0031-899W95J'4702-033 1303 .0010
`PI-IARHADOLDGICAL REVIEWS
`Copyrightfl 1995 by The Aineriun Society for Pharmacology and Experimental Therspeutiu
`
`Vol. 67. N0. 2
`Prinlcd is U..5'.A.
`
`The Search for Synergy: A Critical Review from a
`Response Surface Perspective"
`
`WILLIAM R. GRECO. GREGORY BRAVO, AND JOHN C. PARSONS
`
`Department offiomothsmoiics, Roswell Park Cancer Institute, Buflizlo, New York
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`. Introduction .
`. Review of reviews .
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`. General overview of methods from a response surface perspective . . .
`. Debate over the beat reference model for combined-action . . . .
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`. Comparison of rival approaches for continuous response data . . . . . .
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`A. Isobologram by hand . . .
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`Fractional product method of Webb (1963) . . .
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`Method of Valeriote and Lin (1975) .
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`Method of Drewinko et al. (1976) . . .
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`Interaction index calculation of Berenbaum (1977) . . . . .
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`Method of Steel and Peckham (1979) . .
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`L-:-FF3F‘=F‘5!5.0F’
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`Median-effect method of Chou and Talalay (1984) . . . . . . . . . . .
`Method of Berenbaum (1985) . . .
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`Bliss (1939) independence response surface approach . . . . . . . .
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`K. Nonparametric response surface approaches . .
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`1. Bivariate spline fitting (Siihnel, 1990) . .
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`L. Parametric response surface approaches . .
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`1. Models of Greco et al. (1990) . . . .
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`2. Models of Weinstein et al. (1990) .
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`. Comparison of rival approaches for discrete successlfailure data . . . .
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`A. Approach of Gessner (1974) . .
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`B. Parametric response surface approaches . . . . . . . . . . . . .
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`1. Model of Green and Lawrence (1988) . .
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`2. Multivariate linear logistic model .
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`. Overall conclusions on rival approaches . .
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`General proposed paradigm . . . .
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`Appendix A: Derivation of a model for two mutually nonexclusive noncompetitive inhibitors for a
`second order system . . . . .
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`A. Motivation . . . . . . . . . .
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`. Elements of the derivation of the mutually nonexclusive model for higher order systems from
`Chou and Talalay (1981) .
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`. Assumptions of the derivation of the model for mutual nonexclusivity for two noncompetitive
`higher order inhibitors . .
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`. Derivation . . . . .
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`. Possible rationalization of the mutually nonexclusive model of Chou and Talalay (1981) . . . . 379
`. Appendix B: Problems with the use of the median effect plot and combination index calculations
`to assess drug interactions . . .
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`A. Nonlinear nature of the median effect plot for mutual nonexclusivity . . . . . . . .
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`B. Incorrect combination index calculations for the mutually nonexclusive case . . . . . . . . .
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`C. Nonlinear nature of the median effect plot for mutual exclusivity with interaction . . . . . .
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`XII. References .
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`Alkermes, Ex. 1045
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`332
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`G-RECO ET AL.
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`I. Introduction
`
`The search for synergy has followed many tortuous
`paths during the past 100 years, and especially during
`the last 50 years. Claims of synergism for the efiects,
`both therapeutic and toxic, of combinations of chemicals
`are ubiquitous in the broad field of Biomedicine. Over
`20,000 articles in the biomedical literature from 1981 to
`1987 included “synergism” as a key word (Green and
`Lawrence, 1988). 'I‘ravelera on the search for synergy
`have included scientists from the disciplines of Pharma-
`cology, Toxicology, Statistics, Mathematics, Epidemiol-
`ogy, Entomology, Weed Science, and others. Travelers
`have independently found the same trails, paths have
`crossed. hitter fights have ensued, and alliances have
`been made. The challenge of assessing the nature and
`intensity of agent interaction is universal and is espe-
`cially critical in the chemotherapy of both infectious
`diseases and cancer. In the mature field of anticancer
`
`chemotherapy, with minor exceptions, combination che-
`motherapy is required to care all drug-sensitive cancers
`(DeVita, 1989). For the nascent field of Antiviral Che-
`motherapy, combination chemotherapy is of great re-
`search interest because of its great clinical potential
`(Schinazi, 1991). Our review should aid investigators in
`understanding the various rival approaches to the as-
`sessment of drug interaction and assist them in choosing
`appropriate approaches.
`We will make no attempt to offer advice on the use of
`a discovery of synergy. The interpretation of the impact
`of both qualitative and quantitative measures of agent
`interaction is dependent upon the field of application. At
`the very least, an agent combination that displays mod-
`erate to extreme synergy can be labeled as interesting
`and deserving of further study. (Inventors may use proof
`of synergy as support for the characteristic of “unohvi-
`ousness,” which will assist them in receiving a patent for
`a combination device or formulation with the United
`States Patent Ofiice.)
`There have been many previous reviews of this con-
`troversial subject of agent
`interaction assessment.
`These critiques are summarized in the next section.
`However, our review is unique in several ways. First,
`our bias is toward the use of response surface concen-
`tration-efiect models to aid in the design of experiments,
`to use for fitting data and estimating parameters, and to
`help in
`the results with graphs. In fact, be-
`cause a major strength of response surface approaches is
`that they can help to explain the similarities and difi'er-
`ences among other approaches, the entire review is from
`
`‘ Supported by grants from the National Cancer Institute,
`CA46732, CA16-056 and RR107-I2.
`1'Abbreviations: 3-D, three-dimensional; 2-D, two-dimensional;
`Eq., equation; vs., versus; see table 2 for mathematicslfstatistical
`abbreviations.
`'
`
`To whom correspondence should be addressed: Dr. William R.
`Green, Department of Biomathematics, Roswell Park Cancer Insti-
`tute, Bufialo, NY 14263
`
`:1 response surface perspective. [Response surface meth-
`odology is composed of a group of statistical techniques,
`including techniques for experimental design, statistical
`analyses, empirical model building, and model use (Box
`and Draper, 1987). A response surface is a mathematical
`equation, or the graph of the equation, that relates a
`dependent variable, such as drug effect, to inputs such
`as drug concentrations] Second, two common data sets,
`one with continuous responses and one with discrete
`successffsilure responses, are used to compare 13 spe-
`cific rival approaches for continuous data, and three
`rival approaches for binary success/failure data, respec-
`tively. Third, many detailed criticisms of many ap-
`proaches are included in our review; these criticisms
`have not appeared elsewhere.
`Itshouldbenotedthatthegoalofthisreviewisto
`underscore the similarities, diflhrences, strengths. and
`weaknesses of many approaches, but not to provide a
`complete recipe for the application of each approach.
`Readers who need the minute details of the various
`
`approaches should refer to the original articles. A good
`compendium of recipes for many of the approaches in-
`cluded in this review is the fourth chapter of a book by
`Cslabrese (1991). It should also be noted that many of
`the approaches were originally written as guidelines,
`not detailed algorithms. Therefore, our specific imple-
`mentations of several of the methods may have difi'er-
`ences from the approaches actually intended by the orig-
`inal authors.
`
`There is no uniform sgreement on the definitions of
`agent interaction terms. Sources for extensive discus-
`sions of rival nomenclature include the following: Baren-
`baum (1989); Calabrese (1991); Copenhaver et al.
`(1987); Finney (1952, 1971); Geasner (1988); Hewlett
`and Plackett (1979); Loewe (1953); Kodell and Pounds
`(1985; 1991); Valeriote and Lin (1975); Unkelbach and
`Wolf (1984); and Wampler et al. (1992). It is our view
`thatmanyofthenamingschernesareunnecessarily
`complex. We will use a simple scheme that was the
`consensus of sin scientists who debated concepts and
`terminology for agent interaction at the Fifth Interna-
`tional Conferencc on the Combined Effects of Environ-
`
`mental Factors in San-iselka, Finnish Lapland, Septem-
`ber 6 to 10, 1992 (Greco et al., 1992). The six scientists,
`from the fields of Pharmacology, Toxicology and Biome-
`try, comprised a good representative sample of advo-
`cates of diametrically opposing views on many issues.
`Table 1 lists the consensus terminology for the joint
`action of two agents, the major part of the so-called
`Saarisellra agreement. The foundation for this set of
`terms includes two empirical models for “no interaction‘
`for the situation in which each agent is effective alone.
`(Even though the term “interaction” has a mechanistic
`connotation when applied to agent combinations. it will
`be used throughout this article in a purely empirical
`sense. Also, the less-mechanistic term, “combined-
`action” will he often substituted for "interaction" when
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`SEARCH FOR SYNERGY
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`TABLE 1
`Consensus terminology for two-agent combined-action concepts
`
`Both agents eflictive
`individually; Eq. 5 is
`the reference model
`
`Both agents sflective
`individually; Eq. 11 or 14
`is the reference model
`
`Only one agent
`afiective
`
`Neither agent
`efiective individually
`
`Loewe synergism
`
`Bliss synergism
`
`synergism
`
`Loewe sdditivity
`
`Bliss independence
`
`inertism
`
`coalism
`
`inertism
`
`Loewe antagonism
`
`Bliss antagonism
`
`antagonism
`
`Combination efiect. greater than
`predicted
`Combination efiect equal to
`prediction from reference model
`Combination effect less than
`predicted
`
`feasible.) The mathematical details of these two models
`are described in Section III, and the debate over which of
`these is the best null reference model is the subject of
`Section IV. The first model is that of Loewe sdditivity
`(Loewe and Muiscbnek, 1926), which is based on the
`idea that, by definition, an agent cannot interact with
`itself. In other words, in the sham experiment in which
`an agent is combined with itself, the result will be Loewe
`additivity. The second model
`is Bliss independence
`(Bliss, 1939), which is based on the idea of probabilistic
`independence; i.e., two agents act in such a manner that
`neither one interferes with the other, but each contrib-
`utes to a common result. The cases in which the ob-
`
`served effects are more or less than predicted by Loewe
`additivity or Bliss independence are Loewe synergism,
`Loewe antagonism, Bliss synergism, and Bliss antago-
`nism. respectively. The use of the names Loewe and
`Bliss as adiectives emphasizes the historical origin of
`the specific models and deemphasizes the mechanistic
`connotation of the terms additivity and independence.
`Both Loewe aclditivity and Bliss independence are in-
`cluded as reference models, because each has some log-
`ical basis, and especially because each has its own cote-
`rie of staunch advocates who have successfully defended
`their preferred model against repeated vicious attacks
`(see Section IV). As shown in table 1, when only one
`agent in a pair is effective alone, inertism is used for “no
`interaction,’ synergism (without a leading adjective) for
`an increased elfect caused by the second agent, and
`antagonism for the opposite case. Alternate common
`terms for the latter two cases are potentiation and inhi-
`bition. When neither drug is effective alone, an ineffec-
`tive combination is a case of inertism, whereas an effec-
`tive combination is termed coalism.
`
`For the cases in which more than two agents are
`present in a combination, it may not always be fruitful to
`assign special names to the higher order interactions. It
`may be better to just quantitatively describe the results
`of a three-agent or more complex interaction than to pin
`a label on the combined-action. However, in some fields,
`such as Environmental Toxicology, it may be useful to
`assign a descriptive name to a complex mixture of chem-
`icals at specific concentrations. Then, six of the above-
`mentioned terms have clear, useful extensions to higher
`order interactions: Loewe additivity, Loewe synergism,
`
`Loewe antagonism, Bliss independence, Bliss syner-
`gism, and Bliss antagonism. Note also that all ten terms
`are defined so that as the concentration or intensity of
`the agentls) increases, the pharmacological efi'ect mono-
`tonically increases. This is wby the lower right-hand cell
`of table 1 is missing; a pharmacological efiect less than
`zero is not defined. However, because in the field of
`
`chemotherapy it is common for increased concentrations
`of drugs to decrease the survival or growth of infectious
`agents or of tumor cells, most of the concentration-efi'ect
`(dose-response) equations and curves in this review will
`assume a monotonically decreasing observed effect (re-
`sponse), such as virus titer. The dependent response
`variable will be labeled as effect, % effect, % survival, or
`% control in most graphs and will decrease with increas-
`ing drug concentration. In contrast, ID,‘ values such as
`ID25 will refer to the concentration of drug resulting in
`X92: of phsrmamlogical efi'ect (e.g., 25% inhibition, leav-
`ing 75% of control survival). The above definitions and
`conventions will become clearer in later sections with
`
`the introduction of defining mathematical equations.
`The emphasis of this review will be on approaches to
`assess combinations of agents that yield an unexpect-
`edly enhanced pharmacological eifect. Loewe additivity
`and Bliss independence will be used as references to give
`meaning to claims of Loewe synergism and Bliss syner-
`gism,
`respectively. Loewe antagonism will be only
`briefly discussed, as will synergism, antagonism, and
`coalism. Most concentration-elfect models and curves in
`
`this review will be monotonic. Therapeutic synergy in in
`vivo and in clinical systems, which involves a mixture of
`efficacy and toxicity, and which often involves nonmono-
`tonic concentration-effect curves for each agent individ-
`ually and for the combination, will not be discussed.
`The preceding discussion referred to global properties
`of agent combinations; i.e., it was implied that a partic-
`ular type of named interaction, such as Loewe syner-
`gism, appropriately described the entire 3-Dl concentra-
`tion-eifect
`surface. Some agent combinations may
`demonstrate different types of interaction at dilferent
`local regions of the concentration-effect surface. When
`this occurs, the interaction terms in table 1 can be used
`to describe well defined regions. However, it is impor-
`tant to dilferentiate true mosaics of different interaction
`
`types from random statistical variation andfor artifacts
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`334
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`GRECO ET AL.
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`caused by faulty data analysis methods. Unfortunately,
`rigorous methods to identify true mosaics are not yet
`available.
`
`II. Review of Reviews
`
`We have divided reviews on the subject of synergy into
`three classes: (a) whole books, some of which include
`new methodology, and some of which do not; (b) book
`chapters and journal articles entirely dedicated to re-
`view; and (c) book chapters and articles with noteworthy
`introductions and discussions of combined-action assess-
`
`ment, but which also include new specific methodology
`development or data analyses. Books include: Brtmden
`et al. (1988); Calabrese (1991); Carter et al. (1983); Chou
`and Rideout (1991); National Research Council (1988);
`Pooh (1993); and Vollmar and Unkelbach (1985). Book
`chapters and articles dedicated to a review of the field
`include: Berenbaum (1977, 1981, 1988, 1989); Copen-
`haver et al. (1987); Fiuney (1952, 1971); Gessner (1988);
`Hewlett and Plackett (1979); Jackson (1991); Kodell and
`Pounds (1991); Lam et al. (1991); Loewe (1953, 1957);
`Rideout and Chou (1991); and Unkelbach and Wolf
`(1984). Book chapters and articles that include signifi-
`cant reviews of various approaches, but which also in-
`clude either new methodology development andfor anal-
`yses of new data include: Chou and Talalay (1983, 1984);
`Gennings et al. (1990); Green (1989); Greco and Dembin-
`ski (1992); Hall and Duncan (1988); Kodell and Pounds
`(1985); Prichard and Shipman (1990); Siihnel (1990);
`Syracuse and Green (1986); Tallarida (1992); and
`Machado and Robinson (1994).
`
`Although not exhaustive, this list includes a compre-
`hensive, redundant account of the interaction assess-
`ment literature. This list includes critical and non-
`
`semantics,
`reviews of history, philosophy,
`critical
`approaches advocated by statisticians, and approaches
`advocated by pharmacologists. Most of the reviews are
`biased toward the respective authors’ point of view, and
`many of the reviews harshly criticize the work of rival
`groups. Our review is no exception. A subset of these
`reviews, which along with our own, will provide a com-
`prehensive, but not overly redundant view of the field
`include: chapters 1 to 4 of Calabrese (1991), which pro-
`vide a relatively noncritical recipe-like description of
`concepts, terminology, and assessment approaches, in-
`cluding many disagreements with our review; chapters 1
`to 2 of Chou and Rideout (1991), which also provide a
`contrasting view to our review on many issues; Copen-
`haver et al. (1987), which accents the approaches devel-
`oped by statisticians: Berenbaum (1981, 1988, 1989),
`which critically review the approaches developed by
`pharmacologists; Gessner (1988), which examines ap-
`proaches developed both by statisticians and pharmacol-
`ogists; and Kodell and Pounds (1991), which may be the
`best source for a rigorous comparison of rival concepts
`and nomenclature.
`
`III. General Overview of Methods from a
`
`Response Surface Perspective
`
`Figure 1 is a schematic diagram of a general approach
`to the assessment of the nature and intensity of drug
`interactions. This scheme includes all of the approaches
`examined in later sections. This is because, in essence,
`figure 1 describes the scientific method. A formal statis-
`tical response surface way of thinking underlies all of
`this section. With such an orientation, the similarities
`and differences among rival approaches for the assess-
`ment of drug interactions, both mathematically rigorous
`ones and not-so-rigorous ones, can be readily explained.
`Step 1 is to choose a good concentration-efi‘ect (dose-
`response) structural model for each agent when applied
`individually. A common choices is the Hill model (Hill,
`1910), which is also known as the logistic model (Wand
`and Parker, 1971; Waud et al., 1978). The Sigmoid-
`Emax model (Holford and Sheiner, 1981), is equivalent
`to a nonlinear form of the median-eifect model (Chou
`and Talalay, 1981, 1984). However, the equivalence of
`the median-efiect and Hill models is disputed by Chou
`(1991). The Hill model is shown in figure 2 and as Eq. 1
`for an inhibitory drug. Symbol definitions are listed in
`tal:Ile2.
`
`E
`
`E=
`
`(“Yfill}-‘Ej
`
`IC
`
`5°
`
`hflfidlnn.
`III:-aulssplilllhn
`iillflh
`wmphxfillfl.
`fijhfltifll.
`
`FIG. 1. Schematic diagram of a general approach to the assess-
`ment of the nature and intensity of agent interactions, which in-
`cludes all specific approaches.
`
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`SEARCH FOR SYNERGY
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`335
`
`Emox + B. Eq. 4 is the exponential concentration-efl'ect
`model, which can also be parameterized with an ICE.
`Because real experiments rarely generate data that
`fall on the ideal curve, Step 2 in figure 1 is to choose an
`appropriate data variation model. Model candidates in-
`elude the normal distribution for continuous data, such
`as found in growth assays in which the absorbance of a
`dye bound to cells is the measured signal; the binomial
`distribution (Larson, 1982) for proportions of failures or
`successes, such as in acute toxicology experiments; and
`the Poisson distribution for low numbers of counts, such
`as in clonogenic assays. A composite model is formed
`from one structural model plus one data variation model
`and eventually used for fitting to real experimental
`data. This concept, called generalized nonlinear model-
`ing (McCu1lagh and Nelder, 1989) is illustrated in figure
`3, with the Hill model as the structural model, and the
`normal, binomial, and Poisson distributions (respective-
`ly from lefi to right) as the random models. (Note that
`only one random component is usually assumed for a
`particular data set. Graphs of three random components
`are pictured in figure 3 to illustrate the universal nature
`of the approach. The lower equation in the figure is a
`variant of the Hill model, and the upper one is for the
`binomial distribution. These equations will be described
`in detail in Section VI.)
`
`In Step 3, most approaches can be categorized into one
`of two main strategies. In Step 3a, a structural model is
`derived for joint action of two or more agents with the
`assumption of “no interaction” {Loewe additivity, Bliss
`independence, or another null reference model). Then,
`after the experiment is designed and conducted, data
`from the combination of agents is compared with predic-
`tions of joint action from a null reference combined-
`action model. This comparison can be made with formal
`statistical rejections of null hypotheses, or by less formal
`methods. In contrast, in Step 3b, a structural model is
`derived for joint action that includes interaction terms.
`Then, alter the experiment is designed and conducted,
`the full combined-action model is fit to all of the data at
`
`once, and interaction parameters are estimated. Both
`the left-hand and right-hand strategies end in a set of
`guidelines for making conclusions.
`Examples of approaches that use the left-hand strat-
`egy include: the classical isobologram approach (Loewe
`and Muischnek, 1926); the fractional product method of
`Webb (1963); the method ofValeriote and Lin (1975); the
`method of Drewinlto (1976); the method of Steel and
`Peckhani (1979); the method of Gessner (1974); the
`methods of Berenbaum (1977, 1935); the median-efiect
`method (Chou and Tslslay, 1981, 1934); the method of
`Prichard and Shipman (1990); and the method of Laska
`et al. (1994). Examples of approaches that use the right-
`hand strategy include the universal response surface
`approach (Green et al., 1990; Grace and Lawrence, 1988;
`Greco, 1989; Greco and Tu.ng, 1991; Syracuse and Greco,
`1986); the response surface approaches of Carter’s group
`
`D {Drug Concentration!
`
`FIG. 2. Graph ofthe Hill (1910) model, which is also referred to as
`the Bigmoid-Ema: model (e.g.. Holford and Sheiner, 1981], and
`which is also a nonlinear form of the median-effect equation {Chou
`and Talalay, 1984).
`
`In Eq. 1, E is the measured effect (response), such as the
`virustiterremaini.nginaculturevesselafterd.rugex-
`posure; D is concentration of drug; Ema.-r is the full
`range ofresponse that can be aifected by the drug; Dm or
`ICE, is the median effective dose (or concentration) of
`drug (or H350, ED“, L050, etc.); and m is a slope param-
`eter. When In has a negative sign, the curve falls with
`increasing drug concentration; when m is positive, the
`curve rises with increasing drug concentration. The con-
`centration-effect curve in figure 2 can be thought of as
`an ideal curve formed by data with no discernible vari-
`ation, or as the true curve known only to God or to
`Mother Nature, or as the average curve formed by an
`infinite number of data points at each of an infinite
`number of evenly spaced concentrations. Equations 2 to
`4 are additional candidate structural models for single
`agents.
`
`D In
`E°°" 10,,
`
`1., 1'"
`IO“,
`
`E‘-'
`
`D on
`
`(ECO?! —
`1+ D M
`10,,
`
`+B
`
`m..(§)
`E = Econ exp(a.D) = Econ exp 10so
`
`1
`
`[4]
`
`In Eqs. 2 and 3, the parameter Econ is the control effect
`(or response when no inhibitory drug is applied). When
`there is no B (background response observed at infinite
`drug concentration), than Econ is equivalent to Enter, as
`in Eq. 2. However, when there is a finite B, then Econ =
`
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`
`GREGG ET AL.
`
`TABLE 2
`Mathematical {statistical symbol definitions
`Definition
`
`Measuredefi'ect(orresponse}.inthisreview.nsuallyamessumofsurfival
`Transformed response variable. continuous or discrete
`A particular value of Y
`Probability that the function in parenthesis is true
`Mean orezpectedvnlueofatransfonnedresponse
`Number ofsuccessesinabinomialtriel
`Number of attempts in a binomial trial
`Concentration (or does) of drug, drug 1, drug 2
`Inhibitor concentrations for an inhibitor, inhibitor 1, inhibitor 2
`Control efi‘e-ct (or response)
`MaximI.n:I1eiTect(response).isequaltoficonforaninhihitorydrugintheabaanceofa
`background. B
`Background efiect (response) observed at infinite concentration for an '
`Fraction of effect afiected
`Fraction of efiect unaflected
`F‘ract:l'on enzyme velocity inhibited
`Concentration (or dose) ofdrug resulting in 50% inhibition ofE'ma.r, ofdrug 1, of
`drug 2
`Median eflective dose (or concentration) of drug. of drug 1. of drug 2, of a combination
`ofdnzgs1and2inaconstsntratio(equivalentto!C,,,)
`Concentration {or dose) ofdrugrssultinginxiv inhibition ofEmax,ofdrug1. ofdrug
`2,oracom.hix_1ationofdrugs1and2inaconstantratio
`1: inhibition
`SIopeps:raJ:netar,fordrug1,i'ordrug2.foracombinationofdrugs1and2ina
`constantrafio
`Syneruism-antagonism interaction parameter
`Empirical parameters for exponential concentration-afiect model
`Interaction parameters of model 29
`Interaction parameter of model 30
`Enrpiri'‘cal parameters for probit and logistic’' models
`Interaction index of Berenbauln (197?)
`Combination index of Chou and Tslalsy (1934)
`Ratio of D, to D,
`
`'hitory drug
`
`-moo. Icon ICI0.1o 10»:
`
`DUI, Dru‘. Dnbg ml:
`
`mm Dav -mm {Dian D1,. mxa. 33:. D112
`
`X
`mo ml: mm mm
`
`cc
`a, 5
`P01: P01» bpll bl’!
`‘fl
`
`[Hon B]: 32- 31:
`CI
`R
`
`and Robinson (1994). The method proposed by Siihnel
`(1990) has elements ofboth the lefi:-hand and right-hand
`strategies.
`Although most, and possibly all, approaches for as-
`sessing agent combinations may fall under the scheme
`presented in figure 1, the different approaches differ
`from each other in many respects. The approaches de-
`veloped by pharmacologists usually stress structural
`models, e.g., the median-effect approach (Chou and Ta-
`lalay, 1984). whereas the approaches developed by stat-
`ususlly stress data variation models, e.g., the
`approaches of Finney based on probit analysis (Finney,
`1952). There are differences in the definitions of key
`terms, especially that of “synergisrn.” Some approaches
`only yield a qualitative conclusion (e.g., Loewe syner-
`gism, Loewe antagonism, or Loewe additivity), such as
`the classical isohologram approach. whereas others also
`provide a quantitative measure of the intensity of the
`interaction, such as the universal response surface ap-
`proach. There are differences in the degree ofmathemat-
`ical and statistical rigor, i.e., some approaches are per-
`formed entirely by hand (e.g., the classical isohologram
`approach}, whereas others require a computer (e.g., uni-
`versal response surface approach). Some approaches use
`
`or
`
`5
`2
`1
`on
`the concentration iplfllfl scale)
`
`so
`
`Fla. 3. General scheme for the dissection of a generalized nonlin-
`ear model into random and s'trnctu.rsl components for a concentra-
`tion-eflect curve for a single drug.
`
`(Carter et ai.. 1983, 1986, 1988; Gennings st 111., 1990);
`the response surface approach of Weinstein et. a1. (1990);
`the generalized linear model approach of Lam at al.
`(1991): and the response surface approach of Machado
`
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`SEARCH FOR. SYNERGY
`
`33?
`
`parametric models (e.g., Greco et al., 1990), whereas
`others emphasize nonparametric models (e.g., Siihnel,
`1990; Kelly and Rice, 1990). The suggested designs for
`experiments differ widely among the dilferent ap-
`proaches. It is therefore not surprising that it is possible
`to generate widely differing conclusions on the nature of
`a specific agent interaction when applying different
`methods to the some data set. This will be illustrated
`
`dramatically in Sections V and VI.
`We are highly biased in our view that the right-hand
`strategy in figure 1 for assessing agent interactions is
`superior to the left-hand strategy when used for the
`cases in which an appropriate response surface model
`can be found to adequately model the biological system
`of interest. However, for preliminary data analyses for
`all systems, for the final data analyses of complex sys-
`tems, and for cases in which the data is meager, the
`left-hand approaches are often very useful.
`The derivation oi'Eq. 5, the flagship equation for two-
`agent combined-action developed by our group, is pro-
`vided in detail in Greco at al. (1990). Although we do not
`put forward Eq. 5 as the model of two-agent combined-
`action, it is a model of two-agent combined-action that
`has proved to be very useful for both practical applica-
`tions (Greco et al., 1990; Greco and Dembinski, 1992;
`Gaumont et al., 1992; Guimaras at al., 1994) and meth-
`odology development (Syracuse and Greco, 1986; Greco
`and Lawrence, 1988; Greco, 1989; Greco and Tang, 1991:
`Khinkis and Greco, 1993; Khinkis and Greco, 1994;
`Greco at al., 1994). Eq. 5 will be used throughout this
`review to illustrate concepts of combined-action and to
`assist in the comparison of rival data analysis ap-
`proaches. Eq. 5 was derived with an adaptation of an
`approach sugge