI and F_{a}DRI plots with the actual combination data points. Furthermore, it also allows computer simulation of CI and DRI values at all effect levels (see sections IV and VI). It is further recommended that for an earlystage study, the constant combination ratio experiment should be carried out at an equipotency ratio [e.g., (IC_{50})_{1}/(IC_{50})_{2} ratio] so that the contributions of effects of each drug to the combination would be roughly equal (Table 5) (see also Chou and Talalay, 1984; Chou JH, 1991).
C. The Nonconstant Ratios of Drug Combinations
Tests of the combination at other combination ratios can also be carried out to determine the optimal combination ratio for maximal synergy. In some special cases, experiments are carried out at nonconstant ratios [e.g., keep (D)_{1} constant while varying the (D)_{2} doses]. As long as the m and D_{m} values for each single drug are available, the CI values for each combination data point for the nonconstant ratio design can still be calculated. In these cases, the CI values for each data point on the F_{a}CI plot will be shown, but the plot cannot be simulated (for examples, see section VI and Supplemental Data Appendices II and III). In addition, the normalized isobologram can be constructed for the nonconstant ratio combination design, but only the constant ratio combination design can yield the classic isobologram (see Fig. 8b and 8c).
D. The Optimal Combination Ratio for Maximal Synergy
Chou and Talalay (1983, 1984) recommend that the combination ratio be kept at an equipotency ratio (e.g., IC_{50} or ED_{50} ratio). This way, the contribution to the combination by each drug would be approximately equal. However, doing so is not absolutely required. A drug can be emphasized or deemphasized in many special situations, such as a severe type of toxicity (such as cardiotoxicity or renal toxicity), a narrow toxicity dose range for one drug but not another, or limited availability of a drug, such as source or price or poor solubility. Thus, in extended studies, one can set an arbitrarily particular desired ratio, carry out the experiment, and see what happens. One can also set the equipotent 1:1 dose ratio, add 1:3 and 3:1 dose ratios and find out which ratio yielded better synergy. While looking to synergy, one should also beware that the dose range is feasible in vivo and that the concentration or dose range is in a therapeutically effective range. The use of a computer software for determining the optimal combination ratios for maximal synergy has been proposed (Chou et al., 1986).
E. Combination Designs for Three or More Drugs
The equations for three or more drug combinations have been given in eq. 20, which shares the general form of the twodrug combinations (eq. 16). For a threedrug combination, D_{1}, D_{2}, and D_{3}, the recommendation is to use their IC_{50} ratios a:b:c. Therefore, D_{1} + D_{2} + D_{3} is at (a:b:c). It is also recommended that twodrug combinations be carried out at the same time; thus, D_{1} + D_{2} (a:b), D_{2} + D_{3} (b:c), and D_{1} + D_{3} (a:c) (Fig. 10a). These, in fact, serve as a dissection of threedrug combination into twodrug combinations (Fig. 10b). In this way, not only are the corresponding CI values determined, but also determines the secondary CI (e.g., how the third drug affects the two drug combination can also be determined). In effect, three drugs are considered as if they were two drugs. Therefore, in this experimental design, we have the doseeffect curves for D_{1}, D_{2}, and D_{3} alone, for twodrug combinations D_{1} + D_{2}, D_{2} + D_{3}, D_{1} + D_{3} and for threedrug combination D_{1} + D_{2} + D_{3}. In this case, the twodrug combinations are not absolutely required, although they dissect the threedrug combinations (e.g., D_{1} + D_{2} may be moderately synergistic, D_{2} + D_{3} may be antagonistic, D_{1} + D_{3} may be strongly synergistic, and D_{1} + D_{2} + D_{3} may be synergistic). These dissections provide more insight into the pharmacological interactions of their components.
In theory, the CI values can be determined for ndrug combinations (see eqs. 19 and 20). To avoid experimental variability due to variables in assay conditions, it is recommended that doseeffect curves for each single drug and its combinations be carried out at the same time, whenever possible. This practical consideration would limit the experimental size that is feasible to manage, although the availability of highvolume microwell high throughput assays in vitro would minimize part of the problem. Most drug combinations reported on so far with quantitative CI analysis have been twodrug combinations, although some threedrug combinations (Johnson et al., 1990; Kong et al., 1991; Pan et al., 1992; Kahan et al., 1993; Merrill et al., 1996; Tremblay et al., 1999; HofmannLehmann et al., 2001a,b) and a few fourdrug combinations have been reported on as well (Li et al., 1998; Baba et al., 2000; Kitabwalla et al., 2003). By using computer software, such as CompuSyn (Chou and Martin, 2005), the sevendrug combination (with some component combinations) has been carried out in the author's laboratory (Fig. 9a). Some of the above representations for multidrug combinations are illustrated in sections VI.A.3., VI.C.3., Supplemental Data, and Supplemental Data Appendix IV using the actual experimental designs and crude data, as examples for the computerized analysis.
The traditional Chinese herbal medicines pose a complex and challenging situation for drug combination studies. So far, no systematic analyses have been carried out or have been proposed. The medianeffect principle and the combination index method may provide a scientific quantitative approach to these problems, since, in theory, the CI for ndrug combination can be determined. The main task is to separate, divide, and reduce by means of split elimination to deduce meaningful or useful conclusions.
F. Drug Combination in Vitro, in Vivo, and in Clinics
Experimental conditions for drug combination in vitro can be easily defined, fixed, or standarized, and the drug concentrations can be maintained at a relatively constant during the course of the experiment. Usually, it takes approximately 1 to 2 weeks to complete an in vitro drug combination study based on the CIisobol method of Chou and Talalay (1984). It is quick, accurate, and economical, and, therefore, most drug combination studies in biomedical literature have been conducted in vitro, with doseeffect curves each consisting of five to eight data points for each drug alone and their combinations.
Compared with in vitro studies, determining synergism or antagonism in vivo using animals is obviously more time consuming and more costly and greater variability in measurements will be encountered. Therefore, in vivo drug combination studies are usually carried out only for selected drugs, after in vitro combination studies, and/or before clinical development.
For a twodrug combination therapeutic studies against human tumor xenografts in nude mice (Chou et al., 2005a), they usually consist of four groups: 1) the control group treated with solvent vehicle only, 2) the group treated with drug A at three to four doses, 3) the group treated with drug B at three to four doses, and 4) the group treated with a combination of drugs A and B, at a constant combination ratio (e.g., at the ED_{50} ratio) with several combination dose levels or at nonconstant combination ratios (e.g., keep drug A at constant dose and vary the dose of drug B at several levels). Each dose is used in four to five mice. Thus, approximately 60 nude mice would be needed for the experiment, if one already has preliminary information about the maximal tolerated dose (MTD), route of administration, and treatment schedule for each drug. The length of the study would be approximately 2 to 3 months, not including the followup studies, such as tumor remission or relapse. An example of a detailed in vivo drug combination study in nude mice based on the CIisobol method is found in Chou et al. (2005a). Application of the combination index method in the design of clinical trials has been discussed by Mildvan et al. (1990), Chou et al. (1994), and Chou (1998). Combination therapy of HIV1 infections has been widely implemented in clinical settings (Zhang et al., 1999).
Quantitative determination of the synergism or antagonism of two drugs in clinical trials is very difficult to carry out and, in most cases, practically impossible to perfect. This is apparently the reason why many drug combination clinical trials, such as the combination of anticancer natural product drug(s) + MDRreversing agent(s), have failed or reached inconclusive results. These disappointing results point to the conclusion that the in vitro and animal drug combinations should not be overlooked (Chou, 1994, 1998; Chou et al., 1994, 1998a, 2005a; Chang et al., 2006). The complexities of clinical drug combination trials can be due to a variety of reasons: 1) the patient population varies in terms of sex, age, race, stage of disease, and the history of past treatments; 2) it is not ethical to treat patients with a placebo or suboptimal therapy of doses as required for the drug combination study design; and 3) death should not be allowed in clinical studies as a toxicity endpoint as in animal studies. Therefore, the success of a drug combination clinical trial would very much rely on the information provided by the in vitro and/or animal drug combination studies, which would include dose, schedule, route of administration, efficacy, toxicity, synergism/antagonism, and schedule dependence. There will be little or no flexibility in clinical trials. A comparison of drug combinations for a pair of antitumor agents in vitro, in animals and in clinics is shown in Table 6. This comparison is based on the combination index perspectives (e.g., dose range, dose density, D_{m} value, m value, r value, and quantitation by CI, DRI, and/or isobol) and based on the population size, population variability, as well as ethical considerations. This demonstrates that a clinical trial for drug combination is very difficult to carry out properly. One important issue raised from Table 6 is whether it is ethically acceptable or scientifically correct to conduct a drug combination study in humans without doing any drug combination studies in vitro or in animals that will take only 1 to 2 weeks and 2 to 3 months to complete, respectively. Another issue is whether humans should be subjected to receipt of drug combination treatments under suboptimal conditions or subjected to trialanderror testing or to the chance of for better or for worse results in clinical trials in the absence of any supporting evidence or established rationale for the specific combinations. These issues should be confronted by clinicians who conduct drug combination trials, as well as regulatory agencies, such as the U.S. Food and Drug Administration, which approves investigational new drug applications for drug combinations.
G. Schedule Dependence
For the combination of two drugs, A and B, they can be administered simultaneously, or one after another (e.g., A → B or B → A). The time gap between two drugs can be varied, depending on the need and the design. In sequential administrations, such as A → B, administration of A can be considered as preconditioning or pretreatment for sensitization, desensitization, or preventing toxicities, etc. If A or B by itself has no effect, then the potentiation (enhancement, augmentation) or suppression (inhibition) can be presented as percent inhibition or fold potentiation. If both A and B have an effect, then synergism, antagonism, or an additive effect can be determined using the combination index or the isobol method. In this case, it is important that the assay conditions (such as the incubation time) for “each drug” alone in the sequential administration and for “its combinations” should be corresponding and identical, so that the doseeffect parameters (m and D_{m} values) can be used without any biases.
The schedule dependence studies for three or more drugs should follow the same rule, but it is expected to be more complex than the twodrug sequential combinations. However, sometimes we can lump together two or several drugs as a group and then conduct sequential drug combinations among the groups, such as (A + B) → (B + C), for determining the temporal significance on synergism or antagonism. The schedule dependence drug combination studies using the CI method have been carried out in various fields of research, such as those of Chang et al. (1987), Eron et al. (1992), Chou et al. (1996), Candinas et al. (1997), Longo et al. (1998), Rigas et al. (1999), Takahashi et al. (2002), and others.
H. ConditionDependent Synergism or Antagonism and Combination of Drugs with Different Modalities, Different Units, and Mechanisms
The combination index theorem for two or more drugs (eqs. 16 and 20) as indicated above is valid for combinations of different entities of the drug with the same or different modes of actions. Because all terms in the equations are ratios, all the dose units in eqs. 13, 14, 15, 16, 17, 18, 19, 20 are canceled out and become dimensionless quantities. Therefore, in drug combinations, drug A can be in micromolar concentrations, drug B can be in micrograms per milliliter, drug C can be in international unit, and so forth.
Although the most common combinations are drugdrug combinations (see section V.), the theorem for combination can be applicable to other conditions, such as drug + oxygen tension (Durand, 1990), drug + radiation (Leonard et al., 1996), drug + virus (Aghi et al., 1999; Bennett et al., 2004), drug + biomodulator (Hamashima et al., 1995; Finch et al., 2000), drug + antibodies (Wang et al., 1996; Li et al., 1997), or radiation + oxygen tension (Seo et al., 2006), and radiation + virus (Adusumilli et al., 2004), as long as the medianeffect principle is followed by the doseeffect relationship (i.e., reasonably good r values in the medianeffect plot). Whether it is applicable to drug + pH (Smith et al., 1989), drug + temperature (LakhdarGhazal et al., 1986; Hegedus and Khachatourians, 1996), and drug + magnetic field (Liang et al., 1997) remains to be further explored.
IV. Computerized Automation, Graphic Simulation, and Informatics
A. Computer Software
With the general equations of MEP and CI available, it becomes a logical consequence to develop a massaction law basedcomputer software for automated data analysis. The first software of this kind was developed and demonstrated by a 13 year old, Joseph H. Chou, who presented it at the American Society of Pharmacology and Experimental Therapeutics annual meeting in Philadelphia (Chou et al., 1983). The first edition was for the Apple II computer (Chou and Chou, 1985) followed by the IBMPC edition (Chou and Chou, 1987). Thereafter, the concept of DRI was introduced (Chou and Chou, 1988), and its applications were incorporated into a new version of software called CalcuSyn (Chou and Hayball, 1997), which was presented in Windows format. The most updated software, also for IBMPC, is called CompuSyn (Chou and Martin, 2005). CompuSyn generates better quality graphics that are ready for publication (Chang et al., 2006), more options and flexibility and improved statistics and contains updated developments over the earlier versions (for more information, contact combosyn{at}gmail.com). Distinct from earlier versions of the software, CompuSyn is able to handle the data analysis of largescale drug combination studies all at once. For example, 31 sets of data for 5 drugs have been analyzed simultaneously, and a 90page report was generated, which includes 60 color graphics, many tables, and an overall summary (see Supplemental Data at http://pharmrev.aspetjournals.org/cgi/content/full/pr.58.3.10/DC1).
B. The MedianEffect Plot and the Simulation of DoseEffect Curve
The general dose and effect theory as depicted by the medianeffect equation (eqs. 7, 8, 9) has been estimated to have been tested in >20,000 sets of experimental data in >2600 scientific papers for single drugs and in combinations since its introduction in 1976. In the author's laboratory alone, >5000 sets of data have been used for analysis. The usefulness of the medianeffect equation can be mathematically illustrated with the bold numbers given in Table 7, and its utility can be confirmed by the computer generated doseeffect curves and the medianeffect plot in Fig. 11, colormatched with the numbers in Table 7.
The doseeffect relationship based on the medianeffect equation (eq. 7) at m = 1, 3, and 5 and with D_{m} values at 0.5, 1, 2, 4, 6, 8, and 16 are given in Table 7. After entering the doses (D_{x}) and the corresponding f_{a} values on rows 4 to 6 into the computer, CompuSyn generates (Fig. 11) doseeffect curves (a) and the medianeffect plot (b), which accurately manifest D_{m} = 1 μM (i.e., antilog 0 = 1), and the slopes of 1, 3, and 5. After entering the doses (D_{x}) and the f_{a} values on rows 1, 4, 7, 10, 13, 16, and 19 into the computer, CompuSyn again generates doseeffect curves (c) and the medianeffect plot (d) with an accurately depicted slope of 1 with D_{m} values at 0.5, 1, 2, 4, 6, 8, and 16 μM, respectively.
It should be noted that even just entering iteratively “any two data points of each row” into the computer, CompuSyn will still generate a nearly identical full doseeffect curves faithfully. Thus, it is possible to generate a doseeffect “curve” with only two data points, if the data are accurately measured. This feature defies the commonly held belief that two data points can only draw a connecting straight line. Thus, the MEP of the massaction law provides the rational basis for using a small (or a reduced) number of data points under the improved precision of measurement. This dosenumber reduction can be translated into using a small number of animals and thus not only leads to costeffective benefits, but also markedly reduces the unnecessary use of animals (Chou et al., 1984) (see also section VI.D.). Therefore, the medianeffect principle of the massaction law does a service to humanity.
C. Simulation of the F_{a}Combination Index Plot
Incorporating the medianeffect equation (eq. 8) into the combination index equation (eq. 16) can generate an algorithm used for simulating the calculated CI values at different effect (f_{a}) levels. For the combination of two drugs (D)_{1} and (D)_{2} at the combination ratio of (D)_{1}:(D)_{2} = P:Q, in the combination, (D)_{1,2} = (D)_{1} + (D)_{2}, we get (D)_{1} = (D)_{1,2} × [P/(P + Q)] and (D)_{2} = (D)_{1,2} × [Q/(P + Q)]. The combination index equation indicates that for a given effect of (f_{a})_{x} for x% inhibition of the dose, D_{x}, the combined additive effect for the sum of the fractional doses of each drug, (D)_{1}/(D_{x})_{1} and (D)_{2}/(D_{x})_{2} should be equal to unity. This statement is simple, but the real merit is its actual derivation and its generality in that the relationship holds regardless of the mechanism of each drug and regardless of the types of the drug actions, as shown in the mathematical induction and reduction (Fig. 2). The denominators of eq. 16, (D_{x})_{1} and (D_{x})_{2} for each drug “alone” can be expressed by eq. 8: D_{x} = D_{m} [(f_{a})/(1  f_{a})]^{1/}^{m}, and the numerators of eq. 16 for the “combination” are the actual experimental doses (D)_{1} and (D)_{2} that in combination give rise to (D)_{1,2}. Because synergism is the effect that is more than additive (i.e., requires less dose for a given effect) and antagonism is an effect that is less than additive (i.e., requires more dose for a given effect), therefore, when CI = 1, it indicates an additive effect, CI < 1, it indicates a synergistic effect, and CI > 1, it indicates an antagonistic effect.
Thus, by increasing the doses for the combinations, the effect levels can be increased to different effect (f_{a}) levels (Fig. 8a). The computer using the CompuSyn software will iteratively calculate the CI values at different (f_{a})_{x}. In this way, the entire spectrum of CIs at different f_{a} values can be simulated. The simulation of the F_{a}CI plot and its interpretation are shown in Fig. 8a. Note that both the F_{a} and the CI for the x and yaxes are dimensionless quantities. This algorithm of multiplestep logic (Fig. 7) can be applied for ndrug combination, as shown by eq. 20.
D. Construction of the Classic and Normalized Isobologram
An isobologram is a graph of equipotency doses for two drug combinations. This concept has more than a century of history and has been subjected to decadeslong intensive studies by Loewe (1928, 1953, 1957) and Berenbaum (1977, 1981, 1989). The construction of an isobol used to be carried out manually using graph paper for interpolations and extrapolations. The isobologram equation was formally derived and introduced by Chou and Talalay (1981, 1984) simply by setting the CI equation (eqs. 17 and 18) equal to 1. Thus, Chou and Talalay were able to construct an isobol at any effect levels with a keystroke after the doseeffect data entries. The algorithm for isobol construction is identical to the F_{a}CI plot, except that the isobol is doseoriented and the F_{a}CI plot is effectoriented graphs. Neither graph is dependent on the mechanisms and sites of drug actions, since they are derived from mathematical induction and deduction (see section II.). The practical advantages of the F_{a}CI plot over the isobologram have been discussed in section II.C.3. The typical appearance and interpretations for both the classic and normalized isobolograms are illustrated in Fig. 8, b and c, respectively. The specific real data analyses are given in section VI. and Supplemental Data Appendices II to V.
For an experimental design using a constant ratio [e.g., the diagonal scheme with multiple doses in (IC_{50})_{1}/(IC_{50})_{2} ratio], the m_{1}, (D_{m})_{1}, r_{1}; m_{2}, (D_{m})_{2}, r_{2}; and m_{1,2}, (D_{m})_{1,2}, and r_{1,2} can be determined and thereby allow the isobol to be constructed at any effect (f_{a}) levels. In these calculations, only the m and D_{m} values are used and the r value is only for statistical justification. However, when the nonconstant ratio combination design is used, m_{1,2} and (D_{m})_{1,2} cannot be determined, so only the dosenormalized isobologram can be constructed (Fig. 8c). Theoretically, F_{a}CI plot, classic isobol, and normalized isobol should yield exactly identical conclusions in terms of synergism or antagonism, since they are based on the same MEP general equation (Chou, 1976) and the same CI theorem (Chou and Talalay, 1983, 1984). This theoretical prediction was confirmed using the real experimental data of Yonetani and Theorell (1964) (see Tables 8 and 9 and their analysis) as illustrated by Chou and Talalay (1981), by using a pocket calculator before the term “combination index” was introduced (Chou and Talalay, 1983, 1984). Now, 26 years later, the same sets of experimental data are being analyzed with the fully automated computer software CompuSyn (Chou and Martin, 2005) as illustrated in detail in section VI.A.1. and by the CompuSyn analysis given in Supplemental Data Appendices II and III. Excellent agreements between data and theory and the conclusions from the earlier manual and the recent computerized analysis have been obtained.
For each single drug and its combinations, although serial dilution on drug concentrations is recommended, for convenience, the serial dilution can be carried out in different ways. For example, (D)_{1} can be 2fold serial dilution, (D)_{2} can be 3fold serial dilution, and [(D)_{1} + (D)_{2}] or (D)_{1,2} can be 1.5fold serial dilution if (D)_{1} has a nearly hyperbolic doseeffect curve (m ≈ 1), (D)_{2} has a rather flat doseeffect curve (m < 1) and [(D)_{1} + (D)_{2}] has a steep or sigmoidal doseeffect curve (m > 1). In fact, random dilutions are also allowed for each drug as long as m and D_{m} (and r) values are determined. For the combination, even one data point can be used and the CI value can still be determined.
E. Simulation of the F_{a}DrugReduction Index Plot
Using the MEP and CI principle/DRI equation and using the same logic of the algorithm for the F_{a}CI plot, one can automatically construct a F_{a}DRI plot by simulation; this will be illustrated in section VI. with several examples. In the case of strong synergism, the DRI value can be very large and outofscale. To condense the scale, the F_{a}log(DRI) plot will also be simulated by default. The general appearance for F_{a}DRI plots and their interpretations are given in Fig. 8d.
The applications of the DRI equation and the simulation of the F_{a}DRI plot have toxicological imprints since reducing the dose (while maintaining the same effect) would lead to reduced toxicity. DRI > 1 indicates the reduced dose for a given drug combination compared with the dose of that drug alone. It should be noted that DRI is not an index for synergism or antagonism, whereas CI serves as an index. For example, when CI for two drugs indicates synergism (CI < 1), the DRI for drug 1, (DRI)_{1} can be >1 whereas the DRI for drug 2, (DRI)_{2}, can be <1, depending on the properties of each drug, as well as the drug combination ratio that is being used.
In practical situations, because DRI is most relevant to toxicity in vivo, whether drug 1 and drug 2 have overlapping or nonoverlapping toxicity is of concern (e.g., such as gastrointestinal toxicity, cardiotoxicity, renal toxicity, and neurotoxicity) Furthermore, when determining synergism for “efficacy” in vivo, one should also consider whether there is synergism of “toxicity” toward the host in vivo. The best scenarios of two drug combination are 1) both drugs have strong synergism in therapeutic efficacy (CI <); 2) both drugs have strong antagonism in toxicity toward the host (CI > 1); 3) the toxicities of both drugs in the host are not overlapping, and 4) both drugs (or at least one drug) allow significant dose reductions (DRI > 1) for a given effect. The same logic can be extended to three or more drug combinations.
F. StepbyStep Use of CompuSyn Software for Single Drug and for Drug Combination Studies
Detailed procedures for using CompuSyn for automated doseeffect analysis for parameters of each drug and its combinations for quantitation/simulation of synergism or antagonism are given in the users guide for CompuSyn (Chou and Martin, 2005). In section V., examples of applications in various fields of biomedical sciences, using old and new software, are given. In section VI. and Supplemental Data Appendices II to V, real data samples of applications using CompuSyn are shown.
After data entry (for doses and effects), the CompuSyn software generates, for a single drug, the doseeffect table, the doseeffect curve, the doseeffect parameters (m, D_{m}, and r values), the medianeffect plot, and the dose and effect interchange option. For multiple drugs (in addition as to the single drug) it generates the combination index table, the F_{a}CI plot (with simulation curves if constant ratio combinations), the F_{a}log(CI) plot, the dosereduction index table, the F_{a}DRI plot (with simulation curves if constant ratio combinations), the F_{a}log(DRI) plot, the classic isobologram (for constant ratio combinations), the normalized isobologram (for nonconstant ratio combinations), and the polygonogram (for three or more drugs with constant ratio combinations for each pair of drugs). The CI table and DRI table at actual experimental combination data points are also automatically generated. The software also provides options for classic isobolograms and polygonograms at different effect levels, and options for F_{a}CI plots with or without SDA. The software also provides a brief summary at the end of analysis indicating the project title, drug(s) name, abbreviation, file name, date, parameters, and, if applicable, combination ratio(s) and highlights of CI values at ED_{50}, ED_{75}, ED_{90}, and ED_{95}. In brief, the procedures for the analysis are summarized as follows:

Data Entry (takes 210 min depending on the size of the experiment).

Enter a single drug dose effect (each dose for each effect) for Drug A alone and for Drug B alone (usually five to eight doses or concentrations for an in vitro experiment for each drug).

Enter a combination doseeffect (each combination dose for each effect) for (Drug A + Drug B) mixtures, either at a constant combination ratio [usually the (IC_{50})_{1}/(IC_{50})_{2} ratio is used with serial dilution for several doses] or at a nonconstant combination ratio (e.g., in combinations, keep Drug A at a constant dose while varying Drug B doses). It should be noted that the user should assign a proper unit for each drug (e.g., millimolar, micromolar, or nanomolar) so that the numerical dose entered will not be too small or too big, when using it for scaling the graphics. The computer recognizes (and calculates) only numbers. The user's assigned unit(s) will be used for report generation. It should also be noted that the computer takes into account each data point entered equally importantly, not just emphasizing the points close to IC_{50} or ED_{50}, which is usually done with eye inspections. Extremely low effect(s) or extremely high effect(s) that are beyond the accuracy of assay (or the determination method) should not be entered (e.g., f_{a} < 0.01 or f_{a} > 0.999). On the logarithmic scale, 0.01 and 0.001 or 0.999 and 0.9999 are 10fold different and log 0 is negative infinity.


Report Generation (takes approximately 13 min depending on the speed of the computer): One will be asked to choose the items that will be included in the report (such as Single Drugs, Drug Combos at Constant Ratio, and Drug Combos at Nonconstant Ratios,...). Once the report has been created, it will automatically open in one's default Web browser (e.g., Internet Explorer or Mozilla). At this point, one is no longer in the CompuSyn application. When the generated report is saved, it is essential to delete the “^{*}” or “^{*}cse” and type a new file name. To make changes to the selected choices previously made in the generated report, CompuSyn must be opened first and one must click on “Recall Experiment.”

Printing the Report (may consist of 790 pages depending on the size of the experiment): To print a report, simply choose Print from the Web browser's file menu. The exact dialog box and options vary from browser to browser. Depending on the experimental size, design, and print selections, the printout (in color) consists of the following items and in the following default order:

Experiment title, date, file name, and description note.

The dose and effect tables for each single drug and its combinations, an acknowledgment of the number of data points entered, and the calculated parameters: m, D_{m}, and r.

The doseeffect curves for each drug and their combinations and corresponding medianeffect plots.

The F_{a} and CI tables for f_{a} = 0.05 to 0.97 for each combination (including total dose and combination ratio) as well as the CI values for each actual combination data point; F_{a}CI plots for f_{a} = 0.05 to 0.97 for each combination with computer simulation as well as actual combination data points in the graph, if combinations are in constant ratio or with no computer simulation but with actual combination data points in the graph if combinations are in nonconstant ratios.

The F_{a} and DRI tables for f_{a} = 0.05 to 0.97, the dose required for each drug alone at a given effect (f_{a}) and the fold dose reduction if combined for a given effect (f_{a}); DRI values for each of the actual combination data points; F_{a}DRI plot for each drug and for each combination with computer simulation, as well as the actual combination data points located on the graphs; F_{a}log (DRI) plot for each drug and for each combination with computer simulation as well as the actual combination data points on the graphs.

Isobolograms: The classic isobolograms will be automatically shown for each pair of drug combinations, if a constant ratio combination design is used (default at f_{a} = 0.5, 0.75, and 0.9; other options available). The normalized isobolograms will be automatically shown for each pair of drug combinations, if the nonconstant ratios combination design is used.

Polygonogram for more than three pairs of drug combinations, involving three or more drugs (default sets f_{a} = 0.9, other options available).


Summary Table: The summary table includes title, date, file name, description note; parameters (m, D_{m}, and r) for each drug and their combinations; the CI table for each combination at ED_{50}, ED_{75}, ED_{90}, and ED_{95} levels; and doses for each drug and their combinations required to produce ED_{50}, ED_{75}, ED_{90}, and ED_{95} (option for calculating ED_{x} is also available).

Options for Generating Report: The following selections are made from the menu in Report Option, the specified report will be generated by clicking Generate Report and by checking the items that one wants for tailoring the report. After checking OK, the full report will be generated and will appear on the screen. It may take a while to generate a report for a largescale experiment. The generated report can be printed or saved. If one wants one, two, or more figures for making slides or for publication, then go back to Generate Report. Check off those items not needed, leave the items that are wanted, and click OK. The desired figures will appear on the screen almost instantly.

Some Features and Options.

Error entry of dose or effect for each drug and their combinations may be edited or deleted.

In constant ratio combinations, because the combination ratio has already been entered, enter the dose of any Drug or the Total Dose, and CompuSyn will automatically fill in all the other doses.

For nonconstant ratio combinations, one has to input the Dose and Effect of each drug individually.

It is recommended that one select not more than 10 Single Drugs and Drug Combos to be included in the report, as having too many items on each graph tends to make it overcrowded and unreadable.

All numbers are stored and calculated using 80 bit IEEE standard double floating point precision. This means that the largest gap between representable numbers is about 2 × 10^{16}.

he numbers displayed in the report are rounded to seven decimal points.

“NaN” (Not a Number) appears in the place of a number, indicating that a mathematical error was made, such as dividing by 0 or f_{a} > 1.

Options for selecting “f_{a} for a value” to calculate the corresponding dose, to construct the corresponding isobologram, or to construct the corresponding polygonogram are available.

Options for automatically using colors for each doseeffect curve or for each line in a plot are available; in a polygonogram, synergism is shown as a green solid line and antagonism is shown as a red dashed line.

Split large graphs (on by default). When there are more than two Drugs and two Combos, the doseeffect plot and the medianeffect plot will be automatically split into two graphs to avoid crowding.

For Scaling Window, everything is set to 0 by default, which tells CompuSyn to do the scaling automatically. If one wishes to use a custom scale on one of the graphs, he or she can manually override CompuSyn by entering something in the scale entry for a given graph. Note that for plots with multiple frames, such as Isobologram, the scale on all frames is affected.

G. Statistical Considerations
The derivation of the general medianeffect equation as well as the CI equation is based on the physicochemical principle of the massaction law and mathematical induction and deduction. These derivatives do not invoke any statistical principle, method, or assumptions. Therefore, the methods per se are deterministic not probabilistic. But for their applications, the variability in methods of assay measurements and the variability in biological systems cannot be avoided, and, therefore, they are subject to statistical considerations.
There are three levels of statistics available for conducting a drug combination analysis using the CompuSyn software:

The r value, i.e., the linear correlation coefficient of the medianeffect plot (Chou, 1976) signifies the conformity or goodness of fit of the experimental data with the medianeffect principle. This firstline statistics is a default function of computer software such as CompuSyn.

The SDA, also known as serial deletion analysis, allows for an iterative analysis that deletes one data point (of a dose) at a time. These repeated reruns of calculations from crude data at the beginning to the CI results at the end allow the determination of variability of data, which generate results at different effect levels for the F_{a}CI plot with vertical bars, indicating 95% confidence intervals. A typical example of a F_{a}CI plot is given in Fig. 12.

An easy thirdline statistics is to repeat the drug combination experiment several times and then run CompuSyn several times before calculating the regular statistics of mean ± S.D. or mean ± S.E. for parameters (m, D_{m}, and r), the CI values, and the DRI values.
V. Selected Examples of Cited Applications
Based on the PubMed or ISI Web search, >2600 scientific papers in the biomedical literature have used the medianeffect equation or the combination indexisobologram equation in analysis of data for single drug or for drug combination studies.
In the following, selected examples of applications are categorized and the references are listed under References. Wherever appropriate, brief notations on the names of drugs, the number of drugs studied, and the type of the target of inhibitions are given immediately after the cited references. All examples of applications are from in vitro studies, unless otherwise specifically indicated, in the case of in vivo or animal studies.
A. Cited Methods and Evaluation of Single Drug and Drug Discovery
For the evaluation of pharmacological properties of a new compound in vitro or in vivo for potential drug development, the medianeffect equation and plot provide three pieces of basic information in terms of quantitative parameters: 1) the D_{m} value signifies “potency,” such as IC_{50}, ED_{50}, and LD_{50} (see Fig. 3); 2) the m value signifies the “shape” of doseeffect curve with m = 1, > 1, and < 1 indicating hyperbolic (such as the MichaelisMenten kinetics or simple receptor binding), sigmoidal (such as allosteric enzyme kinetics or oxygenhemoglobin kinetics), and flat sigmoidal (such as negative cooperativity in allosteric kinetic systems), respectively; and 3) the r value signifies the “conformity” of the data set to the medianeffect principle of the massaction law. These three parameters can be automatically obtained using the computer software, CompuSyn (Chou and Martin, 2005) or an earlier version of the software (Chou and Chou, 1985; Chou and Hayball, 1997), after entry of doses and the corresponding effects. The algorithm for the parameter determination is that the antilog of the xintercept of the medianeffect plot gives the D_{m} value, whereas the slope of the medianeffect plot gives the m value, and the linear correlation coefficient of the medianeffect plot gives the r value.
This method is distinct from other methods in biomedical literature by having the following features: 1) Is uses quantitative parameters for potency, shape, and conformity. 2) It may use a smaller number of data points for a doseeffect curve. This feature is particularly useful for costeffective and ethical considerations for animal studies (see sections VI.A.1. and VI.D.1.; for detailed illustrations, see Supplemental Data Appendices IIIV). 3) All data points are equally weighted, without emphasis on the points on the curve nearest to the medianeffect dose. 4) It allows calculation of the dose for any given effect (eq. 8) or calculation of the effect for any given dose (eq. 9) by using a pocket calculator or CompuSyn software. The dose calculated in the form of concentration (e.g., micromolar) for a desired effect (e.g., for 95% or 99% inhibition) provides an estimation of whether the desired plasma or tissue drug concentrations are feasible to achieve at the intended dose or at a MTD in animals or humans.
Researchers in the author's laboratory have used the MEP and its software for evaluating new compounds for the Drug Discovery/Development Program at Memorial Sloan Kettering Cancer Center (MSKCC) and for collaborative research with other institutions during the past 20 years. It is estimated that >20,000 sets of doseeffect data have been subjected to the MEP method of data analysis. Excellent conformity results have been obtained and published, including results for chemicals, drugs, biomodifiers, carcinogens, and radiation. This fact alone attests to the general applicability and usefulness of the MEP method for evaluating efficacy and toxicity of a drug, or drugs, or entities. In conjunction with colleagues at the BioOrganic Chemistry Laboratory at MSKCC and the organic chemistry laboratories elsewhere, our work has resulted in the discovery of many potentially useful new compounds, as indicated by U.S. patents 5,053,431 (1991) for chrysophanol; 5,340,818 (1994) for plant ingredients; 5,354,864 (1994) for acridinylanilines; 5,476,952 (1995) for cyclopentanthraquinones; 5,622,958 (1997) for enediynequinones; 5,939,428 (1999) for acridinylamino5hydroxymethylanilines, 6,147,076 (2000) for ardeemins. 6,204,388 (2001), 6,242,469 (2001), 6,284,781 (2001), 6,300,355 (2001), 6,369,234 (2002), 6,656,961 (2003), 6,656,961 (2003), 6,723,854 (2004), and 6,828,340 (2004) for epothilone derivatives; and 6,355,639 (2002) for the reverse prenyl compounds. Some of the epothilones are in phase I and phase II clinical trials in cancer patients. Some are in different stages of preclinical developments. One of the most remarkable compounds discovered through this process of evaluations is fludelone (26trifluoro9,10dehydro12,13desoxyepothilone B), a microtubule targeted stabilization epothilone that allows the first fully documented report of a “therapeutic cure” against human mammary and colon carcinoma xenografts in nude mice with complete tumor remission without any relapse for >6 months (Chou et al., 2005b).
Examples of specific applications of the MEP method for single drugs (Chou, 1976) are given below.
1. Exploration of Potency, Toxicity, Parameters, and StructureActivity Relations for New Compounds.

Cytotoxic anticancer agents

DNA topoisomerase inhibitors: Kong et al. (1992a), Su et al. (1992a), Su and Chou (1994), Conti et al. (1996), Scarborough et al. (1996), Luo et al. (1996), Josien et al. (1997).

DNA intercalculating agents: Koyama et al. (1989), Su et al. (1995, 1999), Rastogi et al., (2002), Chang et al. (2003).

Alkylating agents: Kong et al. (1992a), Kohler et al. (1993), Shair et al. (1994, 1996); Kim et al. (1996), Shan et al. (1999), Vijayaraghavan et al. (2003), Bacherikov et al. (2004), Liang et al. (2004), Su et al. (2005).

Microtubule depolymerization agents: Shan et al. (1999).

Microtubule stabilization agents: Balog et al. (1997), Su et al. (1997a,b), Meng et al. (1997), Chou et al. (1998a,b, 2001, 2003, 2005b), Stachel et al. (2000), Lee et al. (2000), Rivkin et al. (2003, 2005).

Protein synthesis inhibitors: Su et al. (1992b).

Inhibitors of protein kinase and other enzymes: Steckel et al. (1983), Link et al. (1996).

Antiviral agents

AntiHSV/HBV agents: Kong et al. (1992c), Prochaska et al. (1993).

AntiHSV agents: Chou et al. (1987), Fox et al. (1988), Kong et al. (1992b).

Antihuman hepatitis B virus agents: Fox et al. (1988).

Anticytomegalovirus agents: Yang et al. (1990).

Antisimian varicella virus agents: Soike et al. (1987, 1990).

AntiEpsteinBarr virus agents: Lin et al. (1989).


Cytodifferentiating agents: Kong et al. (1987, 1988), Breitman and He (1990), Laneuville et al. (1994).

α_{1}Adrenergic blockers: Somers et al. (1989), Breslin et al. (1993a,b).

Immunosuppressants: Vathsala et al. (1990, 1991), Zucker et al. (1997).

Insecticides: Chou and Talalay (1983, 1984), Alzogaray et al. (1998).

Hormones and modulators: Sharom et al. (1998).

Antifungal, antimalarial, and anthelmintic agents: Young et al. (1992), Berger et al. (1995, 1996, 1997), GebreHiwot and Frommel (1993).

Tumor promotors: Kopelovich and Chou (1984), Ramel (1986), Porter et al. (1997).

Radiation, radioimmunoassay, and magnetic field: Maisin et al. (1987), Cohen et al. (1991), GriffonEtienne et al. (1996), Liang et al. (1997).

Photodynamic agents: Rezzoug et al. (1998).

Enzymeligand interactions: Kremer et al. (1980), Steckel et al. (1983), Rahier et al. (1989), Barrie et al. (1989), Chou and Chou (1990), Huang et al. (1992), Conner et al. (1992), Dawson et al. (1995), Trzaskos et al. (1995), Jez et al. (1996).

Antiasthmatic and antiallergic agents: Matsushita et al. (1998).

Receptorligand and interactions: Friedman and Skehan (1980), Finotti and Palantini (1981), Zahniser and Molinoff (1983), Clark et al. (1988), Bylund et al. (1988), Bylund and RayPrenger (1989), Baker et al. (1986, 1988), Munson and Rodbard (1988), Dewar et al. (1989), Schoepp et al. (1990, 1992, 1994, 1996, 1997), Breslin et al. (1993), Jin et al. (1994), Katz et al. (1994), Zuckerman et al. (1999), Boileau et al. (1998, 1999).

Antiangiogenic agents: Qian et al. (2004), Vucenik et al. (2004).

Thermo effects: Hegedus and Khachatourians (1996).

Environmental contaminants: Vogiatzis and Loumbourdis (1998).
2. LowDose Risk Assessment for Carcinogens and Radiation:
Chou (1980, 1981, 1991), Chou and Talalay (1984, 1987) (for more information, see sections VI.B.1 through VI.B.2).
3. Calculation of K_{i} from the IC_{50} Value:
Chou (1974, 1977), Chou and Talalay (1981), Murphy and Snyder (1982), Baker and Posner (1986), Goldstein and Barrett (1987), Bylund et al. (1988), Bylund and RayPrenger (1989), Price et al. (1989), Clark et al. (1989), Schoepp and Johnson (1989a,b, 1993), Schoepp et al. (1996), Boileau et al. (1998). For more information, see section VI.B.6.
4. Exclusive and Nonexclusive Inhibitors and Topology of Binding Sites:
Chou (1974, 1977a), Steckel et al. (1983), Chou and Chou (1988, 1990a), Lombardini et al. (1989) (for more information, see section VI.A.2).
5. Drug Resistance Evaluation and Other Applications:
6. Cellular Pharmacological Studies:
Chou et al. (1977, 1982, 1983, 1984, 1994, 1996, 1998a,b, 2003, 2005a,b,c), Long et al. (1982), Kufe et al. (1984), Takemura et al. (1985), Chang et al. (1985), Colombani et al. (1989), Fykse et al. (1989), Dewar et al. (1989), Christensen et al. (1990), Gonzales et al. (1991), Traversa et al. (1994), Cheng et al. (1995), Schultz et al. (1998), Vaskinn et al. (1999).
7. Tissue Pharmacological Studies:
Chou et al. (1977), Sharma and Klein (1988), Puig et al. (1988), Seo et al. (1989), Chu et al. (1989), Gonzales and Moerschbaecher (1989), Schoepp and Johnson (1989a,b), Wiener and van Os (1989), Zemelman et al. (1989), Woodward and Gonzales (1990), Woodward and Blair (1991), Woodward and Harms (1992), Noonan et al. (1992), Woodward and Cueto (1993), Schoepp et al. (1995, 1996), Desai et al. (1995), Mathis et al. (1999).
8. Cardiovascular Pharmacological Studies:
Sharma and Klein (1988), Dong et al. (1988), Herman et al. (1989), Lyu et al. (1992), Heim et al. (1995).
9. Pharmacological Studies on Animals:
Chou et al. (1975, 1981, 1998a,b, 2001, 2005a,b,c), Schinazi et al. (1986).
10. Behavioral Studies:
LakhdarGhazal et al. (1986), Altman et al. (1987), Barrie et al. (1997), Jin et al. (1994), Katz et al. (1994), Schoepp et al. (1995), RyanMoro et al. (1996), Zuckerman et al. (1999).
11. Cancer Prevention Agents:
DinkovaKostova et al. (2005), triterpenoid analogs.
B. Examples of Cited Applications in Drug Combinations
The combination index method that involves both the medianeffect equation (Chou, 1976) and the multiple drugeffect equation (Chou and Talalay, 1984; Chou 1991) has been widely used in biomedical research. One review article alone by Chou and Talalay (1984) has been cited in >1294 scientific papers. These applications are most noticeable in the three areas of biomedical research. The first area is anticancer drug combinations. Many of these studies were carried out at MSKCC and other academic medical institutions in North America, the United Kingdom, Germany, France, Italy, the Netherlands, Spain, Japan. and many other countries. The second area is antiviral agent combinations, especially for antiHIV agents. Many of these studies were carried out in Prof. Martin S. Hirsch's laboratory at Massachusetts General Hospital (MGH), Harvard Medical School. During the 19 years of research collaboration between Hirsch and the author, 21 articles have been published, which covered most of the new antiHIV agents, shortly after their introduction (see below). These studies have helped generate information that led to the discovery of antiHIV cocktails by Prof. David Ho, who was a former trainee in Hirsch's laboratory at MGH (Fischl et al., 1995; Perelson et al., 1997; Cameron et al., 1999; Zhang et al., 1999). In attempts to establish animal models and to develop antiHIV vaccines, researchers in Prof. Ruth M. Ruprecht's laboratory at DanaFaber Cancer Institute, Harvard Medical School, have conducted many studies on multiclade MAbs combinations in vitro and in vivo in macaques. Since the mid1980s, the combination index method has made impacts in AIDS clinical trials, as indicated by the publications by the MGH group and others (Mildvan et al., 1990; Fischl et al., 1995; D'Aquila et al., 1996; Chou and Zhu, 1997). During the 16 years of research collaboration between Prof. Ruprecht and the author, they have published 12 articles, which include combinations of as many as four monoclonal antibodies (see below). The third area is immunosuppressants combinations. Many of these studies were carried out in Prof. Barry D. Kahan's laboratory at the Division of Organ Transplantation, Department of Surgery, University of Texas at Houston. During Prof. Kahan's 16 years of research collaboration with the author, 19 articles have been published, which covered mostly new and some old immunosuppressants in in vitro and in vivo studies, including heart, kidney, and small intestine transplantations in rodents (see below).
Applications of the MEPCI method for combination studies have also been reported in other areas, such as molecular biology, gene therapy, cancer prevention, insecticides, radiation, drug combinations in animals, and clinical protocol design. Some examples of specific applications that have been published are given below. Also given below are sample articles that highlight the observation of strong antagonism, a phenomenon that has been rarely reported. A report that highlights the strong antagonism between AZT and ribavirin (Vogt et al., 1987) was published in Science, before these drugs were used in AIDS patients. The applications of the CIisobol method of Chou and Talalay are also categorized and presented below. Most in vivo studies were not carried out to the full extent and were based on the in vitro findings by using CIisobol methods. Chou et al. (2005a) carried out full in vivo therapeutic studies in xenograft tumorbearing nude mice. They were able to demonstrate doseeffect parameters and construct the F_{a}CI plots and isobolograms. With the constant ratio diagonal design (Table 5), only 50 nude mice were used for each in vivo xenograft drug combination studies. Both antitumor effect and toxicity were analyzed for synergism and/or antagonisms in vivo.
1. Anticancer Agent Combinations.

Cytotoxic agents (CAs).

Two CA combinations: Chou et al. (1993), edatrexate + cisplatin; Koechli et al. (1993), paclitaxel (Taxol) + doxorubicin (Adriamycin); Figul et al. (2003), temozolomide + didox; Balzarotti et al. (2004), temozolomide + other cytotoxic agents; Honore et al. (2004), discodermolide + paclitaxel; Horvath et al. (2004), didox + carmustine; Chauhan et al. (2005), oral proteosome inhibitor + bortezomib; Tanaka et al. (2005a,b), oxaliplatin + CPT11; Shanks et al. (2005), gemcitabine + various antitumor agents; Fischel et al. (2001), irinotecan, 5FU, and oxaliplatin ternary combination; Harris et al. (2005a,b), XR 5944 + carboplatin or doxorubicin.

Twodrug and threedrug combinations with different mechanisms of action: Chou et al. (1994), paclitaxel (Taxol), topotecan, cisplatin, vincristine, and VP16 (polygonograms were used; see detailed sample analysis in section Ii.e. for two and threedrug combinations of anticancer agents and Supplemental Data Appendix II for details and the CompuSyn printout); Fischel et al. (2001), irinotecan + 5FU + oxaliplatin.

CAs + cytodifferentiating agent: Kong et al. (1988).

CAs + MDRreversing agents: Perez et al. (1994), carboplatin resistance; Chou et al. (1998a), MDR and multidrug resistance protein resistance reversal by ardeemins; Chang and Chou (2000) and Chou and Chang (2002), MDR; Chou et al. (2005c), MDR reversal by ningalins; Chang et al. (2006), nordihydroguaiaretic acids + doxorubicin or paclitaxel.

CAs + modulators: Sacks et al. (1995), retinoic acid; Finch et al. (2000) GCSF + LiCl/alltransretinoic acid; MullerTidow et al. (2003), IFN + lovastatin/bcrabl^{+} cells.

CA + virus: Bennett et al. (2004), mitomycin C + oncolytic herpes virus.

CA + enzyme: Romanini et al. (1989), trimetrexate + carboxypeptidase G_{2}.


Modulator combinations or modulator + hormone: Bregman and Meyskens (1986), difluoromethylornithine + biological modifiers; Durand and Goldie (1987), etoposide + cisplatin, in spheroid model; Triozzi et al. (1989), IFN + steroids; Durand (1990), cisplatin + CCNU in spheroid cells (oxygen tension and distance from spheroid surface); Raje et al. (2004), rapamycin + thalidomide analog; Whitmore et al. (2004), polyrI:rC + CpGoligodeoxynucleotides; Algur et al. (2005), radiation + zoledronic acid; Bozec et al. (2005), dual antiEGFR + radiation; Cosaceanu et al. (2005), radiation + insulinlike growth factor receptor inhibitor; Dai et al. (2005), paclitaxel (Taxol) + amifostine (a normal tissue protection agent); Gemmill et al. (2005), Iressa + rapamycin; Horvath et al. (2005), freeradical scavenger + AraC; Mohammed et al. (1995), platinum drugs + tamoxifen.

Radiation combinations: Leonard et al. (1996), paclitaxel (Taxol) + radiation; Donson et al. (1999), tamoxifen + radiation and radiation + virus; Adusumilli et al. (2004) ionizing radiation + oncolytic virus.

Cancer prevention combinations and cell differentiation combinations.

Khafif et al. (1998), epigallocatechin3gallate + curcumin.

Soriano et al. (1999), chemopreventive agents + cytotoxic agents.

Finch et al. (2000), GCSF/retinoic acid receptor α + LiCl/alltransretinoic acidtreated cells.

2. Antiviral Agent Combinations.

AntiHIV agent combinations: For crude data analytical illustrations, see Kong et al. (1991). For a largescale antiHIV drug combination study (Chou and Zhu, 1997) involving two to fivedrug combinations of seven antiHIV agents, see section VI.C.3. Details of the CompuSyn report (90 pages) are found in Supplemental Data (http://pharmrev.aspetjournals.org/cgi/content/full/pr.58.3.10/DC1). For other applications, see Hartshorn et al. (1986, 1987), rIFN_{α} + phosphonoformate and AZTrIFN_{α}, the first antiHIV combinations; Vogt et al. (1987, 1988), DDC + rIFN_{α}; Johnson et al. (1990, 1991, 1992), threedrug combination of AZT + CD4 + IFN_{α} and AZT + DDI or IFN_{α} for HIV1/AZT, protease inhibitor RO131 + AZT, DDC, or IFN_{α} for HIV/AZT; Richman et al. (1991), BIRG587 +AZT for HIV/AZT; Chou et al. (1991), oligonucleotide SdC28 + AZT, IFN, or dextran sulfate; Kong et al. (1991), twodrug and threedrug combinations, AZT, phosphonoformate, and 3′azido3′deoxythymidine; Pan et al. (1992), threedrug combination, 3′fluoro3′deoxythymidine, CD4, and IFN_{α}; Tilley et al. (1992), hMAb combinations; Eron et al. (1992), AZT + DDC for HIV1/AZT and schedule dependence; Merrill et al. (1996), twodrug and threedrug combinations, lamivudine, stavudine, and AZT; Zhu et al. (1996), AZT, stavudine, and nevirapine for HIV/AZT; Deminie et al. (1996), protease inhibitors + reverse transcriptase inhibitors; Tremblay et al. (1999, 2000, 2002), twodrug and threedrug combinations or protease inhibitors for HIVresistant isolates and fusion inhibitor T20 + CXCR4 blocker AMD3100; CCR5 antagonist SCHC + other antiHIV agents; Hostetler et al. (2000), phosphonoformate analogs + AZT; Xu et al. (2001), hMAbs (clade B) combinations against HIV (clade C); Kollmann et al. (2001), nevirapine + efavirenz; Kitabwalla et al. (2003), hMAbs (clade B) combinations against HIV (clades A and D), a fourdrug combination; Zhang et al. (2005), MAbs as antiHIV vaccine.

Antisimianhuman HIV combinations: Li et al. (1997, 1998), human antiHIV1 envelope MAb, globulin combination against simian immunodeficiency virus/HIV1; Baba et al. (2000) hMAbs of IgG1 combinations against simianhuman immunodeficiency virus.

Prevention of HIV infection in vivo: Ruprecht et al. (1990), AZT + IFN against retroviral viremia in mice; Baba et al. (2000) hMAbs of IgG1 combinations against simianhuman immunodeficiency virus infection; HofmannLehmann et al. (2001a,b), passive immunization combinations against oral AIDS virus transmission, threedrug combinations of hMAbs against oral simianhuman immunodeficiency virus in macaques.

AntiHSV combinations: Schinazi et al. (1986), in vitro and in vivo combinations; Gong et al. (2004), betulin + acyclovir.

Anticytomegalovirus agent combinations: Yang et al. (1990), twodrug combinations againstguinea pig cytomegalovirus in vitro. Manion et al. (1996), twodrug combinations.

AntiEpstein Barr virus agent combinations: Lin et al. (1989), IFN_{α}, IFN_{γ} + AZT against EpsteinBarr virus in vitro.

Antisimian varicella virus combinations: Soike et al. (1987, 1990), guanine analog + IFN_{β} and 1(2deoxy2fluoro1bdarabinosyl)5ethyluracil + IFN_{β} against simian varicella virus infection in monkeys.
3. Immunosuppressant Combinations for Organ Transplantations.

In vitro studies.

Bone marrow transplant by purging progenitor cells: Chang et al. (1985), 4hydroperoxycyclophosphamide + etoposide; Chang et al. (1987), combinations of various cytotoxic agents.

Cyclosporine (C_{S}A), rapamycin (Rapa, Sirolimus), FK506 (tacrolimus) combinations: Kahan et al. (1991a,b), C_{S}A + Rapa; Vathsala et al. (1991), C_{S}A + Rapa; Kahan et al. (1993), combinations in vitro and in vivo, brequinar + C_{S}A or + Rapa, twodrug and threedrug combinations; Knight et al. (1994) C_{S}A + MAb.

Scheduledependent combinations: Candinas et al. (1997), leflunomide + C_{S}A or + FK506; Podder et al. (2001), sirolimus + C_{S}A.


In vivo combinations and organ transplantations.

Heart transplantation: Stepkowski et al. (1994), C_{S}A, Rapa, and brequinar combinations, mice cardiac allografts; Kaji et al. (1994), 15desoxyspergualin + C_{S}A, rat cardiac allografts; Tu et al. (1995), C_{S}A, Rapa, and brequinar combinations, mice cardiac allografts; Hamashima et al. (1995), donor antigen + C_{S}A or + C_{S}A/sirolimus, rats heart allografts.

Kidney transplantations: Kahan et al. (1993), in vitro and in vivo, brequinar + C_{S}A or + Rapa, twodrug and threedrug combinations; Chou et al. (1994), C_{S}A and other combinations; Kahan et al. (1993), C_{S}A and other combinations.

Small intestine transplantations: Wang et al. (1996), brequinar + C_{S}A, and brequinar + MAb, rat small intestine allografts; Stepkowski et al. (1996), oral sirolimus + C_{S}A, rat heart allografts; Grochowicz et al. (1997), castanospermine + tacrolimus, rat cardiac allografts; Stepkowski et al. (1997), sirolimus + C_{S}A, rat heart and kidney allografts; Candinas et al. (1997), leflunomide + C_{S}A or FK506, rat cardiac allografts.

Islet transplantation: Gores et al. (1994), 15desoxyspergualin + C_{S}A, rat islet allografts.

Immunosuppressant combination reviews: Kahan et al. (1992); Chou et al. (1994), protocol design; Chou and Kahan (2001), book chapter.

4. Schedule Dependence of Combinations.

Antitumor combination schedules.

In vitro combinations: Chang et al. (1987), αdifluoromethylornithine + CisPt; Perez et al. (1993), edatrexate + CisPt; Chandrasekaran et al. (1995), AZT + 5FU, cellcycle dependence; Chou et al. (1996), paclitaxel (Taxol) or docetaxel (Taxotere) + edatrexate; Takahashi et al. (2002), schedule of ecteinascidin743 + paclitaxel (Taxol); Hubeek et al. (2004), sequential combinations of GCSF, fludarabine, and AraC; De Luca et al. (2004), gemcitabine + vinorelbine; Levis et al. (2004), FLT3 inhibitor + cytotoxic agents; Tanaka et al. (2005a,b), paclitaxel (Taxol) + oxaliplatin; Fischel et al. (2005a,b), docetaxel + capecitabine/5′deoxy5fluorouridine; Harris et al. (2005a,b), XR5944 + irinotecan or 5FU.

In vivo combinations: Rigas et al. (1999), paclitaxel (Taxol) + edatrexate, Phase I trials; Takahashi et al. (2002) schedule of ecteinascidin743 + paclitaxel (Taxol).


Antiviral combination schedules.

In vitro combinations: Mazzulli et al. (1994), various antiHIV combinations.

In vivo combinations: Schinazi et al. (1986), various antiHSV agent combinations.

5. Drug Combinations That Highlight Antagonism.

Antitumor combinations: Kong et al. (1988), cytodifferentiating agent + cytotoxic agents, AraC, Adriamycin (doxorubicin), and harringtonine; Chou et al. (1994), paclitaxel (Taxol) + vincristine, CisPt + VP16; paclitaxel (Taxol) + vincristine + VP16, in polygonograms; De Luca et al. (2004), gemcitabine + vinorelbine.

AntiHIV combinations: Vogt et al. (1987), AZT + ribavirin; Chou et al. (1991), various agent combinations; Merrill et al. (1997), protease inhibitors, indinavir, saquinavir.

Immunosuppressant combinations: Vathsala et al. (1991), lowdose FK506 + C_{S}A.
6. Topological Analysis of Multiligand Bindings.

Chou (1974), the distribution equation for ligandbinding sites.

Chou (1977a), availability of ligandbinding sites in a steadystate system.

Chou and Chou (1988), ligand exclusivity and competitiveness.

Lombardini et al. (1989), binding site combination kinetics.

Chou and Chou (1990), topological assessment with the aid of a computer.
7. Selectivity of Synergism.

Berman et al. (1989), HIV versus normal bone marrow progenitor cells.

Chang et al. (1985, 1987), sequential versus reverse sequential administration.

Chou et al. (1996), sequential dependence.

Chang et al. (1987), leukemic cells versus normal hematopoietic precursors.

Peters et al. (1991), cisplatin plus 4hydroperoxycyclophosphamide, influence of glutathione.

Durand and Goldie (1987), etoposide + cisplatin, spheroids; Durand (1990), cisplatin + CCNU, effects of distance and oxygen tension on multicell spheroids.

Duffy et al. (1998), anticancer drugs + nonsteroidal antiinflammatory drugs, effect of multidrug resistance protein.

Norberg and Wahlstrom (1986), hexobarbital + thiopental, agespecific, in vivo.

Martin and Symonds (2002), toremifene + IFN_{α} dependence on IFN_{α} subtype.

Pei et al. (2004), flavopiridol + Bcl2 inhibitor, free radical and Jun NH_{2}kinasedependent.

Konecny and Pegram (2004) gemcitabine + trastuzumab or cisplatins with HER2 overexpression cells.
8. Gene Therapy or Molecular Biology by Combinations.

Aghi et al. (1998), ganciclovir/TK and 5fluorocytosine/cytidine deaminase.

Aghi et al. (1999), oncolytic virus/oxazaphosphorine/cytochrome + ganciclovir/HSVTK.

Rainov et al. (2001), temozolomide + ganciclovir/HSVTK.

Singh et al. (2002), p53 induction/zinc + phosphatidylinositol 3′kinase inhibition/LY294002.

Formento et al. (2004), Iressa + trastuzumab.

Kaliberov et al. (2004), gene therapy mediated via adenovirus + radiations.

Lee et al. (2004), insulinlike growth factor binding protein3 + farnesyl transferase inhibitor.

Qian et al. (2004), antiangiogenics + antihistone deacetylase.

Park et al. (2004), Iressa + paclitaxel (Taxol).

Yee et al. (2004), FLT3 inhibitor + cytarabine or + daunorubicin.

Yen et al. (2004), Targretin + paclitaxel (Taxol).

David et al. (2005), farnesyl transferase inhibitor + proteosome inhibitor.

Fischel et al. (2005a), Iressa + cetuximab.

Gu et al. (2005), imatinib + mycophenolic acid.

Lunghi et al. (2005), arsenic trioxide + mitogenactivated protein kinase kinase1 inhibitor.

Mullerad et al. (2005), oncolytic HSV + mitomycin C.

Rahmani et al. (2005), heat shock protein90 inhibitor + histone deacetylase inhibitor.

Tseng et al. (2005), imatinib + phosphoinositidedependent.

Van Schaeybroeck et al. (2005), gefitinib + cytotoxic agents.

Aghi et al. (2006), alkylating temozolomide + oncolytic HSV, in vitro and in vivo.

Bruzzese et al. (2006), EGFR kinase inhibitor + gefitinib or + IFN_{α}, in vitro and in vivo.

Kaliberov et al. (2006), adenoviraldirected enzyme/prodrug + immunotherapy, in vitro and in vivo.

Zhou et al. (2006), combinations targeting EGFR and ErbB2 simultaneously, in vitro and in vivo.
9. Combinations of Other AntiInfectious Disease Agents.

Anticryptosporidium combinations: You et al. (1998), paromomycin + lasalocid.

Antifungal combinations: Kullberg et al. (2004), antifungals + cytokines.

Mycotoxin combinations: Koshinsky et al. (1991), toxin combination against CO_{2} release in yeast.

Antimalarial combinations: Coutaux et al. (1994), chloroquine + monoamine reuptake inhibitors.

Antisevere acute respiratory syndromecoronavirus combination: Morgenstern et al. (2005), ribavirin + IFN_{β}.

Antibovine viral diarrhea virus combination: Yanagida et al. (2004), mizoribine + IFN_{α} against bovine viral diarrhea virus
10. Cardiovascular Drug Combinations:
Bennett et al. (1984), aorta/relaxation drug combinations.
11. Combination for Animal Growth:
Nakagawa et al. (1996), synthetic growth hormone releasing peptide + growth hormone inducer.
12. Anesthetic Combinations:
Norberg and Wahlstrom (1986), hexobarbital + thiopental in vivo; Bansinath et al. (1992), ketamine + halothane, in vitro; Naguib (1994), rocuronium bromide + mivacurium chloride, neuromuscular relaxants.
13. Radiation and Drug Combinations:
Potmesil et al. (1986), radiation + doxorubicin analogs in vitro; Seo et al. (2006), combination of radiosensitizers in vitro.
14. Antiparasitic Combination:
Davoudi et al. (2005), antileishmania vaccine using HSVTK conferring increased sensitivity to ganciclovir and 5fluorocytosine.
15. Segmental Reviews for MedianEffect Principle and Combination Index Methods.

Journal review articles.

Chou (1980), on carcinogen risk assessment.

Chou and Talalay (1983), on a new look at a very old problem.

Chou et al. (1994), on clinical protocol design.

Chou et al. (2006), on MDR reversal and clinical protocol design.


Monograph chapters.

Chou and Talalay (1984), a stateoftheart overview of MEP and CI.

Chou and Talalay (1987), on drug combination in chemotherapy.

Chou (1991), a updated overview of MEP and CI.

Chou (1994), on clinical perspectives.

Chou et al. (1994), on immunosuppressants combinations.

Chou and Kahan (2001), on modern immunosuppressives and therapy.

Chou and Chang (2002) and Chang et al. (2006), on MDR reversal and clinical protocol design.


Encyclopedia chapters.

Chou and Fanucchi (1988), in Encyclopedia of Medical Devices and Instrumentation, on computer software as a medical device.

Chou et al. (1991a, 1997), in Encyclopedia of Human Biology, a brief overview on MEP and CI.

Bertino and Chou (1997), in Encyclopedia of Cancer, a brief overview on drug combinations.


Editorial: Chou (1998), “Drug Combinations: From Laboratory to Practice.”

Computer Software and Users' Guides.

Chou and Chou (1985), for Apple II computer.

Chou and Chou (1988), for IBMPC.

Chou (1991a) and Chou JH (1991), review, theory, and equations.

Chou and Hayball (1997), CalcuSyn for PC Windows.

Chou and Martin (2005), CompuSyn for PC Windows interface, CDROM, Web browser, graphic improvements, statistic updates, and polygonograms.

VI. Illustrations of Real Data Analysis with MassAction LawBased Computer Software
A. SingleDrug, TwoDrug, and ThreeDrug Combination Analysis with Computer Software
1. SingleDrug Analysis and TwoDrug Combinations.

Inhibition of alcohol dehydrogenase with ADP using Yonetani and Theorell (1964) data. Data given in Table 8 are analyzed for ADP as a single drug (points [1][5]). After entry of dose and effect, dose_{2} and effect_{2},..., the computergenerated report indicates that m = 1.0423, D_{m} = 1.6561 μM, and r = 0.99964. These results indicate excellent conformity to the massaction law principle. Other single drugs in the table, such as ADPR and ophenanthrolene, can be analyzed similarly.

Inhibition of alcohol dehydrogenase with ADP and ADPR in nonconstant ratio combinations, using “all data points” in Table 8 with constant ratio and nonconstant ratios.

Inhibition of alcohol dehydrogenase with ADP and ADPR in a constant ratio (1:180) using diagonal data points [1]∼[5] only. For details of the CompuSyn analysis on section VI.A.1.a.c., see Supplemental Data Appendix II.

Inhibition of alcohol dehydrogenase with ADP and ophenanthrolene (Table 9). For details of the CompuSyn analysis on section VI.A.1.d., see Supplemental Data Appendix II.

Conclusions and comments for section IV.A.1.a.d.:

In entering experimental data (Yonetani and Theorell, 1964) into the derived multiple drugeffect equations (Chou and Talalay, 1981), using manual calculations in 1981 (before the computer software for these became available), there was excellent conformity between the medianeffect equation and the multiple drugeffect equation from the laboratory of a highly esteemed Nobel Laureate, Axel Hugo Theodor Theorell (Chou and Talalay, 1981).

In the present computerized analysis for either one of three compounds (ADP, ADPR, or ophenanthrolene), the r values ranged from 0.99 to 0.999 and for their constant ratio combinations (points [4][5] in Tables 8 and 9), the r values ranged from 0.99 to 0.999. Again, the experimental results conformed to the massaction law excellently.

. The m values are nearly 1, indicating that both inhibitors and their combinations follow firstorder kinetics, which is exactly what was presented by MichaelisMenten kinetics (Dixon and Webb, 1964).

The computerized CI analysis indicates that ADP and ADPR have an additive effect, whereas ADP and ophenanthrolene exhibit a synergistic effect. Most importantly, Supplemental Data Appendices II and III both indicate that virtually identical conclusions can be reached when data analyses are executed in the following ways:

Using the full contents of the data in a checkerboard (or Latin square) (i.e., in both Tables 8 and 9, there are 36 datapoints, 5 for D_{1}, 5 for D_{2}, and 25 for D_{1} + D_{2} in various ratios, plus a control).

Using only the constant ratio design: Table 8, points [1][5] in 1:190; and Table 9, points [1][5] in 1:17.4.

Using only the various partial data for the combination analysis, either in a constant ratio or nonconstant ratios.

Using only one single data point of combination. Still we can obtain a nearly identical conclusion. However, this single dose design for combination is not recommended as a common practice, since most biological studies are not as accurately measured as enzyme assays.


The above results indicate that by using the combination index method, we can still reach a nearly identical conclusion, even though the data from different experimental designs or very few data points were used, provided that the data obtained are from accurate measurements. However, it is recommended that the constant ratio design (i.e., the diagonal design in Tables 8 and 9, for points [1][5], excluding the two triangles in the tables) is the most efficient and economic way to conduct drug combination studies, since, as shown in the above examples, it saves 80% of data points (in two triangles) in the table and yet it allows computerized simulation of the F_{a}CI plot and the F_{a}DRI plot with the full scope of effects and the construction of an isobologram at any desired effect level. It also allows construction of a classic polygonogram if the experiment is expanded to three or more drugs. The saving of data points for a combination study can be translated into saving animals, saving costs, and saving time in in vivo studies (Chou et al., 2005a) (see also section VI.D.). The prerequisite of using this method is to conduct reliable and accurate assays, both for single drugs and their combination(s).
2. Topological Analysis for the Multiple Ligand Sites in the SteadyState System.

By application of the LineweaverBurk plot, ADP and ADPR are known to be competitive inhibitors with NAD, and ophenanthrolene is known to be a noncompetitive inhibitor (Yonetani and Theorell, 1964). By contrast, the medianeffect plots for ADP and ADPR and (ADP + ADPR) at a constant ratio yield three parallel lines, indicating that the bindings for ADP and ADPR are “mutually exclusive.” The medianeffect plots for ADP, as well as those for ophenanthrolene, yield parallel lines. However, ADP + ophenanthrolene at a constant ratio yields a steeper and slightly concave upward curve, indicating that the bindings for ADP and ophenanthrolene are “mutually nonexclusive” (Chou and Talalay, 1981). Note that competitiveness can be determined by the LineweaverBurk plot (Lineweaver and Burk, 1934) plot, whereas exclusiveness can be determined by the medianeffect (Chou, 1976) plot. These plots have somewhat different meanings, with the latter having topological implications (Chou, 1974, 1977a; Chou and Chou, 1988, 1990a). These topological analyses are useful for simple systems, such as enzymes or receptors. Some complexity is expected in conditions with combinations of three or more ligands.

The availability of the ligandbinding site of an enzyme or receptor in the steady state can be described by eq. 3, K_{i}/I_{50} = E_{x}/E_{t} (Chou, 1974, 1977a), where K_{i} can be determined with the LineweaverBurk plot (1934), and the IC_{50} value can be determined by the medianeffect plot (Chou, 1976) (see also Fig. 3). A method of topological analysis for the binding of two or more inhibitors has been proposed (Chou and Chou, 1988, 1990a). This proposed method for the steady equilibrium state is illustrated in Fig. 13. By contrast, the tight bindings can be analyzed by physicalmechanical means, such as cofraction in chromatography and cozoning of electrophoresis.
3. Two and ThreeDrug Combinations against Cancer Cell Growth and the Construction of Polygonograms.

Crude data: Numerical crude data are obtained from a study reported in Chou et al. (1994) on inhibition of teratocarcinoma cell growth by paclitaxel (Taxol), cisplatin, topotecan, vincristine, and etoposide individually and in their two and threedrug combinations (Table 10). These experiments used a constant ratio design. All five drugs that have been studied have different pharmacological mechanisms of action. The polygonogram was used for the first time; however, this term was not formally introduced until 4 years later (Chou and Chou, 1998).

Computer printouts: The full computer printout using the CompuSyn software for analysis of data in Table 10 is shown in Supplemental Data Appendix IV. Computer printouts for (A) paclitaxel (Taxol), (B) cisplatin, (C) topotecan, and A + B, B + C, A + C, and A + B + C in a constant ratio of A:B:C = 1:100:10 and for their twodrug components are included. Printouts for other combinations using (D) etoposide and (E) vincristine, such as A + D, A + E, B + D, B + E, A + B + D, and A + B + E are not shown in Supplemental Data Appendix IV, because of the size of the report, but the conclusions can be found in the original article (Chou et al., 1994).

Summary, conclusions, and comments for the above examples: Table 11 gives a brief summary of the results for paclitaxel (Taxol), cisplatin, and topotecan combinations on the CI and DRI values at ED_{50}, ED_{75}, ED_{90}, and ED_{95}. Conclusions are the following:

Overall, the doseeffect relationships for single drugs (r = 0.9050.993) (Table 9) and for combinations (r = 0.9840.999) (Table 8) follow the massaction principle excellently.

The in vitro cellular studies usually have m > 1; however, for animal studies m >> 1 is common (e.g., see Tables 11, 12, 13).

. It has been reported that paclitaxel (Taxol) is a microtubule stabilizer, vincristine is microtubule depolymerizer, and topotecan is a DNA topoisomerase I inhibitor, whereas etoposide is a DNA topoisomerase II inhibitor and cisplatin is an alkylation agent. The CI results shown in Tables 8 and 10 clearly indicate that, in most cases, there is no way to predict synergism or antagonism from the mechanisms of drug actions. Synergism or antagonism needs to be determined and not be predicted.

. In this study, we frequently observed antagonism occurring at low effect levels and synergism occurring at high effect levels for a given combination. For cancer chemotherapy, we need to kill cancer cells 90, 99, or 99.9%. Synergism at high effect levels should be more relevant to therapy than those at low effect levels. It should be noted that the synergism scale of CI is from 0 to 1, and the antagonism scale of CI is from 1 to infinity. The F_{a}log (CI) plot places synergism and antagonism on symmetrical and equal footings.

B. Other Applications of the MedianEffect Principle of the MassAction Law
1. Estimating LowDose Risk of Carcinogens.

Introduction: Estimating the effect of a very low dose of carcinogen that may affect only [1/10,000] or [1/100,000] of a population is difficult to accomplish without extrapolation. The extrapolation process involves a great deal of magnification on a mathematical scale and in one's imagination. Ideally, extrapolation should be based on sound principles, and one should not merely rely on arbitrary mechanical tools or visual inspection. Because carcinogenic data conform to the massaction law excellently, as indicated by the r values (i.e., r ≈ 1), it is therefore proposed that the medianeffect equation of the massaction law be used for a lowdose risk assessment. Thus, in 1980, Chou applied the medianeffect equation using a pocket calculator and analyzed two sets of experimental data for chronic carcinogenic exposure from Peto (1974) and Peto et al. (1975) and three sets of experimental data of acute carcinogenic exposure from Bryan and Shimkin (1943). Later, these data were analyzed using computer software (Chou and Chou, 1985; Chou and Talalay, 1987). These data have now been further analyzed with CompuSyn (Chou and Martin, 2005)and along with the calculated parameters are given in Table 12.

Analytical conclusions: For “chronic exposure” to the skin of Swiss albino female mice (using benzo[a]pyrene as a carcinogen), starting at 10 weeks and starting at 55 weeks (Peto et al., 1975), there was little difference in carcinogenic effect, with D_{m} = 39.469 and 41.120 weeks, respectively. There were also no significant differences in the m values of 4.5693 ± 0.0630 and 4.6225 ± 0.1337, respectively; which both showed very steep (i.e., very sigmoidal) doseeffect curves. Remarkably, both give excellent conformity to the massaction law with r = 0.9981 and r = 0.9917, respectively.
For “acute exposure” of C3H male mice via subcutaneous injection with benzo[a]pyrene and methylcholanthrene and dibenz[a,h]anthracene (Bryan and Shimkin, 1943), it yielded somewhat less sigmoidal shapes for the three carcinogens (m = 1.3879, 1.930, and 1.7720, respectively) than chronic exposure, as indicated above. Again, excellent conformity to the massaction law with r values of 0.9973, 0.9992, and 0.9984, respectively, have been observed (Fig. 14). Even at low doses, there is no apparent tendency for a systematic deviation from linearity. This conformity to the medianeffect principle provides a rationale for lowdose risk assessment for carcinogens.
By using the benzo[a]pyrene data from both chronic and acute exposure in Table 12 as examples, it is possible to estimate the risk of a lowdose carcinogen (e.g., 0.50.001 D_{m}) for both acute and chronic exposures, as shown in Table 13. These findings have interesting and divergent implications for risk at low and high doses. For a given low total dose of carcinogen, chronic exposure is much less hazardous than an acute single injection. Thus, at 0.1 and 0.01 D_{m}, chronic exposure is 10^{3} and 10^{6} times less hazardous, respectively. In contrast, extrapolation to high (cumulative or single) doses indicates that chronic exposure is more hazardous than a single injection. The medianeffect plots indicate that both the D_{m} values and m values are greatly affected by the mode or route of exposure. The m value, which describes the increment of dose versus the increment of effect, is a reflection of the basic intrinsic characteristics of the doseeffect relation of a carcinogen (Chou, 1976, 1980). Although these analyses have been carried out for single carcinogens only, similar methods using the multiple drug equation are apparently applicable to the effects of multiple carcinogens (including cocarcinogens) and the analysis of their interactions (e.g., synergism, additive effect, or antagonism).
2. Risk Assessment for Radiation.

Crude data: Data for radiationinduced leukemia incidence among Hiroshima atom bomb survivals during 1950 to 1957 (Heyssel et al., 1960; Upton, 1961) were subjected to MEP analysis (Chou and Chou, 1985). These data, as shown in Table 14, have now been further analyzed with CompuSyn (Chou and Martin, 2005).

Analytical conclusions and comments.

The following parameters were obtained: m = 0.8956, a slightly flat doseeffect curve suggesting a negative cooperativity or repair process; D_{m} = 2,769,289 (rad), the dose required to induce leukemia in one half of a million people; and r = 0.98963, a very good correlation coefficient despite the fact that it was obtained from vivo human studies.

From these parameters, we can get an estimated risk at a very low dose by using eq. 9: f_{a} = 1/[1 + (D_{m}/D)^{m}], thus: ⇓

Similarly, we can calculate the required dose for producing a given low risk, e.g., 1 × 10^{6}, 1 × 10^{5}, 1 × 10^{4} by eq. 8: D_{x} = D_{m} [f_{a}/(1  f_{a})]^{1/}^{m}, thus: ⇓

3. Therapeutic Index and Safety Margin

Introduction: Depending on the type of drugs and type of experiments, the therapeutic index is usually defined as the ratio of LD_{50}/ED_{50} or TD_{50}/ED_{50}, and the safety margin can be defined as TD_{10}/ED_{90} or, more restrictively, TD_{5}/ED_{95}, where L, E, and T represent lethal, therapeutically effective, and toxicity, respectively. The requirement of these indexes for beneficial therapy is usually low for cytotoxic anticancer agents (e.g., therapeutic index = 210) and very high for drugs that are used for treating nonlifethreatening diseases. The endpoint of the measurements can also influence the magnitude of an index. Although the endpoint for lethality (for LD) is clearcut and universal, the choice of endpoints for therapeutic effect (for ED) and for the toxic effect (for TD) are somewhat arbitrary and circumstantial. For a less quantitative context, the therapeutic index is called the therapeutic window.

Crude data: An antitumor agent has been studied in mice bearing L1210 leukemia. The fractional leukemic cell survival is calculated from a calibration curve for increasing the size of leukemic cell inoculation and lifespan shortening as proposed by Skipper (1974). These data are given in Table 15.

Analytical conclusions: The parameters for therapeutic effect are m = 9.7139, an extremely steep doseeffect curve (i.e., a very highly sigmoidal shape); D_{m} = 3.19097; and r = 0.9940 and for lethality to the host are m = 7.3700, an extremely steep doseeffect curve; D_{m} = 47.4524; and r = 0.9757. By using the above parameters and eq. 8,
CompuSyn will provide D_{x} values at any f_{a} levels. Thus, Therapeutic index = LD_{50}/ED_{50} = 47.452/3.1909 = 14.870 Safety margin = LD_{10}/ED_{90} = 35.219/4.0009 = 8.803 Similarly,
The doses (in the same unit) and fractional effect pairs (for the therapeutic and the toxic effects) are needed to calculate m and D_{m} (and r) parameters. These parameters are then used to calculate the D_{x} values by using the medianeffect equation.
4. AgeSpecific Cancer Incidence Rate Analysis.

Introduction: The application of MEP to the agespecific cancer incidence or mortality rate in humans was not originally planned since the derivation of the medianeffect equation at a steady state and at equilibrium state did not include a time factor. However, it is reasonable to assume that carcinogenic insults are proportional to age (Chou, 1978; Chou and Miller, 1980). The proportionality is implicated in the m value, which is the slope of the doseeffect curve. Chou and Chou (1985) have used the U.S. Third National Cancer Survey from 1975 and have randomly selected agespecific colon cancer incidence data among U.S. white males, 1969 to 1971, as an example for an analysis (Table 16). These data have now been further analyzed with CompuSyn (Chou and Martin, 2005). Remarkably, the r value is 0.9958, which shows excellent conformity between data and theory and thus points to useful epidemiological applications.

Crude data and analytical results: The analysis has used eq. 9: (f_{a}) × 10^{5} = [1 + (D_{m}/age)^{m}]^{1} × 10^{5} (for a population of 100,000). Sample data are given in Table 16. The medianeffect plot yields m = 4.982 ± 0.137, a very high m value (i.e., a highly sigmoidal shape); D_{m} = 254.92 (years), calculated by D_{m} = antilog (y_{intercept}/m); and r = 0.9958. As shown in the last two columns of Table 16, the observed incidence per 10^{5} population [(f_{a})_{obs} × 10^{5}] and the calculated incidence per 10^{5} population [(f_{a})_{cal} × 10^{5}] are in close agreement. From the parameters and the equation above, it can be calculated that at age 65, a U.S. white male would have a 1.103 × 10^{3} chance of getting colon cancer. At the age of 80, the chance of colon cancer among U.S. white males would be increased to 3.098 × 10^{3} (a 2.8fold increase, although age increased only 23%).
5. Epidemiological Applications (Chou, 1978; Chou and Miller, 1980):
The medianeffect equation and parameters have also been applied to other epidemiological data for trend studies:

Epidemiological trend analysis for different time periods, e.g., different decades for various diseases can be analyzed in terms of massaction law parameters.

Social and economic effect analysis, e.g., changes in the population who smoke are reflected in the m values (e.g., decreased smoking leads to decreased m values and increased D_{m} values) for lung cancer among males and females and among racial groups.

Physiological status effect analysis, e.g., sudden changes in the m value (e.g., a break) are seen in breast cancer in females at the time of menopause.

Other applications, e.g., the use of the medianeffect equation for the analysis of the environmental factor effects.

Changes in the massaction law parameters for the cancer mortality rate at different decades may reflect therapeutic improvement and/or the early detection of cancer over decades for trend analysis.

The above analyses will be quantitative, not only qualitative.
6. Calculation of K_{i} from IC_{50}.

Introduction: During the early stages of deriving the medianeffect equation, Chou (1972, 1974) used a systematic approach to derive equations in enzyme kinetic systems. The following relationship was derived (Chou, 1974).
or
for the competitive inhibition of a single substrate (A) in a firstorder reaction, where K_{A} is the enzymesubstrate dissociation constant (K_{m}) and E_{x} is the proportion of total enzyme (E_{t}) that is available for inhibitor binding. This relationship also indicates that the K_{i} (the inhibitorenzyme dissociation constant) can never be greater than IC_{50}.
Apparently eq. 24 can be used for calculating K_{i} from IC_{50} in receptor binding system, which yields where L is the primary or substrate ligand concentration and K_{D} is the ligandreceptor dissociation constant. K_{D} can be determined by the Scatchard (1949) plot and IC_{50} (for the inhibitor or secondary ligand) can be determined by the medianeffect plot. Currently, eq. 25 is one of the most widely cited methods in the biomedical literature, especially in neuroscience and pharmacology. The following is an example for analysis.
By using an illustrative inhibitor doseeffect relationship given in Table 17, the m, D_{m}, and r parameters for an inhibitor can be determined with the medianeffect plot. These, in conjunction with K_{D} determination using the Scatchard plot at the fixed concentration of a primary ligand, the K_{i} values can be determined from the IC_{50} (i.e., the D_{m} value) by using eq. 24 (Cheng and Prusoff, 1973; Chou, 1974), as illustrated in the footnote of Table 17. Many other examples for calculating K_{i} from IC_{50} are given in section V.A.3.
C. Sample Analysis of Drug Combination Data with Computerized Summaries
1. Synergism of Two Insecticides on Houseflies.
Le Pelley and Sullivan (1936) studied the lethality of rotenone, pyrethrins, and their mixture on houseflies. In this study, 900 to 1000 adult flies were used for each dose of these drugs at five dose levels. These data, as shown in Table 18, are of historical interest, since during the past 70 years, researchers from at least five different laboratories (Finney, 1947, 1952; Chou and Talalay, 1987; among others) have attempted to answer the question of whether or not there is synergism between these insecticides. A complete CompuSyn report (printout) for both 1:5 and 1:15 combinations is given in Supplemental Data Appendix V (http://pharmrev.aspetjournals.org/cgi/content/full/pr.58.3.10/DC1).
Table 18 reveals that the r values are greater than 0.993 and nearly 1.0, indicating that the applicability of the MEP method to these in vivo data is excellent. The LD_{50} values for rotenone, pyrethrins, and their 1:5 mixture calculated from the medianeffect plot are 0.1505, 0.8932; and 0.4497 mg/ml, respectively (Table 18; see also Supplemental Data Appendix V for details). These values are in close agreement with those obtained from the probit analysis by Finney (1952), who obtained 0.156, 0.918, and 0.455 mg/ml, respectively.
The original authors interpreted their results as indicating no striking antagonistic or synergistic effect of the mixture. Richardson, quoted by Finney (1952), used a method for the mixture equivalent to the similar action law and asserted that there was pronounced synergism. Bliss, as indicated by Finney, supported Richardson's conclusion, and Finney after a new analysis of data, also agreed that there was evidence of synergism (Finney, 1952). The present analysis of the same data with the combination index equation and with the computer simulation (Table 18; Supplemental Data Appendix V) indicates that rotenone and pyrethrins (1:5) yielded nearly an additive effect with slight synergism (the CI values from ED_{50} to ED_{95} are 0.91760.9152). The 1:15 combination yielded slightly less synergism than the 1:5 combination since the former yielded CI values from ED_{50} to ED_{95}, which ranged from 0.9543 to 0.9187. CompuSyn generated a F_{a}CI plot [and F_{a}log (CI) plot] and a F_{a}DRI plot [and F_{a}log (DRI) plot]; the classic isobolograms for rotenone and pyrethrins are given in Supplemental Data Appendix V.
2. Antagonism between Methotrexate and Arabinosylcytosine.
The inhibition of the incorporation of [^{3}H]dUrd into DNA of L1210 leukemic cells by methotrexate (MTX) and AraC antimetabolites was analyzed by Chou and Talalay (1984). The experiment was carried out in a constant molar ratio of 1:0.782, and the data are given in Table 19.
The computerized analysis of these data yields the following results. For MTX, m = 1.091, D_{m} = 2.554 μM, and r = 0.9842; for AraC, m = 1.0850, D_{m} = 0.06245 μM, and r = 0.9966. For the combination of MTX and AraC (1:0.782): m = 1.1296, D_{m} = 0.2496 μM, and r = 0.9995. The combination index shows an antagonism at all effect levels with the CI value at IC_{50} to IC_{90} ranging from 1.809 to 1.626.
3. SevenDrug Combination against Human Immunodeficiency Virus and Their Polygonograms.
a. Introduction.
In this large scale experiment, seven antiHIV agents were used for two, three, four, and fivedrug combination to compare the results under the standarized conditions with P24 enzymelinked immunosorbent assays in MT4 cells infected with HIV1III_{B}. The preliminary results of this study were disclosed earlier (Chou and Zhu, 1997). In all, 21 sets of twodrug combinations, 20 sets of threedrug combinations, 10 sets of fourdrug combinations and 2 sets of fivedrug combinations have been analyzed using CalcuSyn (Chou and Hayball, 1997). The purposes of this study were 1) to rank synergism and antagonism within the group and among the groups for the degrees of synergism and antagonism in terms of CI values, 2) to compare the DRI values within the group and among the groups for the benefit of dose reduction at different effect levels, 3) to illustrate the automated construction of a polygonogram for a largescale experiment, 4) to demonstrate the usefulness of a polygonogram when presenting a massive amount of results, semiquantitatively, for a simple visual inspection, and 5) to predict the synergism or antagonism, semiquantitatively, of a higher number of drugs in combinations (that have not yet been carried out) from the CI results of the lower number of drugs in combinations.
The CompuSyn (Chou and Martin, 2005)generated computer report for two to fivedrug combination antiHIV studies consists of 90 pages of printout, which is available as Supplemental Data (http://pharmrev.aspetjournals.org/cgi/content/full/pr.58.3.10/DC1). Both software programs yielded vertically identical results except for slight differences after several digits of the decimal point. These slight differences are due to the slight differences in digits of the decimal points used for data entries. Using CalcuSyn, data sets need to be divided into many groups for analysis, whereas CompuSyn can handle largescale data sets at once, as shown in Supplemental Data. In addition, CompuSyn has better graphics, new features such as polygonograms. and more flexible options (see section IV.).
b. Summaries of results.
The massaction law parameters for each drug alone are given in Table 20. The CI values for 21 sets of twodrug combinations at ED_{50}, ED_{75}, ED_{90}, and ED_{95} are given in Table 21; the CI values for 20 sets of threedrug, 10 sets of fourdrug, and two sets of fivedrug combinations at ED_{50}, ED_{75}, ED_{90}, and ED_{95} are given in Table 22. The selected examples for the relationship of CIs and DRIs in the two to fivedrug combinations are given in Table 22. As expected from eq. 16, DRI and CI are somewhat inversely related. However, DRI for each drug can be influenced by the combination ratios of the experimental design. The polygonograms for twodrug combinations of the seven drugs are given in Fig. 9a.
c. Conclusions.
Conclusions are as follows:

The rank orders of potency based on IC_{50} values (in micromolar concentrations, except for IFN in kallikrein units per milliliter) (Table 20) are AZT > ABT538 > D4T > NEV > IFN > DDC >> DDI. The combination ratios for these seven drugs that have been used in experiments are 1:4:10:16:40:50: 1600, respectively.

For twodrug combinations, the rank orders for synergism based on the weighted CI values in Table 21 are AZT + NEV > IFN + NEV ≥ DDI + ABT538 > DDI + NEV ≥ AZT + ABT538, IFN + ABT538. The rank orders for antagonism based on weighted CI values are AZT + D4T > DDI + D4T ≥ DDI + DDC > AZT + DDC.

For threedrug combinations in Table 22, the rank orders for synergism are AZT + IFN + NEV > DDI + IFN + NEV ≥ AZT + IFN + ABT538 ≥ AZT + NEV + ABT538 ≥ AZT + DDI + NEV > DDI + NEV + ABT538 ≥ DDI + IFN + ABT538 ≥ AZT + DDI + ABT538. The ranks for antagonism are AZT + DDC + D4T ≥ AZT + D4T + ABT538.

For fourdrug combinations in Table 22, the synergism rank orders are DDI + IFN + NEV + ABT538 > AZT + IFN + NEV + ABT538 ≥ AZT + DDI + IFN + ABT538 ≥ AZT + DDI + IFN + NEV. The rank orders for antagonism are AZT + DDC + D4T + ABT538 > AZT + D4T + IFN + ABT538.

For fivedrug combinations in Table 22, the synergism is AZT + DDI + IFN + NEV + ABT538 and the slight antagonism is AZT + DDC + D4T + NEV + ABT538.

Results in Table 23 indicate that two, threefour and fivedrug combinations all showed favorable dosereduction from ED_{50} to ED_{95}. The higher number of drugs in combination yields more favorable dose reduction than the lower number of drugs in combination.

The polygonogram (Fig. 9a) for twodrug combinations of seven drugs provide semiquantitative simple predictions of synergism or antagonism in three, four and fivedrug combinations as indicated in the experimental observations in Table 22.

The results in Table 21 for twodrug combinations offer good predictions for synergism or antagonism for the three, four, and fivedrug combinations in Table 22. There are 120 possible combinations for 7 drugs and 21 combinations for twodrug combinations in Table 21 and yet the software produced a great deal of predictive power for combinations involving three, four, and five drugs. The results for threedrug combinations also provided good predictions for the outcomes of four and fivedrug combinations.

AZT, DDC, DDI, and D4T are nucleoside analogs that are known to inhibit HIV reverse transcriptase whereas NEV is a nonnucleoside reverse transcriptase inhibitor. ABT538 is known to be a HIVassociated protease inhibitor, and IFN possesses rather complex mechanisms. The results in Tables 21 and 22 suggest that combinations consisting of different classes of antiHIV agents (e.g., attacking different stages of the HIV life cycle) are more likely to produce better synergism than those consisting of several antiHIV agents from the same class.
D. Approaches for the Conservation of Laboratory Animals
Millions after millions of animals have been used annually for biomedical research in academia and industry for doseeffect studies of drug efficacy, toxicity, and drug combinations. During the past decades, much attention has been focused on humane treatment of animals as indicated in federal legislation and regulations (U.S. Department of Agriculture Animal Welfare Act and Regulations, 2002; U.S. Public Health Service Policy on Humane Care and Use of Laboratory Animals, 2002,). It should be stressed, however, that attaining research aims with a reduced number of animals or avoiding the necessary use of animals is not only economically costeffective but also an ethical blessing for humanity. The cost of using animals in preclinical research not only involves animal purchasing, maintenance, facilities, and equipment but also involves personnel time and their efforts, in addition to frequently unavoidable pain, discomfort, suffering, or death of the laboratory animals. The popular methods in the 1940s to the 1960s for determining LD_{50} values using a large number of animals for the sake of fulfilling statistical requirements is no longer considered necessary or ethical, especially when using large animals.
During past decades, Chou JH, Chou, and Talalay, (Chou JH et al., 1983, 1984) had advocated reducing the numbers of animal used by scientific means, such as improving experimental design and using a computer and software based on the massaction law. A featured special report on Chou's computer approach in saving animal's lives was presented by Cusack (1983) that highlighted the team of father and son's crusade of saving animals with a cartoon.
Conservation of laboratory animals can be achieved by these and other means. We repeatedly advocated use of the medianeffect principle (MEP) of the massaction law, its efficient experimental design, and computer simulation as well as SDA and polygonogram methods (Chou and Chou, 1985, 1989; Chou and Hayball, 1997; Chou and Martin, 2005). These aims can be illustrated on several fronts, as indicated below:
1. The MedianEffect Principle.
With the medianeffect principle [i.e., the medianeffect equation (eqs. 7, 8, 9) and the medianeffect plot], we are using animal data to “conform” the fundamental massaction law. We are not using animal data to “fit” an empirical and unfounded doseeffect curve. These are illustrated by the theoretical analysis given in Figs. 3 and 11, as well as by actual experimental data in vitro given in Tables 8, 9, 10, 11 and 20, and the in vivo data given in Tables 12, 13, 14, 15, 16 and 18, and in Fig. 14 (and their computerized analysis, parameters, and graphic printouts). There is no doubt (i.e., from their r values) that the experimental data, both in vitro and in vivo, conform to the MEP excellently. At the theoretical extreme, we can even obtain a doseeffect curve from only two data points, but this requires accurate measurements and uses of uniformed test subjects. By using fewer data points (either theoretical or experimental, in vitro and in vivo), virtually identical results or conclusions can actually be verified by arbitrarily deleting one, two, or even three data points from the existing data points, rerunning the analysis by using a computer, and compare the resulting conclusions. The results and conclusions are usually similar (see Supplemental Data Appendices IIV). For the study of microtubule stabilization of epothilones as antitumor agents, we have used only three to six nude mice for each dose against various xenograft tumors and depending on the experiment purposes, we have used a very few doses (e.g., three to four doses for the doseeffect relationships in vivo) (Chou et al., 1998a,b, 2001, 2005a,b,c).
2. Experimental Design.
For drug combination studies, there are different ways to combine the drugs. I recommend using a constant ratio combination (e.g., equipotent combination, such as IC_{50} or ED_{50} ratios) as illustrated in Tables 5 and 19 and Fig. 10. This type of diagonal scheme design for in vivo studies can easily save 60 to 80% or more animals than the checkerboard design and yet allows the determination of potency, shape, and conformity parameters (D_{m}, m, and r, respectively) as well as automated construction of doseeffect curves, medianeffect plots, the F_{a}CI plot, the F_{a}DRI plot, and isobolograms. Remarkably, before use of the combination index method and formal derivation of the isobologram equation, earlier scholars, such as Le Pelley and Sullivan (1936) (see Table 18 and Supplemental Data Appendix V) had already used 1:5 and 1:15 constant combination ratios for rotenone and pyrethrins against houseflies (8001000 houseflies/dose). Equally remarkable, Yonetani and Theorell (1964) carried out elegant studies on alcohol dehydrogenase with outstanding accuracy (as manifested by r values) and had also used inhibitors in combination at constant ratio(s) (see Tables 8 and 9 and Supplemental Data Appendices II and III).
In vivo anticancer drug combination studies were carried out by Chou et al. (2005a) on the basis of the combination index method and the constant ratio experimental design. Taxotere and T900607 were combined at 1:2 ratios against a human mammary carcinoma MX1 xenograft in nude mice. There were three doses for each drug alone and in combination. Each dose was tested in five mice. A total of 60 mice, 50 mice for the drug combination experiment [15 mice each for the single drug alone and its combination (three groups) and 5 mice for the untreated control group] plus 10 mice for the preliminary exploratory studies, had been used, and the authors were able to obtain the abovementioned parameters (including ED_{50}) and plots (e.g., doseeffect curves, medianeffect plots, F_{a}CI plots, F_{a}DRI plots, and isobolograms).
3. Serial Deletion Analysis.
In the doseeffect relationship studies, there is no fixed rule as to how many doses should be used for a given drug. Usually, for in vitro studies, we used five to eight concentrations and for in vivo studies we used three to five doses. As indicated above, the medianeffect plot can generate a doseeffect curve and calculate massaction law parameters with two or more doses. For example, for pyrethrins in Table 18, 5 doses, 0.5, 0.75, 1, 1.5, and 2 μg/ml, were used, for which the generated effect yielded m = 2.387, D_{m} = 0.893 μg/ml, and r = 0.995 (see Supplemental Data Appendix V). If we rerun these data by deleting one dose each time, a series of m, D_{m}, and r values to subject to statistical treatments can be obtained. Similarly, the same principle can be applied for rotenone. Each corresponding D_{m} and m value for each drug can be introduced into the combination index equation for a series of CI determinations, which can also be subjected to statistical treatment. Thus, the mass lawbased equation has unique properties that allow treating one experiment as if it were several experiments for maximal utilization of data. Consequently, intelligent design and analysis may reduce experimental size, reduce time and effort needed for experiments, and reduce animal usage for both single drug and drug combination studies.
4. Polygonogram.
Although, testing of extra largescale drug combinations are difficult to carry out in animals for practical and economical considerations, theoretically, as shown in Fig. 8a, seven drugs may have 120 different drug combinations. Inspection of a polygonogram with only 20 sets of twodrug combinations of seven antiHIV agents allows semiquantitative prediction of what would happen to the combinations with higher number of drugs before the experiments are actually carried out. The usefulness of these predictions has been largely confirmed by experiments that are actually carried out for three, four, and fivedrug combinations. This predictive utility can be translated to fewer experiments, less time and effort, and less animal usage when the experiments are carried out in vivo.
Appendix I: Derivation of the Multiple DrugEffect Equation
Detailed descriptions for the derivation of multiple drugeffect equation based on the medianeffect principle (Chou, 1976) have been given in Chou and Talalay (1977, 1981, 1984). This process has been outlined in Fig. 2. The early steps of the derivations can be summarized as the following, which can then lead to the combination index equation (Chou and Talalay, 1983).
A. Summation of the Effects
The summation of the effects (additive effects) of multiple inhibitors of various types on the initial velocity of enzyme systems obeying MichaelisMenten kinetics is described by the general relation (Chou and Talalay, 1977): wherein v_{1,2,3...} _{n} is the velocity of reaction in the simultaneous presence of n inhibitors, v_{i} is the velocity observed in the presence of each individual inhibitor, and v_{0} is the velocity in the absence of inhibition. The derivation is based on the assumption that each enzyme species can combine with no more than one of the inhibitors (i.e., the inhibitors are mutually exclusive). The above relationship holds irrespective of the number of inhibitors, the type of inhibition (competitive, noncompetitive, or uncompetitive), or the kinetic mechanism (sequential, pingpong, or random) of the enzyme reaction under consideration. Derivations from this equality define synergism or antagonism of inhibitors, depending on whether the value on the left side of the above equation is greater or smaller than the right side, respectively. Knowledge of the kinetic constants for substrates and inhibitors is not required.
If we define f_{v} = fractional velocity = v_{n}/v_{0} and f_{i}, the fractional inhibition, = (1  f_{a}), then the analysis can proceed as below:
B. Alternative Equations for Multiple Inhibitors in FirstOrder Systems
These can be obtained, and several useful forms of eq. A1 can be derived as follows. Because the fractional velocity (f_{v}) is v_{n}/v_{0} and the fractional inhibition (f_{i}) is (1  f_{v}), then multiplying both sides of eq. A1 by v_{0} and taking the reciprocal, we obtain Because by definition f_{i} + f_{v} = 1, eq. A2 can be converted to Again, because by definition (f_{i})_{1,2... }_{n} = 1  (f_{v})_{1,2... }_{n}; therefore Based on the distribution equation of enzyme species, it was shown by Chou (1974) that the fractional velocity of MichaelisMententype enzyme reactions with different number of substrates, different reaction mechanisms (e.g., sequential or pingpong), and different types and mechanisms of inhibition (e.g., competitive, noncompetitive, or uncompetitive) by a single inhibitor, which binds to a given enzyme species, can be expressed by where E_{t} is the total amount of enzyme and E_{x} is the fractional availability of the enzyme species with which the inhibitor may combine.
Furthermore, the availability of the ligandbinding site, E_{x}/E_{t}, can be described by the distribution equation (Chou, 1974, 1977) where I_{50} is the medianeffect concentration of the inhibitor, i.e., that required to inhibit the enzyme reaction by 50%.
Hence, eq. A6 can be transformed to and Note that eq. A8, which is derived for a reference ligand (e.g., inhibitor), has the same form as the MichaelisMenten (1913) equation for a primary ligand (e.g., substrate).
Equations A7 and A8 give which is the medianeffect equation for firstorder kinetics (Chou, 1975, 1976).
In the presence of N inhibitors, a combination of eqs. A3 and A9, gives From eqs. A4, A5, and A10, we obtain (Chou and Talalay, 1981)
C. Inhibition of HigherOrder Kinetic Systems by a Single Inhibitor
If we consider the case where m molecules of inhibitor, I, interact with one molecule of an enzyme species to form a complex, then by extending the firstorder medianeffect equation to higher orders, the following relationship is obtained (Chou, 1975, 1976): The validity of eq. A12 is independent of the mechanism of inhibition and of the mechanisms of the enzyme or the nature of the enzyme species that binds to the inhibitor. An explicit and formal derivation for eq. A12 is given in Chou (1976) and Chou and Talalay (1981, Appendix II).
Because f_{i} + f_{v} = 1, an alternative form of eq. A12 is By definition of terms: Thus, the logarithmic form of the medianeffect equation (eq. A12) may be written as A graphic representation of eq. A15 for different m values and IC_{50} = 1 is given in Fig. 5. The slope of each line represents m and the medianeffect axis, where log [(f_{i})^{1}  1]^{1} = 0 (i.e., f_{i} = 0.5), intersects each line at the I_{50} value. This plot is designated as the “medianeffect plot.”
This equation for the “reference ligand” (such as inhibitor) is arithmetically similar to the currently formulated versions of the Hill equation for the “primary ligand” (S): where n is the Hill coefficient and K is a constant that was undefined. A comparison of the medianeffect equation and the Hill equation is given in Table 2.
D. Inhibition of the HigherOrder Kinetic Systems by Mutually Exclusive Inhibitors
In the presence of n inhibitors, each with m binding sites and if interactions follow Hilltype kinetics (i.e., the inhibitor binds to the enzyme sites in one step only), the general equation can be obtained by relating the firstorder (eq. A11) to the mth order relationship of f_{i}/f_{v} = (I/I_{50})^{m} (eq. A12) or (f_{i}/f_{v})^{1/}^{m} = I/I_{50} to give (Chou and Talalay, 1981) Taking the mth root gives When m = 1, eqs. A17 and A11 become the same. The usefulness of the medianeffect concentration (I_{50}) in analyzing the effect of multiple inhibitors is apparent from eq. A17, because when (f_{i})_{1,2... }_{n} = (f_{v})_{1,2...} _{n} = 0.5, the left term of the equation is equal to unity for any value of m and even when the m values for the inhibitors are different.
An alternative form of eq. A17 is The behavior of two mutually exclusive inhibitors in a secondorder system (m = 2) may now be presented graphically, e.g., a plot of f_{i} with respect to the concentration of inhibitor(s) or the medianeffect plot of log [(f_{i})^{1}  1]^{1} with respect to log (I).
E. Multiple Inhibitions by Mutually Nonexclusive Inhibitors
1. First Order.
In an earlier study (Chou and Talalay, 1977), we reported that multiple inhibitions of (firstorder) MichaelisMenten kinetic systems with noncompetitive, mutually nonexclusive inhibitors in single substrate reactions may be described by
By using the general principles described above, we can develop a simple shorthand method for deriving equations for multiple inhibitions by mutually nonexclusive inhibitors in firstorder and the simplified higherorder kinetic systems. To illustrate this procedure, it is convenient to first consider a mutually exclusive case by examining the distribution of enzyme species and then extending the analysis to mutually nonexclusive cases, as shown below.
a. Case 1.
I_{1} and I_{2} are mutually exclusive inhibitors that bind to the enzyme to form the following species: E + EI_{1} + EI_{2} (but not EI_{1}I_{2}). From the medianeffect equation (eq. A11), we obtain or, rearranging and substituting for f_{i} = 1  f_{v},
Now, by analogy to the inductive reasoning of the distribution of enzyme species, it becomes apparent that the terms in the denominator of eq. A20: 1, (I)_{1}/(I_{50})_{1}, and (I)_{2}/(I_{50})_{2}; or 1, (f_{i})_{1}/(f_{v})_{1}, and (f_{i})_{2}/(f_{v})_{2} correspond to the species E, EI_{1}, and EI_{2}, respectively. It should be noted that this is valid irrespective of the type of inhibition exerted by I_{1} and I_{2}.
b. Case 2.
I_{1} and I_{2} are mutually nonexclusive inhibitors that bind to the enzyme to form the following species: E + EI_{1} + EI_{2} + EI_{1}I_{2}. The enzyme species E, EI_{1}, EI_{2}, and EI_{1}I_{2} should correspond to the terms of 1, I_{1}/(I_{50})_{1}, I_{2}/(I_{50})_{2}, and I_{1}I_{2}/(I_{50})_{1}(I_{50})_{2}, respectively. Therefore, or, rearranging and substituting for f_{i} = 1  f_{v},
This relationship permits us to compare graphically the effects of mutually exclusive and nonexclusive inhibitors in higherorder systems. At all concentrations, the mixtures of mutually nonexclusive inhibitors have more inhibition than the exclusive inhibitors, and at high concentrations, there is more inhibition than with the parent components (i.e., the concave upward doseeffect curves) (Chou and Talalay, 1981).
2. Multiple Inhibitions by Inhibitors with Different Kinetic Orders.
The assumption has been made that all nonexclusive inhibitors follow the same kinetic order, m. As already pointed out (Chou and Talalay, 1981), when (f_{v})_{1,2... }_{n} = (f_{i})_{1,2... }_{n} = 0.5 (i.e., when the combination of n nonexclusive inhibitors produces the median effect), then eq. A21 can be given by Because f_{i} + f_{v} = 1; therefore, f_{i}/f_{v} = (1/f_{v})  1. Thus, eq. A21 becomes which is identical to eq. A19 (at n = 2); i.e., (f_{v})_{1,2} = (f_{v})_{1} · (f_{v})_{2}. Proof of (f_{v})_{1,2} = (f_{v})_{1} · (f_{v})_{2} for case 2 can also be conveniently obtained by (v_{i})_{1,2}/v_{0} = (v_{i})_{1}/v_{0} \m=.\\m=.\\m=.\ (v_{i})_{2}/v_{0}, as shown by Chou and Talalay (1977).
Further examples of cases involving more than two nonexclusive inhibitors and mixtures of exclusive and nonexclusive inhibitors are analyzed in the appendix of Chou and Talalay (1981). All of these analyses provide inductive proof that the combined effects of mutually nonexclusive inhibitors can be described by the product of the fractional velocities observed in the presence of the component inhibitors; i.e.,
3. HigherOrder Multiple Mutually Nonexclusive Inhibitors.
As indicated in eqs. A21 and A22, the third term (product) will result in a concave upward curve compared with the parent compounds. In calculating the CI value, this will have an additional (product) term and thus yield a greater CI value than the mutually exclusive assumption of CI calculation. Consequently, the mutually nonexclusive assumption will project somewhat less synergism (or more antagonism) and hence was termed a “conservative” estimation, since this “intrinsic synergism” has already been taken into account in the overall synergism (Chou and Talalay, 1981). The earlier software (Chou and Chou, 1985; Chou and Hayball, 1997) automatically calculated CI values and generated the F_{a}CI plot under both mutually exclusive and nonexclusive assumptions, since exclusivity of two drugs, in most cases, can be determined by the medianeffect plot. However, with three or more drug combinations, especially at kinetic order m ≠ 1, (e.g., m ≫ 1, m ≪ 1), there can be partial exclusivity or mixed situations that make the complexity of equations beyond practical analysis. To be consistent with the classic isobologram and its equation, we decided that all drug combinations will be analyzed with mutually exclusive assumptions under the same universal standard.
This practice has been manifested in most drug combination studies in the past. The recent software CompuSyn no longer automatically provides CI analysis under mutually nonexclusive assumption (Chou and Martin, 2005). Thus, intrinsic synergy due to nonexclusivity is incorporated into the overall synergism calculation.
Glossary
Additive effect (CI = 1): The combined effect predicted by the massaction law principle in the absence of synergism or antagonism.
Antagonism (CI >1): Smaller than expected additive effect based on the massaction law.
Classic isobol: An equipotent graph with the doses of Drug_{1} and Drug_{2} on x and yaxes, respectively.
Combination index (CI): A quantitative measure based on the massaction law of the degree of drug interaction in terms of synergism and antagonism for a given endpoint of the effect measurement (Chou and Talalay, 1981).
CompuSyn: A computer software for PCs developed by Chou and Martin (2005) that can be used for doseeffect analysis for single drugs using the medianeffect equation and for multiple drug combinations using both the medianeffect equation and the combination index equation.
Dosereduction index (DRI): A measure of how many fold the dose of each drug in a synergistic combination may be reduced at a given effect level compared with the doses of each drug alone (Chou and Chou, 1988).
F_{a}CI plot: A plot of CI on the yaxis as a function of effect level (f_{a}) on the xaxis. The computersimulated F_{a}CI plot displays synergism or antagonism for the entire spectrum of effect levels (e.g., f_{a} = 0.010.99) (Chou and Talalay, 1981, 1984).
F_{a}DRI plot: A plot of DRI on the yaxis as a function of effect level (f_{a}) on the xaxis (Chou and Chou, 1988).
Isobologram (ED_{50} isobol, ED_{75} isobol, ED_{90} isobol, etc.): A graph indicating the equipotent combinations of various doses of two drugs. It can be used to illustrate additive effect, synergism, or antagonism, at different dose levels.
m value: The shape parameter for the doseeffect curve. The m value is the slope of the medianeffect plot. m = 1, m > 1, and m < 1 indicate hyperbolic, sigmoidal, and flat sigmoidal, respectively (Chou, 1976).
Medianeffect dose (D_{m}): The dose that produces 50% effect such as IC_{50}, ED_{50}, or LD_{50}. It is a potency parameter, and it is obtained from the antilog of the xintercept of the medianeffect plot (Chou, 1976).
Medianeffect equation:f_{a}/f_{u} = (D/D_{m})^{m}; a general equation for doseeffect relationship derived from the massaction law principle that takes into account both the potency (D_{m}) of a dose (D) and the shape (m) of doseeffect curve, where f_{a} and f_{u} are the fractions affected and unaffected, respectively. m = 1, m > 1, and m < 1 indicate hyperbolic, sigmoidal, and flat sigmoidal shape, respectively (Chou, 1976).
Medianeffect plot: A plot x = log (D) versus y = log (f_{a}/f_{u}), where f_{a} + f_{u} = 1 and f_{u} = 1  f_{a}. This plot linealizes all doseeffect curves that followed the massaction law principle. The slope gives the m value, and the xintercept antilog gives the D_{m} value (Chou, 1976).
MEP: The medianeffect principle of the massaction law (Chou, 1976, 1991). This includes the medianeffect equation and plot and its parameters, as well as the interconversion of dose and effect.
Mutually exclusive drugs: Two (or more) drugs with similar basic modes of action are considered mutually exclusive drugs. The mutually exclusive condition is the general assumption of the classic isobologram and its equations are accepted as the golden standard for calculating the CI and DRI values.
Mutually nonexclusive drugs: Two (or more) drugs with totally independent modes of action analogous to the binding of one ligand to the receptor will not affect the binding of other ligands at different sites. In the ideal situation, even if D_{1} and D_{2} alone give parallel lines on the medianeffect plot, D_{1} + D_{2} in combination will give a nonparallel slightly concave upward curve. Thus, D_{1} + D_{2} in combination contain an element of “intrinsic synergism,” which may contribute to the overall synergism. If a mutually nonexclusive condition is assumed for the CI calculation, it will contain the third term (i.e., the product of the first two terms) in the CI equation. Consequently, the CI value will be greater (i.e., synergism will be less), and thus it was termed “conservative synergism” (Chou and Talalay, 1984). In real life, partial exclusivity may occur that would be difficult to quantitize. This is especially so if the combination includes more than two drugs, since different pairs of drugs could have different degrees of exclusivity. To be consistent with the classic isobologram assumption and for simplicity of calculation, the present software uses only the mutually exclusive assumption and does not use the mutually nonexclusive assumption for drug combinations (i.e., intrinsic synergism, if it exists, is included in the overall synergism in drug combination analysis).
Normalized isobol: An equipotent graph with the normalized dose of Drug_{1} as [D_{1}/(D_{x})_{1}] and Drug_{2} as [D_{2}/(D_{x})_{2}] on the x and yaxes, respectively. Both x and yaxes are scaled to 1 (Chou and Talalay, 1984).
Polygonogram: A polygonal graphic representation depicting synergism (solid line in red tone), additive effect (thin line in pink tone), and antagonism (broken line in blue tone) for three (triangular), four (tetrahedral), five (pentagonal), or more drug combinations. The degree of boldness (thickness) of the line represents the degree of synergism or antagonism. The “component drugs” in pairs or triplets, etc., in the polygonogram may be considered dissectional components presented in the same graph. This method was first used by Chou et al. (1994), and the term was formally named as “polygonogram” by Chou and Chou (1998). In the current software (Chou and Martin, 2005), each drug pair is represented in the polygonogram. That is, only combinations (at a constant ratio) with two component drugs are shown on the polygonogram. Threedrug component combinations, if carried out, can be manually drawn with triangles; fourdrug component combinations can be drawn with rectangles or squares, etc. The polygonogram provides a simple visual presentation for complicated multidrug combinations. It also provides a rational for projecting the outcome of synergism or antagonism for the multidrug combination experiments that have not yet been conducted.
Potentiation: A condition in which one of the two drugs is not effective by itself, but increases the effect of the other drug. It is usually described by percent potentiation or fold potentiation, a synonym of augmentation or enhancement. No CI can be determined without m and D_{m} values due to an ineffective drug.
r value: The conformity parameter for goodness of fit to the medianeffect principle (MEP) of the massaction law. It is the linear correlation coefficient of the medianeffect plot, where r = 1 indicates a perfect conformity.
Sequential deletion analysis (SDA): An iterative sequential deletion of one dose (or concentration) of a drug at a time for repetitive CI calculations. This is followed by calculating the mean ± 95% confidence interval at each specified effect levels of the F_{a}CI plot (Chou and Martin, 2005).
Synergism (CI < 1): Greater than expected additive effect based on the massaction law.
Acknowledgments
I express my deepest gratitude to my Ph.D. thesis mentor, Professor and American Cancer Society Chair Robert E. Handschumacher of Yale University College of Medicine, and my postdoctoral mentor, Professor and John Jacob Abel Chair Paul Talalay of The Johns Hopkins University School of Medicine, for allowing me, then as a student and junior scientist, to have a freespirit for independent exploration in the research field that is only tangential to theirs. Their generosity and kindness transcended to their continued support of a series of my early theoretical works and encouraged me to be the solo author. Even long after I left Johns Hopkins for the SloanKettering Institute, Prof. Talalay kindly joined me for the work that led to the current combination index theorem. I also thank Professor and Eugene Higgins Chair Lenard J. Savage of Yale Statistics Department, the coauthor of Inequalities for Stochastic Processes: How to Gamble If You Must. During the last 2 years of his life, Prof. Savage helped me solve some combinatorial and number theory problems and served as an examiner in my Ph.D. thesis defense 36 years ago.
This article is dedicated to my mother, ShengMei Chen, for her guidance and loving care after my father died when I was six; to my wife, Dorothy, for her understanding, patience, and support for this multidecadelong project; and to my son, Joseph, for helping me to write the first computer software ever for the massaction lawbased doseeffect analysis and for the drug combinations when he was 13 (before personal computers became available), and for signing an agreement with ElsevierBiosoft, Cambridge, UK, for the software publication.
Footnotes

↵1 Abbreviations: IC_{50} (or D_{m}), concentration (or dose) required to inhibit (or to affect) a system by 50%; CI, combination index; DRI, dosereduction index; HIV, human immunodeficiency virus; MDR, multidrug resistance; MTD, maximal tolerated dose; MEP, medianeffect principle; VP16, etoposide; SDA, serial deletion analysis; MSKCC, Memorial SloanKettering Cancer Center; HSV, herpes simplex virus; MGH, Massachusetts General Hospital; MAb, monoclonal antibody; AZT, 3′azido3′desoxythymidine; CPT11, irinotecan; 5FU, 5fluorouracil; IFN, interferon; GCSF, granulocyte colony stimulating factor; CCNU, lomustine; EGFR, epidermal growth factor receptor; AraC, 1βdarabinofuranosylcytosine; r, recombinant; AZT, zidovudine; DDC, 2′,3′dideoxycytidine; DDI, 2′,3′dideoxyinosine, didanosine; HIV1 or HIV/AZT, zidovudinesensitive or resistant HIV1; RO131, a protein inhibitor from Roche Pharmaceuticals; BIRG587, N11cyclopropyl4methyl5,11dihydro6Hdipyrido[3,2b:2′,3′e][1,4]diazepine6one, nevirapine; C_{S}A, cyclosporine; Rapa, rapamycin (sirolimus); FK506, tacrolimus; CisPt, cisdiamminedichloroplatinum(II); TK, thymidine kinase; LY294002, 2(4morpholinyl)8phenylchromone; ADPR, ADPribose; dUrd, deoxyuridine; MTX, methotrexate; D4T, 2′,3′didehydro2′,3′didesoxythymidine; NEV, nevirapine; ABT538 (A84538), (2S,3S,5S)5[N[N[[NmethylN[(2isopropyl4thiazolyl)methyl]amino]carbonyl]valinyl]amino]2[N[(5thiazolyl)methoxycarbonyl]amino]3hydroxy1,6diphenylhexane (for more specific information on some of the abbreviations used in this article, please see Glossary on page 672).

The author is a principal in and holds the copyright to CompuSyn, the software used for the analyses performed in this article. His son, who contributed to the software's development, is in a position to receive royalties for its publication.

Article, publication date, and citation information can be found at http://pharmrev.aspetjournals.org.

doi:10.1124/pr.58.3.10.

↵ The online version of this article (available at http://pharmrev.aspetjournals.org) contains supplemental material.
 The American Society for Pharmacology and Experimental Therapeutics
References
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵

 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵

 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵