0031-6997/95/4702-0331$03.00/0
`PHARMACOLOGICAL REVIEWS
`Copyright © 1995 by The American Society for Pharmacology and Experimental Therapeutics
`
`Vol. 47, No. 2
`Printed in U.S.A.
`
`The Search for Synergy: A Critical Review from a
`Response Surface Perspective’
`
`WILLIAM R. GRECO, GREGORY BRAVO, AND JOHN C. PARSONS
`
`Departmeni of Biomathematics, Roswell Park Cancer Institute, Buffalo, New York
`
`ah dos tz pela pla apt lal nla eteset alan la teien a ap elute of anabelna wails gueahen baad GonanR 332
`Ds, Wettenbdebitiit Fo ie chanel are ie
`Th Review of Peviewe 252255522655 Feo Sessa eka saa ss doe e eae saneeeeie geese ikea aiees 334
`II. General overview of methods from a response surface perspective ........s50-seeeeeeeeeneeees 334
`IV. Debate over the best reference model for combined-action .......:.ccece cece cere eee etere reese 344
`V. Comparison of rival approaches for continuous response data..............0200. cece eee ee eee 348
`A. Isobologram by hand ............. 2.2002: c ec ee eee eee eee eee eee ee estes eeenennnage 349
`B. Fractional product method of Webb (1963). ....- 2.0.0.2 cece eee eee cece ner en ere eernenee 351
`C. Method of Valeriote and Lin (1975) .........0.0 000 ccc ccc eee eee eee eee eee eenees 352
`D. Method of Drewinko et al. (1976) .......0.. 60 cece eee eee eee tere eee eee e rete eran 352
`E. Interaction index calculation of Berenbaum (1977) .........-...-2.----2e0ece sence een eeee 352
`F. Method of Steel and Peckham (1979). ..... 2.2... 0-0 0s cece eee eee eee reer tener eeeens 353
`G. Median-effect method of Chou and Talalay (1984) ... 2... 2.0. cee cee cece nes 354
`H. Method of Berenbaum (1985). ........0.0.0. 0000s cece cence eee e eect nee neeeeneneenaes 358
`I. Bliss (1939) independence response surface approach ...........-2 5000 eee ee cere n cere nee 360
`J, Method of Prichard and Shipman (1990)... 2... 0... ccs cc cece eee cece eee eee eee ene e eens 360
`K. Nonparametric response surface approaches ........0. 0000s c cece e cece c eee e eee ees 362
`1. Bivariate spline fitting (Stihnel, 1990). ..... 2.2.0. ees 362
`L. Parametric response surface approaches..........2...00eseeec esse eect reece een ce tener eees 363
`1. Models of Greco et al. (1990) ... 0.2.2... 0... cece cee eee ia selepewcanis dopa ceeveea ey 364
`2. Models of Weinstein et al. (1990) ..... 2.2... eee ee eee eee eee eae eeeneees 365
`VI. Comparison of rival approaches for discrete success/failure data .................00eeeeeuaeee 367
`A. Approach of Gessner (1974) . 2.2... ccc cece ree cece eee e eens ee ebeeaeeestvens 369
`B. Parametric response surface approaches... .......6.ccc ee cece eee eee eee eee een eae ne nnee 371
`1. Model of Greco and Lawrence (1988) ........0 0c cece eee ee eee eee e eee ene 371
`2. Multivariate linear logistic model .....-- 2.0.0 0c cece cence neers e eee tere n eee nateees 371
`VII. Overall conclusions on rival approaches .....-. 2... 0.0.00 e eee eee eee eee eee teenie 373
`VIII. Experimental design... ... 2.0.00... cece cece cence ee eee n eee e eee eee ee baeeneee 373
`TX. General proposed paradigm ........... 2220... c cece eee ee een eee rent eee ease eees 376
`X. Appendix A: Derivation of a model for two mutually nonexclusive noncompetitive inhibitors for a
`second Order SYStEM .. 12. cece denne seu re scars ccareseesueesdesreenerererescesnesecusenpes 377
`Ba. MOtivatdenn oo iin sins va ae one De ele vice slag ca tslel dale ap clad ge slenult aaeletaa gad eda eaeea de hee 377
`B. Elements of the derivation of the mutually nonexclusive model for higher order systems from
`Chou'and Tatalay (1982). ooo. cs eid oes beast piaewe cede breeeepabsacaaaedasiadanes 377
`C. Assumptions of the derivation of the model for mutual nonexclusivity for two noncompetitive
`higher order inhibitors. ...55 sce) ia adic ceeescadseesauewurssa masa ean nebre Haas RaMeS vaR So 378
`DE SDSrivaons pyc e5 asd, SS oo ns Sa ee Oo Dees AS See cers Rerags Tase s Vere eevee Ras ahve ds 378
`E. Possible rationalization of the mutually nonexclusive model of Chou and Talalay (1981) .... 379
`XI. Appendix B: Problems with the use of the median effect plot and combination index calculations
`to assess drug interactions ...........6 cece ec eee eee ee eee eee eee ene eee 379
`A. Nonlinear nature of the median effect plot for mutual nonexclusivity...........0000cer rene 380
`B. Incorrect combination index calculations for the mutually nonexclusive case ..........000+- 382
`C. Nonlinear nature of the median effect plot for mutual exclusivity with interaction.......... 382
`All) RGIBVRYOEA Saou vas wndasaiincitavansh saps ouyk's tafscituaa a Salsa nh aaa e ap ten caus sana heads 382
`
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`332
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`GRECO ET AL.
`
`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 effects,
`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 (Greco and
`Lawrence, 1988). Travelers 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 sametrails, paths have
`crossed, bitter 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-
`motherapyis required to cure 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-
`sessmentofdrug interaction and assist them in choosing
`appropriate approaches.
`Wewill 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 uponthefield 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 “unobvi-
`ousness,” which will assist them in receiving a patent for
`a combination device or formulation with the United
`States Patent Office.)
`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-effect models to aid in the design ofexperiments,
`to use forfitting data and estimating parameters, and to
`help in visualizing the results with graphs. In fact, be-
`cause a major strength ofresponse surface approachesis
`that they can help to explain the similarities and differ-
`ences among other approaches,the entire review is from
`
`* Supported by grants from the National Cancer Institute,
`CA46732, CA16056 and RR10742.
`+ Abbreviations: 3-D, three-dimensional; 2-D, two-dimensional;
`Eq,, equation; vs., versus; see table 2 for mathemptical/stetiatical
`abbreviations.
`To whom correspondence should be addressed: Dr. William R.
`Greco, Department of Biomathematics, Roswell Park Cancer Insti-
`tute, Buffalo, NY 14263
`
`a response surface perspective. [Response surface meth-
`odology is composedof a group ofstatistical 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 commondata sets,
`one with continuous responses and one with discrete
`success/failure 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.
`It should be noted that the goal of this review is to
`underscore the similarities, differences, 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
`Calabrese (1991). It should also be noted that manyof
`the approaches were originally written as guidelines,
`not detailed algorithms. Therefore, our specific imple-
`mentations of several of the methods may have differ-
`ences from the approachesactually intendedbythe orig-
`inal authors.
`There is no uniform agreement on the definitions of
`agent interaction terms. Sources for extensive discus-
`sions ofrival nomenclature include the following: Beren-
`baum (1989); Calabrese (1991); Copenhaver et al.
`(1987); Finney (1952, 1971); Gessner (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
`that many of the naming schemes are unnecessarily
`complex. We will use a simple scheme that was the
`consensus of six scientists who debated concepts and
`terminology for agent interaction at the Fifth Interna-
`tional Conference on the Combined Effects of Environ-
`mental Factors in Sarriselki, 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
`Saariselkaé agreement, The foundation for this set of
`terms includes two empirical models for “no interaction”
`for the situation in which each agentis 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 be often substituted for “interaction” when
`
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`TABLE 1
`
`Consensus terminology for two-agent combined-action concepts
`
`SEARCH FOR SYNERGY
`
`333
`
`Both agents effective
`Both agents effective
`re
`‘
`oP
`aaae
`.
`individ
`;
`Eq. 6 is
`individually;
`Eq. 11 or 14
`the aeees
`is the
`ae sandal
`
`Loewe synergism
`Bliss synergism
`
`Only one agent
`ee
`ered
`effective individually
`synergism
`
`3
`Neither agent
`Me
`eae
`affective individually
`coalism
`
`Combination effect greater than
`predicted
`Combination effect equal to
`prediction from reference model
`Combination effect less than
`predicted
`
`Loewe additivity
`
`Bliss independence
`
`inertism
`
`inertism
`
`Loewe antagonism
`
`Bliss antagonism
`
`entagonism
`
`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 additivity
`(Loewe and Muischnek, 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 mannerthat
`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 adjectives emphasizes the historical origin of
`the specific models and deemphasizes the mechanistic
`connotation of the terms additivity and independence.
`Both Loewe additivity and Bliss independence are in-
`cluded as reference models, because each has somelog-
`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
`agentin a pair is effective alone, inertism is used for “no
`interaction,” synergism (without a leading adjective) for
`an increased effect caused by the second agent, and
`antagonism for the opposite case. Alternate common
`terms for the latter two cages 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 namesto the higherorderinteractions.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 ofchem-
`icals at specific concentrations. Then, six of the above-
`mentioned terms haveclear, 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 agent(s) increases, the pharmacological effect mono-
`tonically increases. This is why the lower right-handcell
`of table 1 is missing; a pharmacological effect less than
`zero is not defined. However, because in the field of
`chemotherapyit is common for increased concentrations
`of drugs to decrease the survival or growth of infectious
`agents or of tumorcells, most of the concentration-effect
`(dose-response) equations and curvesin 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, JD, values such as
`ID,, will refer to the concentration of drug resulting in
`X% of pharmacological effect (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 effect. 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-effect 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-D‘ concentra-
`tion-effect
`surface. Some agent combinations may
`demonstrate different types of interaction at different
`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 differentiate true mosaics of different interaction
`types from random statistical variation and/or artifacts
`
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`334
`
`GRECO ET AL.
`
`caused by faulty data analysis methods. Unfortunately,
`rigorous methods to identify true mosaics are not yet
`available.
`
`I. Review of Reviews
`
`Wehavedivided 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: Brunden
`et al. (1988); Calabrese (1991); Carter et al. (1983); Chou
`and Rideout (1991); National Research Council (1988);
`Péch (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); Finney (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 andarticles that include signifi-
`cant reviews of various approaches, but which also in-
`clude either new methodology development and/or anal-
`yses ofnew data include: Chou and Talalay (1983, 1984);
`Gennings et al, (1990); Greco (1989); Greco and Dembin-
`ski (1992); Hall and Duncan (1988); Kodell and Pounds
`(1985); Prichard and Shipman (1990); Sithnel (1990);
`Syracuse and Greco (1986); Tallarida (1992); and
`Machado and Robinson (1994),
`Although not exhaustive, this list includes a compre-
`hensive, redundant account of the interaction assesa-
`ment literature. This list includes critical and non-
`critical
`reviews of history, philosophy,
`semantics,
`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 ofrival
`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-
`haveret 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.
`
`Til. 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 schemeincludesall 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-
`mentof drug interactions, both mathematically rigorous
`ones and not-so-rigorous ones, can be readily explained.
`Step 1 is to choose a good concentration-effect (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 (Waud
`and Parker, 1971; Waud et al., 1978). The Sigmoid-
`Emax model (Holford and Sheiner, 1981), is equivalent
`to a nonlinear form of the median-effect model (Chou
`and Talalay, 1981, 1984). However, the equivalence of
`the median-effect 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
`table 2.
`
`naele=—|
`Ti
`
`=
`
`ge beck to step 3, 4 or 6.
`
`Fic. 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
`
`335
`
`E(Ettect)
`
`Emax + B. Eq. 4 is the exponential concentration-effect
`model, which can also be parameterized with an IC.
`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-
`clude the normal distribution for continuous data, such
`as found in growth assays in which the absorbance of a
`dye boundto cells is the measured signal; the binomial
`distribution (Larson, 1982) for proportionsoffailures 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 (McCullagh 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 left 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 ofnull hypotheses,or by less formal
`methods. In contrast, in Step 3b, a structural model is
`derived for joint action that includes interaction terms.
`Then, after the experiment is designed and conducted,
`the full combined-action modelis 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 Muiechnek, 1926); the fractional product method of
`Webb (1963); the method ofValeriote and Lin (1975); the
`method of Drewinko (1976); the method of Steel and
`Peckham (1979); the method of Gessner (1974); the
`methods of Berenbaum (1977, 1985); the median-effect
`method (Chou and Talalay, 1981, 1984); 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 (Greco et al., 1990; Greco and Lawrence, 1988;
`Greco, 1989; Greco and Tung, 1991; Syracuse and Greco,
`1986); the response surface approachesof Carter’s group
`
`D (Drug Concentration)
`
`Fic. 2. Graph of the Hill (1910) model, which is also referred to as
`the Sigmoid-Emax 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
`virus titer remaining in a culture vessel after drug ex-
`posure; D is concentration of drug; Emax is the full
`range ofresponse that can be affected by the drug; Dm or
`ICgo is the median effective dose (or concentration) of
`drug (or ID59, EDgo, LD, etc.); and m is a slope param-
`eter. When m 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 thoughtof 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
`numberof evenly spaced concentrations. Equations 2 to
`4 are additional candidate structural models for single
`agents,
`
`ICs
`
`ICs
`
`Eco ( aalo
`E= mi D\
`E==aca) +B
`~
`ae PP
`ICs
`
`D im
`
`1Dia(5)
`E=Econexp(aD)=Econ“i Tc
`
`
`
`50
`
`(2)
`[3]
`
`[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), then Econ is equivalent to Emax, as
`in Eq. 2. However, whenthere is a finite B, then Econ =
`
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`GRECO ET AL.
`
`TABLE 2
`Mathematical/statistical symbol definitions
`,
`Definition
`
`Symbol
`
`Measured effect (or response), in this review, usually a measure of survival
`Transformed response variable, continuous or discrete
`A particular value of Y
`Probability that the function in parenthesis is true
`Mean or expected value of a transformed response
`Number of successes in a binomial trial
`Numberof attempts in a binomial trial
`Concentration (or dose) of drug, drug 1, drug 2
`Inhibitor concentrations for an inhibitor, inhibitor 1, inhibitor 2
`Control effect (or response)
`Maximum effect (response), is equal to Econ for an inhibitory drug in the abeence of a
`background, B
`Background effect (response) observed at infinite concentration for an i
`Fraction of effect affected
`Fraction of effect unaffected
`Fraction enzyme velocity inhibited
`Concentration (or dose) of drug resulting in 50% inhibition of Emax, of drug 1, of
`drug 2
`Median effective dose (or concentration) of drug, of drug 1, of drug 2, of a combination
`of drugs 1 and 2 in a constant ratio (equivalent to [C,,)
`Concentration (or dose) of drug resulting in X% inhibition of Emax, of drug 1, of drug
`2, or a combination of drigs 1 and 2 in a constant ratio
`% inhibition
`Slope parameter, for drug 1, for drug 2, for a combination of drugs 1 and 2 in a
`constant ratio
`Synergism-antagonism interaction parameter
`Empirical parameters for exponential concentration-effect model
`Interaction parameters of model 29
`Interaction parameter of model 30
`Empirical parameters for probit and logistic models
`Interaction index of Berenbaum (1977)
`Combination index of Chou and Talalay (1984)
`Ratio of D, to D,
`
`hibitory drug
`
`D, [drug], D,, (drug 1), Dz, {drug 2)
`I, qT, iy
`Econ
`Emax
`
`B f
`
`a
`fu
`fi
`ICso, Igo. ICyo,15 TCso2
`
`Dm, Dm,, Dmy Dmy,
`
`IDx, Dx, ICx, ID.1, Dx, [Dx¢9, DXq, DXyp
`
`xX
`mm, My, Mg, Mig
`
`a a
`
`,6
`PC,, PC;, bpy, bp
`n
`
`Bi Ba By
`
`ci
`R
`
`336
`
`artysinetty
`
` Ss
`
`o2
`
`20
`Ww
`05
`1
`2
`Drug Concentration | pM, log scale)
`
`80
`
`Fic. 3. General scheme for the dissection of a generalized nonlin-
`ear model into random and structural components for a concentra-
`tion-effect curve for a single drug.
`
`(Carter et al., 1983, 1986, 1988; Gennings et al., 1990);
`the response surface approach of Weinstein et al. (1990);
`the generalized linear model approach of Lam et al.
`(1991); and the response surface approach of Machado
`
`and Robinson (1994). The method proposed by Siihnel
`(1990) has elements ofboth the left-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 bystat-
`isticians usually 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 “synergism.” Some approaches
`only yield a qualitative conclusion (e.g., Loewe syner-
`gism, Loewe antagonism, or Loewe additivity), such as
`the classical isobologram 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 isobologram
`approach), whereas others require a computer(e.g., uni-
`versal response surface approach). Some approaches use
`
`6 of 55
`
`Alkermes, Ex. 1045
`
`6 of 55
`
`Alkermes, Ex. 1045
`
`

`

`SEARCH FOR SYNERGY
`
`337
`
`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 different 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 same data set. This will be illustrated
`dramatically in Sections V and VI.
`Weare 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 ofEq. 5, the flagship equation for two-
`agent combined-action developed by our group, is pro-
`vided in detail in Greco et 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;
`Gaumontet al., 1992; Guimaras et al., 1994) and meth-
`odology development (Syracuse and Greco, 1986; Greco
`and Lawrence, 1988; Greco, 1989; Greco and Tung, 1991;
`Khinkis and Greco, 1993; Khinkis and Greco, 1994;
`Greco et 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 suggested by Berenbaum (1985), with the as-
`sumption of Eq. 2 as the appropriate model for each
`agent alone. The interaction parameteris a.
`
`ratherit is an empirical equation that often matches the
`shape of real data (e.g., Gaumontet al., 1992; Greco et
`al., 1990; Greco and Dembinski, 1992; Greco and Law-
`rence, 1988). However, as shown below,it is consistent
`with Eq. 6, the equation for Loewe additivity (Loewe and
`Muischnek, 1926), which is the basis ofmany interaction
`assessment approaches.
`= Ht aie
`IDy,
`IDx2
`
`[6]
`
`For an inhibitory drug, Eq. 6 refers to a particular X%
`(percent inhibition level), e.g., 58% inhibition. ID, ;,
`IDx.2 are the concentrations of drugs to result in X%
`inhibition for each respective drug alone, and D,, D, are
`concentrations ofeach drug in the mixture that yield X%
`inhibition. When the right-hand side of Eq. 6 [equal to
`the Interaction index, J, of Berenbaum (1977) or to the
`combination index, CI, for the mutually exclusive case of
`Chou and Talalay (1984)] is less than 1, then Loewe
`synergism is indicated, and when the right-handsideis
`greater than 1, Loewe antagonism is indicated. When
`Eq. 2 is an appropriate concentration-effect model for
`each drug alone, then Eq. 7, which is a rearrangement of
`Eq. 2 [similar to a rearrangement of the median-effect
`equation from Chou and Talalay (1984), relates the ID,
`value for any X% inhibition to the observed response
`level, E, and the parameters, Econ, IC,o, and m.
`
`IDy=IOnles)
`
`E
`
`Um
`
`(7]
`
`Note that the right-hand expression of Eq. 7 is the same
`as the denominators ofthe first two right-hand terms of
`Eq.5. Therefore, the first two right-hand terms ofEqs. 5
`and 6 are eq