Understanding synergy€¦ · Synergy analyses in energy homeostasis research provide an...

Understanding synergy

Nori GearyZielackerstrasse 10, 8603 Schwerzenbach, Switzerland

Submitted 15 June 2012; accepted in final form 3 December 2012

Geary N. Understanding synergy. Am J Physiol Endocrinol Metab 304: E237–E253, 2013. First published December 4, 2012; doi:10.1152/ajpendo.00308.2012.—Analysis of the interactive effects of combinations of hormones or other manipu-lations with qualitatively similar individual effects is an important topic in basicand clinical endocrinology as well as other branches of basic and clinical researchrelated to integrative physiology. Functional, as opposed to mechanistic, analysesof interactions rely on the concept of synergy, which can be defined qualitativelyas a cooperative action or quantitatively as a supra-additive effect according tosome metric for the addition of different dose-effect curves. Unfortunately, dose-effect curve addition is far from straightforward; rather, it requires the developmentof an axiomatic mathematical theory. I review the mathematical soundness, facevalidity, and utility of the most frequently used approaches to supra-additivesynergy. These criteria highlight serious problems in the two most common synergyapproaches, response additivity and Loewe additivity, which is the basis of theisobole and related response surface approaches. I conclude that there is noadequate, generally applicable, supra-additive synergy metric appropriate for en-docrinology or any other field of basic and clinical integrative physiology. Irecommend that these metrics be abandoned in favor of the simpler definition ofsynergy as a cooperative, i.e., nonantagonistic, effect. This simple definition avoidsmathematical difficulties, is easily applicable, meets regulatory requirements forcombination therapy development, and suffices to advance phenomenological basicresearch to mechanistic studies of interactions and clinical combination therapyresearch.

cooperative effect synergy; energy homeostasis; food intake; Loewe additivity;response surface methodology

“One and one is what I’m telling you . . .” — Debbie Harry (71)

INTRODUCTION

What is Synergy?

INTERACTION AMONG DIFFERENT ENDOCRINE CONTROLS and amongendocrine and nervous controls is a fundamental characteristic ofphysiological regulation. Interactions are evident at all levels oforganization, from genetic to organismic (34, 67, 78, 85, 104,160). For example, the secretion of most hormones is under bothendocrine and neural control, and many endocrine target cells aswell as neurons in a variety of brain sites express receptors tomultiple hormones (34, 67, 85, 104). At the organismic level, thecoordinated actions of nerves and hormones acting in severaltissues are vital to many regulatory functions, including bloodglucose homeostasis, fluid and mineral balance, and energy ho-meostasis, which provides many of the examples in this review (7,54, 68, 83, 85, 110, 151). That many endocrine and metabolicdiseases are treated with combination therapies also reflects theimportance of interactions in physiological regulation (10, 29, 85).

Administration of combinations of drugs often produces unex-pectedly large responses. This is known as synergy. It is oftendescribed as “1 � 1 � 2,” or supra-additive synergy. This is an aptdescription because, as shown below, addition causes the most

complications and misunderstandings in synergy analyses. (Notethat I couch the discussion in terms of “drugs” for simpicity; whatis meant is any quantifiable manipulation, including doses ofhormones, neurotransmitters, neuromodulators, pharmaceuticalagents, etc., as well as amounts of activity, hours of food depri-vation, etc.).

What is the Problem?

Despite the apparent simplicity of the concept of supra-additivesynergy, translating it into a specific quantitative methodologywith formal validity is a thorny problem for which many solutionshave been offered (for a sense of the variety of approaches, seeRefs. 12, 64, and 121). Furthermore, few investigators appreciatethe prerequisites and limitations of supra-additive synergy meth-odologies. These problems are exacerbated by the different evo-lution of the concept in different fields, by the relative paucity oftheoretical accounts of synergy that are not forbiddingly quanti-tative, by the failure of most theorists to clearly distinguish theirapproaches from those of others, by some widely propagatederrors in the existing accounts, and by the momentum of themisdirected literature.

Synergy analyses in energy homeostasis research provide anunfortunate example of inappropriate or incomplete synergyanalyses. Review of the energy homeostasis literature identi-fied 44 studies, including four of my own, that used theresponse additivity approach to supra-additive synergy (2, 8,

Address for correspondence: N. Geary, Zielackerstrasse 10, 8603 Schwerzen-bach, Switzerland (e-mail: [email protected]).

Am J Physiol Endocrinol Metab 304: E237–E253, 2013.First published December 4, 2012; doi:10.1152/ajpendo.00308.2012. Review

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15, 17, 23, 36, 40, 43, 44, 46–48, 53, 59, 61, 70, 75, 80, 86, 87,103, 109, 111, 112, 115, 116, 127–129, 132, 135, 138, 144,145, 147, 149, 158, 161–163, 166–168, 171). None of these 44studies attempted to establish the validity of this approach, and,as explained in RESPONSE ADDITIVITY SYNERGY, it is almost nevervalid. A simple thought experiment shows why. If a drug hasa linear log dose-effect curve, as many do through much oftheir dynamic ranges, then adding a dose of the drug to itselfwill not double the effect, as predicted by response additivity,but increase it only by a factor of 0.3 (i.e., by a factor of log 2).Doubling the log dose is required to double the response. Thissuggests that many response additivity studies that reportadditivity may in fact have shown synergy.

Eleven energy homeostasis studies that used a linear Loeweadditivity approach were identified (9, 13, 50, 84, 91, 131, 134,136, 137, 161, 169). Again, none of these studies attempted toestablish the validity of this approach, and, as explained inLoewe Additivity Synergy, it is rarely valid. Other fields havemore successfully shunned response additivity but still sufferfrom misunderstandings of Loewe addivity. For example, al-though nowhere is the concept of synergy more important andmore discussed than in anesthesiology (45, 72, 77, 108, 142,143), one can find recent examples of uncritical application oflinear Loewe additivity there as well (e.g., Refs. 38, 73, and79).

The gravity of incorrect applications of supra-additive syn-ergy metrics can hardly be overstated. For example, as shownin Interdeterminate Loewe Additivity Solutions, uncritical useof linear Loewe additivity can completely reverse the interpre-tation of combination data, turning what should be antagonisminto synergy.

Why is Understanding Synergy Important?

Improving research practice by understanding synergy isimportant for at least three reasons. First, if synergy were betterunderstood, researchers would make fewer innocent errors insearching for supra-additive synergy. Second, there would befewer misguided attempts to understand the physiologicalbases of what are in fact artifactual synergies. Third, therewould be fewer misinformed efforts to translate apparentlypromising but erroneous basic research synergies into clinicalresearch. These improvements in research practice would leadto substantial savings in human, animal, and financial re-sources.

What is the Solution?

My review of supra-additive synergy in RESPONSE ADDITIVITY

SYNERGY, LOEWE ADDITIVITY SYNERGY, and OTHER ACTIVITIES leadsme to conclude that there is no metric to detect supra-additivesynergy, i.e., 1 � 1 � 2, with adequate formal validity, facevalidity, and general applicability. Therefore, I recommendadoption of a simpler definition of synergy as a cooperativeeffect. As I describe in COOPERATIVE EFFECT SYNERGY, suchsynergy is defined simply as a response to a drug combinationthat is greater than the responses to the drugs individually.Addition is not involved. Figure 1 provides some examples ofcooperative effect synergy and its utility in furthering basic andclinical research.

SUPRA-ADDITIVE SYNERGY IS A FORMAL QUANTITATIVETHEORY

Mathematical Nature of Synergy

As is often emphasized (19, 49, 65, 142), quantitative synergyanalyses are formal mathematical exercises, not wet physiology.There are two main reasons. First, there is not sufficient mecha-nistic knowledge to enable quantitative predictions of the effectsof drug combinations. If, for example, two drugs each boundreversibly to a single receptor with known kinetics and if thedrugs’ individual effects depended only on the number of recep-tors bound, then their individual and interactive effects could becomputed using the laws of mass action (12, 30, 64, 139, 143,155). Similarly, interactive effects could be computed if they wereshown to result directly from the frequency of action potentials insome accessible population of neurons. However, such quantita-tive mechanistic knowledge is not yet available in endocrinologyor in other branches of integrative physiology. The second reasonthat synergy is intrinsically mathematical is that physiologicalmeasurements cannot be used to test synergy metrics. The theo-rems derived in Euclidian geometry can be compared with thecharacteristics of existing objects to determine whether the theorycorresponds to our direct experience of physical reality. In con-trast, supra-additive synergy metrics lead to mathematical predic-tions that have no directly observable physiological counterparts.This has the implication that, unlike more mechanistic models, itis difficult or impossible to apply the criteria of construct andpredictive validity to synergy. How then to judge synergy met-rics? I suggest the criteria of 1) face validity, 2) mathematicalsoundness, and 3) utility with respect to furthering basic andclinical research.

Although my approach to synergy is quantitative, and manyimportant points can be made only mathematically, the math-ematical content of the article is minimal. I present mathemat-ical derivations in the appendices in this article, and I illustratemany points graphically.

Mathematical Notation

The mathematical notation is as follows: a and b denoteparticular amounts or doses of “drugs” A and B, which mightbe hormones or any other quantifiable manipulations; EA andEB are the dose effect functions of drugs A and B, respectively,although the subscripts are omitted in unambiguous situations,situations; E(a) or EA(a) and E(b) or EB(b) are the effects ofdose a of A and b of B when applied individually, and E(a �b) is the effect of a and b applied simultaneously. EADD(a � b)is the theoretical additive effect of the a � b combination. Forclarity, I occasionally denote the effect of a or b alone as E(a �0) or E(0 � b). I assume that the dose-effect curves of both Aand B increase (or decrease) monotonically to a maximum.Other situations bring special problems (4, 5, 26, 27, 157).

RESPONSE ADDITIVITY SYNERGY

According to the response or effect additivity synergy met-ric, the additive or zero-interactive effect of a drug combina-tion is the sum of the individual effects,

EADD�a � b� � EA�a� � EB�b� , (1)

and synergy occurs when the observed E(a � b) is greater thanthis sum (12, 76, 82, 170). This metric is ad hoc based neither

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Fig. 1. Examples of the utility of synergy studiesin energy homeostasis. Three examples demon-strate translations of initial basic research findingssuggesting synergy among inhibitory controls ofeating (A, C, and E) to mechanistic (B and D) andclinical (F) studies. Note that in each case, asimple definition of synergy as cooperative actionwould suffice as well as supra-additive definitionsto motivate followup. A and B: the synergisticeffect of gastric loads (GAS) and cholecystokininagonism (CCK analog) on food intake (141)prompted a neurophysiological study that demon-strated that this interaction occurs at the level ofabdominal vagal afferents (data are extracellularrecordings of action potentials) (140). i.a., Intra-arterial; VEH, control vehicle injections. *Signifi-cantly different from control. C and D: the syner-gistic effect of peripheral CCK and central leptin(LEP) injections on food intake (41) prompted amolecular genetic investigation of the site of theleptin receptors mediating this effect (113). fak/fak,Koletsky rats, which lack functional leptin recep-tors; Ad-LEPR-B, adenovirus expressing these re-ceptors; Ad-lacZ, control adenovirus expressing areporter gene. Ad-LEPR-B and Ad-lacZ were de-livered to the arcuate nucleus of the hypothalamus6 days prior to the effect of intraperitoneal in-jection of CCK on food intake being tested. InC: *significantly different from VEH/VEH and§significantly different from CCK/VEH. InD: *significantly different from VEH injection inwild-type rats (data not shown) and †significantlydifferent from fak/fak Ad-lacZ. E and F: the syn-ergistic effect of chronic amylin and leptin treat-ment on body weight in rats susceptible to dietaryobesity (DIO Prone) (133) prompted a clinicalstudy that demonstrated a similar phenomenon inobese humans (pramlintide and metreleptin areamylin and leptin agonists, respectively) (133). InE: #significantly different from amylin, P � 0.05.in F: significantly different from metreleptin andpramlintide; P � 0.01; ##P � 0.01; ###P �0.001. All parts of the figure reprinted with per-mission.

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on physiology nor on formal axioms. Nevertheless, the no-tion’s simplicity has intuitive appeal and is used frequently. Ilisted 44 examples above from the energy homeostasis litera-ture; examples are also easily found in anesthesiology (e.g.,Refs. 38, 73, and 79) and other fields.

The problem with the method is that it usually violates the“principle of sham combinations,” which was introduced byLoewe and Muischnek (93, 94, 96) but independently conceivedby others (e.g., see Ref. 69). The principle is simply that a drugcannot synergize with itself (or, equivalently, it must add to itselfaccording to the additivity metric). This seems self-evident. Re-sponse additivity violates this principle, because for any drug witha curvilinear dose-effect curve sham combinations indicate syn-ergy in concave-up parts of the dose-effect curve (i.e., 2ndderivative of the dose effect equation � 0) and indicate infra-additivity in concave-down parts (2nd derivative � 0). Figure 2depicts this graphically. Only if both drugs have linear dose-effectcurves with zero intercepts is response additivity synergy validwith respect to this principle. Of course, this is very rarely thecase. Rather, most dose-effect curves in pharmacology and phys-iology are curvilinear (35, 37, 105, 130). Thus, response additivitysynergy is generally invalid if one accepts the principle of shamcombinations.

Synergy is sometimes claimed if combinations of subthresh-old doses of two drugs yield significant effects. This form ofresponse additivity is also invalid. The dose-effect curve of anydrug with a threshold is empirically concave-up in the region ofthe threshold because all subthreshold effects are indistinguish-able from zero and thus lie on a horizontal line. Thus, response

additivity in the threshold region violates the principle of shamcombinations. This is shown schematically in Fig. 2. Themistaken belief that two effects must differ if one is significantand the other is not, which is widespread in neuroscience (117),may contribute to the attraction of “subthreshold response”additvity.

More generally, the principle of sham combinations empha-sizes the critical point that, in the context of synergy analyses,addition refers not simply to the addition of two numbers butto the addition of dose response curves. Thus, supra-additivesynergy requires the development of a theory of addition ofdose-effect curves.

It is also worth noting that factorial analysis of variance of theindividual and combination effects of particular doses of drugs Aand B [i.e., EA(a), EB(b) and E(a � b), respectively] is inappro-priate for assessing synergy (28). The problem is that, in factorialanalysis of variance, deviations from additivity are indicated bysignificant interaction effects, i.e., EA(a) � EB(b) � E(a � b).This is the equivalent of testing Eq. 1 or simple response additiv-ity; i.e., it is not a valid approach to synergy if one accepts theprinciple of sham combinations.

Finally, it is important to note that not all theorists agree onthe necessity of fulfilling the principle of sham combinations.This is the case, for example, in a recent response additvityapproach developed for the detection of synergy in antimicro-bial effects (20, 92, 159) as well as in the Bliss additivityapproach (please see Bliss Aditivity).

LOEWE ADDITIVITY SYNERGY

Theory and Validity

Originally, Loewe additivity referred to a graphic way toanalyze drug interactions that was introduced by Fraser (51,52) and developed in detail by Loewe and Muischnek (93–96).Loewe’s most fundamental contribution was to discern theprinciples underlying the method and recognize that they couldlead to a variety of nonlinear forms of drug additivities. Theprinciples are the sham combination principle and the drugdose equivalence principle.

Grabovsky and Tallarida (63), Tallarida (153–156), and Tal-larida and Raffa (157) articulated these principles into an axiom-atic mathematical model. Loewe additivtiy is sometimes called theisobolographic method, dose additivity, or, in the toxicology litera-ture, concentration additivity (76, 119, 126). It is also the basis ofcomparison in most response surface synergy metrics. Despite themethod’s popularity, its prerequisites and shortcomings are notwidely understood. Therefore, I describe them here in detail.

The principles of sham combinations and drug dose equiv-alence combine to produce additive predictions as follows:dose a of drug A is equivalent to dose ba of drug B if a and ba

have equal effects (principle of dose equivalence), and ba canbe added to any other dose b of drug B to give the additiveeffect of the (a � b) combination (principle of sham combi-nations). Dose equivalence can be computed on the basis oftransforming either from the dose-effect curve of A to that of B(A ¡ B) or from B to A (B ¡ A). That is,

EADD�a � b� � EB�ba � b� � EA�a � ab� , (2)

where EB is measured on the dose-effect curve of drug B andEA is measured on the dose-effect curve of drug A. This isshown schematically in Fig. 3.

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Fig. 2. Response additivity violates the principle of sham combinations byproducing nonsensical predictions for drugs with curvilinear dose-effectcurves. In the example shown, the effect of dose 2 is 1, so according to theprinciple of sham combinations, the predicted effect of dose 2 plus dose 2 is1 � 1 � 2. In fact, due to the concave-up shape of the dose-effect curve in thatrange, the observed effect of dose 4 is almost 4. Thus, response additivityindicates that low doses of this drug synergize potently with themselves. Thesame problem occurs when using response additivity to analyze the effects of2 subthreshold drug doses. If the threshold for a statistically significant effectunder the conditions tested is �3, then the effect of dose 2 will be nonsignif-icant, and adding 2 individually nonsignificant doses of 2 will result in asignificant effect of �3; thus, according to response additivity, drugs willsynergize with themselves around the dose range for detecting statisticallysignificant effects.

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The method is usually used to derive pairs of dose a of drugA and dose b of drug B that should add to a constant effect, i.e.,if both dose Ax of drug A and dose Bx of drug B have effect X(a � b), dose combinations for which

EADD�a � b� � EA�Ax� � EB�Bx� � X. (3)

The plot of doses that fulfill Eq. 3 on a graph whose x-axisis the dose of drug A and whose y-axis is the dose of drug B isthe isobole for effect level X. Synergy (or supra-additivity)occurs if combinations of doses a and b that fall on the isoboleproduce an effect greater than X or, alternatively, if combina-tions of doses a and b that lie below the isobol produce effectX. If (a � b) combinations that lie on the isobol produce effectX, they are additive; if they produce an effect less than X, theyare infra-additive.

If the dose-effect curves of drugs A and B are characterizedmathematically, then formulas for equivalent dose of one drugin terms of the other and additive predictions can be sometimesbe derived algebraically, as discussed in Linear Isoboles are aRarity and Curvilinear Isoboles. However, if the form of thedose-effect curve is complex, a solution may not be possible, asdiscussed in Indeterminate Loewe Additivity Solutions andBoundary Conditions.

Loewe addition has good face validity. It suggests that, at thelevel of intracellular postreceptor effects or postsynaptic neu-ronal processing, the representations of diverse stimuli thatsynergize are transformed, at least in part, into a single repre-sentation. This is exactly how we understand the integrativeactions of intracellular and interneuronal signaling. It is alogical generalization of the classical Sherringtonian principleof neural signal processing (25, 146) that if two stimuli elicit

the same reflex, they must converge into a single final commonpathway. From an information-processing perspective, the in-put signals generated by the two stimuli lose their individualidentities at the point of convergence; i.e., the output could bedriven by either stimulus, which corresponds to the principle ofdrug dose equivalence. In addition, the integrated output signalcan be increased identically by appropriate increases in eitherinput signal, which corresponds to the principle of sham combi-nations. A response in the Sherringtonian context is anabstraction referring to an elemental neural reflex, such asthe iconic scratch reflex. However, it can be extended to anyelemental neuroendocrine response, for example, the secre-tion of insulin by the pancreatic �-cells or the momentaryrate of licking liquid food by a rat. By extension, moreintegrated responses that are driven in part by the elementalresponses, such as blood glucose level or daily food intake,fit the same analysis.

Linear Isoboles are a Rarity

Only in a limited number of situations are isoboles straightlines. The simplest case is if both drugs have linear dose-effectcurves with zero y-intercepts and without maxima in the regionstudied. The isobol for effect level X is the follwing line

1 � a ⁄Ax � b ⁄ Bx, (4)

shown in Fig. 4 (again, doses Ax and Bx alone each lead toeffect level X). The derivation of Eq. 4 from the principles ofdrug dose equivalence and sham combinations in this simplecase is given in APPENDIX 1. As mentioned above, synergy (orsupra-additivity) occurs if combinations of doses a and b thatfall on the isobole produce an effect greater than X or, alter-natively, if combinations of doses a and b that lie below theisobol produce effect X.

Isoboles are also straight lines for dose-effect curves thatapproximate rectangular hyperbolas with equal maxima. Rect-angular hyperbolas have the form

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Fig. 3. Graphic computation of equivalent doses and Loewe additive predic-tions. Dose 10 of drug A has effect 1, which is also given by dose 100 of drugB (dotted lines). Thus, doses 10A and 100B are equivalent. The Loewe additiveprediction for combination of doses 10A and 100B [i.e., EADD(10A � 100B)]can be solved by either B ¡ A or A ¡ B trasnforms (dashed lines). In theformer case, EADD(10A � 100B) � EA(10 � 10) � EA(20) � 1.3; in the lattercase, EADD(10A � 100B) � EB(100 � 100) � EB(200) � 1.3. Please see textfor further details.

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DOSE of A (a)Fig. 4. The equation and graph (solid line) of the linear isobole. The isoboledescribes drug combinations that lead to the same effect. Dose Ax of drug Aand dose Bx of drug B as well as all (a � b) combinations that fall on theisobole lead to identical effects or display Loewe additivity. The isobole islinear as depicted here only if the relative potency of the 2 drugs is constant.Constant relative potency has the consequence that the effects of (a � b)combinations (�Ax � �Bx) for which � � � � 1 have the same effect as Axalone or Bx alone. The dotted line shows this for � � � � 0.5.

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EA�a� � EAMaxa ⁄ �a � A50� , (5)

where EA(a) is the effect of dose a of drug A, EAMax is itsmaximal effect, and A50 is its rate or potency constant, whichis equal to dose producing the half-maximal effect. If drugs Aand B have rectangular hyperbolic dose-effect curves withequal maxima (i.e., EAMax � EBMax), then the isobole is againlinear and described by Eq. 4, as shown in Appendix 2 (156,157). Figure 5 shows typical rectangular hyperbolic dose-effectcurves. Although many physiological dose-effect curves are

well described by rectangular hyperbolas (35, 37, 105, 130),the constraint that the two drugs under consideration have thesame maximal effects substantially reduces the generality ofthe linear isobole.

It is possible, of course, to equate different maximum effectsby expressing data as percent maximal effect, but this is notadvisable in situations where transformation changes the rela-tionship of the variables change to the underlying physiologi-cally relevant variables. For example, if one drug’s maximumweight loss effect is 10 kg and another’s is 20 kg, then 1%maximum response for the former represents only half as muchweight loss in kilograms as 1% maximum response for thelatter. Synergy analyses in percent transforms in such situa-tions seems senseless.

To my knowledge, the only other dose-effect curves thatresult in linear isoboles are the probit and logit functions (153),which are commonly used to describe proportions of successesin a given number of cases.

Additivity is often defined as the ability of doses of twodrugs to substitute for each other in proportion to their poten-cies to produce the half-maximal effect (recall that potenciesare the doses producing the half-maximal effects A50 and B50);that is, for predicted additivity, the dose b of drug B to add witha particular dose a of drug A is

b ⁄ B50 � �A50 � a� ⁄ A50, (6)

which rearranges by simple algebra to Eq. 4. Thus, this for-mulation is not generally valid. Rather, it is valid only fordose-effect curves for which the principles of dose equivalenceand sham combination lead to Eq. 4, linear Loewe additivity.

As described in more detail below, isoboles for many,probably most, drug combinations do not follow Eq. 4 but arecurvilinear (19, 63, 99, 100, 155, 156, 172). For example,rectangular hyperbolic dose-effect curves with different max-ima produce curvilinear isoboles (please see Curvilinear Isobo-les). Equation 4, and therefore linear isoboles, results onlywhen the relative potency of drugs A and B is constant; i.e.,dose-effect curves for which a/b is a constant for all doses a ofdrug A and b of drug B for which the (a � b) combination leadsto the same effect. If the relative potency of drugs A and B isvariable, then the isobole is not linear.

Relative potency is most conveniently tested using logdose-effect curves. Drugs with parallel log dose-effect curveshave constant relative potency. Therefore, if the slopes of twolog dose-effect curves are not significantly different, it isappropriate to analyze synergy with linear isoboles (or theresulting response surfaces; please Loewe Additvity ResponseSurfaces). This criterion is rarely, if ever, applied. Further-more, parallel log dose-effect curves seem to be the exceptionrather than the rule; for example, I have found no suchinstances in interaction studies in energy homeostasis research.Even rather small deviations from constant relative potencycan lead to curvilinear isobolograms (please see CurvilinearIsoboles) and worse, indeterminate solutions (please see Inde-terminate Loewe Additvity Solutions). Without consideration ofthe individual dose-effect curves, deviations from the linearpredictions indicate only that the two drugs are not identicaland are not both agonists of a single receptor (139, 155). Theydo not indicate that they are supra-additive or synergistic.

The misunderstanding of the applicability of linear isobolesstems in large part from influential reviews by Berenbaum (11,

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0 200 400 600 800 1000

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Fig. 5. Examples of rectangular hyperbolic dose-effect curves, showing thatconstant relative potency is associated with parallel log dose-effect curves.A: formulas for 4 rectangular hyperbolic dose-effect curves, drugs W–Z. Theformulas have the form E � EMAX D /(D � D50), where EMAX is the maximumeffect, D is the dose, and D50 is a rate or potency constant that equals the doseyielding half-maximal effect. B: the 4 dose-effect curves in untransformedcoordinates. C: the same dose-effect curves with dose on a log scale. Note thatfor drugs W, X, and Y, EMAX � 100, the rate constants vary by factors of 10,and the nearly linear midranges of their log dose-effect curves are parallel andseparated by 1 log unit. This means that their relative potencies are constant;the effect of any dose of drug W is also the effect of dose 10W of drug X anddose 100W of drug Z. Note also that drug Z has the same rate constant as drugY, but a larger EMAX, so that the midranges of their log dose-effect curves arenot parallel and their relative potencies vary; for small effects, drug Y is nearlyas potent as drug Z, but for larger effects, drug Y is much less potent thandrug Z.

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12) that included derivations supposedly establishing that Eq. 4is true regardless of the form of the drugs’ dose-effect curves.This derivation is incorrect (please see APPENDIX 3). The proofis in fact limited to dose-effect curves whose relative potencyis constant. Either this was not recognized or its implicationswere not understood, and the linear-isobole (and linear re-sponse surface method described in the next section) quicklybecame the gold standard for synergy studies (21, 55).

Although he did not compute additive effects mathemat-ically, Loewe (94) understood and stated clearly that linearisoboles are valid only if the dose-effect curves of the drugsconsidered have constant relative potency and that this israrely the case (“. . . the overwhelming probability is thatthe isobole deviates from the endpoint diagonal with eitherSW- or NE-convexity” and that “. . . heterodynamic isobo-les usually have two isoboles for the same endpoint, onedeviating to NE and the other to SW from the endpointdiagonal;” the endpoint diagonal is the linear isobole; NEand SW are compass directions relative to the y-axis, whichis north; heterodynamic means variable relative potency).Many major reviews of synergy (11, 12, 19, 21, 56, 57, 60,122, 170) cite Loewe (94) without mentioning these points.How this could be is a puzzlement.

Curvilinear Isoboles

In contrast to the situations described above, nonparallel logdose-effect curves generate curvilinear isoboles. This appearsto occur much more frequently than do linear isobols (e.g.,Refs. 63, 99, 100, 155, 156, and 172 and the examples below).It is the case, for example, for rectangular hyperbolic dose-effect curves with different maxima, such as dose-effect curvesW and Z in Fig. 5. For such curves, Grabovsky and Tallarida(63) derived the following equivalent dose formula

a � ba � A50 ⁄ �EAMax ⁄ �EBMax�1 � B50 ⁄ b� � 1�� , (7)

which results in curvilinear isoboles for all effect levels X,

1 � a ⁄ Ax � A50 ⁄ �Ax�EAMax ⁄ EBMax�1 � B50 ⁄ b� � 1�� . (8)

Eq. 8 leads to curvilinear isoboles in which convex orconcave arcs connect points (Ax, 0) and (Bx, 0).

Indeterminate Loewe Additivity Solutions

Perhaps the greatest disadvantage of Loewe additivity is thatoften no clear prediction can be generated. Such indetermi-nancy occurs when the additive prediction based on using theB-equivalent dose of A differs from that based on the A-equiv-alent dose of B so that drug-dose combinations between thetwo solutions are supra-additive according to one transforma-tion and infra-additive according to the other.

Indeterminant Loewe additivity solutions occur frequently. Tal-larida (155, 156) gives an example of indeterminate Loeweadditivity based on the Hill equation (or Hill-Langmuir equation).The Hill equation is a three-parameter equation that results insigmoid functions that provide good fits to a wide range of phar-macological and physiological data. It is considered the basis ofquantitative pharmacology (33, 58, 62). The form of the equation is

EA�a� � AAMaxap ⁄ �ap � A50

p � (9)

where a is the dose of drug A, EAMax is the maximum effect, A50

is dose yielding the half-maximal effect, and p is a constant, often

called the slope parameter (note that if p � 1, Eq. 9 becomes therectangular-hyperbola equation, Eq. 5). Examples are shown inFig. 6.

If the dose-effect curves of two drugs are given by Hillequations with different values of p, solving for the isoboleleads to two different curvilinear solutions, depending onwhether one uses the A ¡ B or B ¡ A transformation (i.e.,whether one uses the B-equivalent dose of A or the A-equiv-alent dose of B). This is shown in Fig. 7. Figure 8 showsgraphically how this can result from combinations of a drugwith a linear dose-effect curve and a drug with a rectangularhyperbolic dose-effect curve.

X: E= 100D0.5/(D0.5 + 500.5)A

Y: E = 100D/(D + 50)( )

Z: E = 100D3/(D3 + 503)

B

CDOSE (D)

Z

X

Y

X

ZY

LOG DOSE (D)

EFF

EC

T (E

)E

FFE

CT

(E)

100

75

50

25

0

100

75

50

25

0

0 200 400 600 800 1000

-1 0 1 2 3 4

Fig. 6. Examples of Hill equation dose-effect curves. The formulas have thesame form as the rectangular hyperbolic equation in Fig. 5, except that eachterm is raised to an exponent, p. This results in the possibility of independentlyvarying the slope, maximal effect (EMAX), and half-maximal dose (A50), thusgreatly increasing the number of dose-effect curves that can be well modeled.A: formulas for three Hill equation dose-effect curves, drugs X–Z. Theformulas have the form E � EMAX Dp/(Dp � D50

p), where EMAX is themaximum effect, D is the dose, D50 is the dose yielding the half-maximaleffect, and p is a constant causing the slope to vary. B: the dose-effect curvesin untransformed coordinates. C: the same dose-effect curves with dose on alog scale.

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The interpretation different outcomes of A ¡ B or B ¡ Atransformations is problematic. Tallarida (155) argues that,because each solution represents additivity, the region con-tained between them is a region of additivity. But each solutionrepresents additivity for only one of the two equally validtransformations and represents infra- or supra-additivity for theother. Thus, it seems more reasonable to interpret both solu-tions and the area between them as indeterminate outcomes.This counterintuitive result can be described as a situation inwhich EA(a) � EB(ba) and EB(b) � EA(ab) but EA(a � ab) �EB(ba � b). Synergy is then restricted to the area below bothsolutions and antagonism to the area above both.

Loewe addition frequently leads to such indeterminate re-gions. Furthermore, the region of indeterminancy is oftenlarge. In a series of studies of combinations of various anti-epileptic drugs in rat models, Luszczki and colleagues (98–101) and Wojda et al. (172) found that A ¡ B and B ¡ Atransformations of the data resulted in indeterminate areas thatfilled �25–75% of the total dose space [i.e., the area boundedby the 4 points (0, 0), (Ax, 0), (0, Bx), and (Ax, Bx)]. Lorenzoand Sánchez-Marín (97) concluded that the gravity and fre-quency of this problem suffice to invalidate Loewe additivityas a general solution to synergy.

Combinations of two drugs with linear log dose-effect curvesthat have different slopes also lead to indeterminate outcomes.The energy homeostasis literature exemplifies this. The 11 Loeweadditivity energy homeostasis studies that I know of all used alinear approach (9, 13, 50, 84, 91, 131, 134, 136, 137, 161, 169).An example is shown in Fig. 9. Note that the slopes of the logdose-effect curves of cholecystokinin (CCK) alone and amylinalone differed by �25%. Correct calculation of Loewe additivityindicated that this difference is sufficient to change the interpre-tation of some of the CCK-amylin combination effects fromsynergy, as these authors concluded, to antagonism. The individ-ual log dose-effect curves in eight of the 10 remaining Loewe

additivity energy homeostasis studies also appeared to differ by�20% (9, 50, 91, 131, 134, 136, 161, 169), suggesting that correctanalyses would also change the interpretation of these studies.

Boundary Conditions

Boundary conditions pose additional problems for Loewe ad-ditivity. If dose equivalence is computed graphically, there is noexact transform for subthreshold doses. A reasonable solution is touse linear interpolation between zero and the smallest dose with ameasureable effect. If dose equivalence is computed mathemati-cally, below-threshold doses are not a problem.

More difficult is when doses of drug A lead to effects exceedingthe maximum effect of drug B (i.e., drug B is a partial agonist).Such doses of drug A cannot be transformed to equivalent dosesof drug B either graphically or mathematically, leaving the upperborder of Loewe additivity undefined. The problem of differentmaxima is common. One example is provided by leptin. Asshown in Fig. 1, C and E, exogenous leptin alone often has onlymodest effects on food intake and body weight in rats butapparently large interactive effects when combined with CCK or

1 A = 3.2 B

A3

2

BA

11 B = 0.2 A

0 1 2 3 4 5 0

3

2

B

EFF

EC

TE

FFE

CT

1

DOSE (A or B)0 1 2 3 4 5

0

EA(1 + 0.2) = EA(1.2) = 2.5

EB(3.2 + 1) = EB(4.2) = 2.8

E(1A + 1B) is equivalent to either:E(1A + 1B) is equivalent to either:

Fig. 8. Graphic application of the dose equivalence principle yielding indeter-minate Loewe additive solutions. A: dose 1 of drug A is equivalent to dose 3.2of drug B (dashed lines), and dose 1 of drug B is equivalent to dose 0.2 of drugA (dotted lines), as explained in Fig. 3 and in the text. B: different additivepredictions are generated for combination of dose 1 of drug A and dose 1 ofdrug B when the drug A-equivalent dose of drug B is used (dashed lines, 1.2units of drug A yields effect 2.5) than when the drug B-equivalent dose of drugA is used (dotted lines, 3.3 units of drug B yields effect 2.8).

ect B

X)yi

elds

effe

a b

” (do

se 2

2

bb a

DO

SE

“B” b a

DOSE “A” (dose 60 yields effect AX)

25

20

15

10

5

00 10 20 30 40 50 60

Fig. 7. Loewe additivity often leads to indeterminate isobolar solutions. Thedotted lines are equally valid alternate solutions for the isobole for the effectof 30% EMAX (yielded by doses Ax and Bx) for drugs A and B with dose-effectcurves given by the Hill equation, Eq. 6, with the same EMAX but differentexponents (i.e., with relative potencies that change as effect level changes).The upper dotted line results from transforming doses of drug A to theirequivalent doses of drug B, and the lower dotted line results from transformingdoses of drug B to their equivalent doses of drug A. The area between the linesis supra-additive according to the A ¡ B transformation solution and infra-additive according to the B ¡ A transformation. Please see text for furtherdetails. From Ref. 155, used with permission.

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amylin. As discussed in Linear Isoboles are a Rarity, normalizingto equate maxima, for example, by percent maximum transfor-mations, is an inadequate solution. Another possibility is to de-velop an alternative supra-additivity metric. Howard and Webster

(76), for example, recently proposed a “generalized concentrationaddition” method that was based on the inverse functions of thedrugs’ dose-effect curves {function f1(x) is the inverse of func-tion f(x) if f1[f(x)] � x}. Although this clever mathematical ployavoids the partial agonist problem, it abandons the axiomaticfoundation of Loewe additivity.

A conceptually similar situation is when in one drug is com-pletely inactive. Much of the synergy literature in clinicalanesthesiology is designed to meet just this problem (e.g., Refs.45, 74, 77, 102, 108, 142, and 143). The basic approach ofcontemporary anesthesiology is to capitalize on the potentsynergistic effects of combinations of a hypnotic drug with ananalgesic drug that alone has little or no hypnotic potency; i.e.,it is a partial agonist (e.g., Refs. 74 and 102). Loewe additivitycannot handle these situations because there are no equivalentdoses. In response, anesthesiologists have developed a varietyof alternative additivity metrics (45, 74, 77, 102, 108, 142,143), frequently not emphasizing that these are no longer trueLoewe additivity analyses. Tallarida (153) suggested a solutionbased on an ad hoc formula to generate pseudoequivalentdoses. Another is to abandon supra-additive synergy, as de-scribed more in Response Surface Modeling and CooperativeEffect Synergy.

Loewe Additivity Response Surfaces

The response surface approach extends the analysis of syn-ergism from a single-effect level to a range of effect levels. Insimple cases, the surface is a three-dimensional contour de-scribing the Loewe additive effects of continuous ranges ofdoses derived mathematically from the principles of shamaddition and dose equivalence in a manner analogous to thederivation of isoboles (153). Doses of drugs A and B areplotted on the x- and z-axes, and the predicted E(a � b) isplotted on the y-axis. The intersection of the response surfacewith a horizontal plane at height z � X is isobole for effectlevel X. If the dose-effect curves of drugs A and B are linearwith zero z-intercepts, rectangular hyperbolas with equal max-ima, or parallel logits or probits, these intersections will belinear. An example is shown in Fig. 10.

Synergy can be assessed in two ways. If the observed E(a � b)is added to the surface plot, then synergy is indicated by pointsabove the surface and infra-additivity by points below thesurface. Alternatively, the observed data can be fit to a surfaceusing second- or higher-order polynomials (57, 64, 108, 153).The observed contour is then compared with the additivepredictions. Dose combinations under areas of the observedcontour that lie above the additive prediction are synergistic.

Response surfaces are more complex, and indeed are notsurfaces in the usual sense at all, if the drugs’ dose-effectcurves lead to indeterminate solutions, as discussed in Indeter-minate Loewe Additvity Solutions. In such cases, single-dosecombinations will approach as a limit the additive solutions ofmany effect levels, not just one. This can be envisioned withreference to Fig. 7, which shows the additive solutions foreffect X, which is obtained with dose 60 of drug A alone anddose 22 of drug B alone. If one now imagines the additivesolutions for effect 0.8X given by dose 50 of drug A alone anddose 18 of drug B alone, it should be clear that the loweradditivity solution for X and the upper additivity solution for0.8X will intersect; i.e., the dose combinations represented by

A

Δn ≈ 25%

Dose (nmol/kg)

ntak

e (%

)0

min

Foo

d In

ecre

ase

in 3

0D

e

COMBINATION EFFECTSCOMBINATION EFFECTS

DOSE LOEWE OBSERVED (Amylin, CCK; ADDITIVITY EFFECT

nmol/kg) (%) (%)

B

0.77, 0.09 ≈ 17 – 19 33 ± 130.26, 0.88 ≈ 30 – 42 58 ± 100.77, 0.88 ≈ 42 – 56 54 ± 10 0.26, 2.62 ≈ 57 – 62 39 ± 11

100

80

60

40

20

0

-20

Zero 0.1 1 10 100

Amylin+ CCK

CCK

Amylin

Fig. 9. An example of indeterminate Loewe additivity solutions and erroneousinterpretations arising from improper application of Loewe additivity. A: theacute eating-inhibitory effects of intraperitoneal injections of amylin (Œ) andcholecystokinin (CCK; �). The authors smoothed the raw data with 4-param-eter logistic fits, which they used to interpolate isoeffective doses of amylinand CCK. Whether the observed combination effects (o) were synergistic wasassessed with linear isoboles and response surfaces based on the isoeffectivedose estimations; this analysis suggested that almost all amylin-CCK dosecombinations resulted in synergy. However, note that the amylin and CCK logdose-effect curves are nonparallel; the slopes differ (n) by �25%. Thus,analysis of the combination data with linear isoboles and response surfaces wasinappropriate. Graph from Ref. 13, reproduced with permission. B: graphicestimation of Loewe additivity predictions for dose combinations near thecurves’ linear portions indicates that additive solutions were indeterminate forconsiderable effect range. For example, for combination of 0.26 nmol/kgamylin with 0.88 nmol/kg CCK, Loewe additivity was indeterminate from�30 to 42% decrease, and for combination of 0.77 nmol/kg amylin with 0.88nmol/kg CCK, Loewe additivity was indeterminate from �42 to 56%. As aresult, although some dose combinations indeed appeared synergistic (e.g.,combinations of 0.77 nmol/kg amylin with 0.09 nmol/kg CCK and 0.26nmol/kg amylin with 0.88 nmol/kg CCK), others were infra-additive ratherthan synergistic (e.g., combination of 0.26 nmol/kg amylin with 2.62 nmol/kg)or indeterminate (e.g., combination of 0.77 nmol/kg amylin with 0.88 nmol/kgCCK). Note also that the maximum effect of CCK exceeded that of amylin,which also poses a problem for the computation of Loewe additivtiy. Theeffects described in this article prompted a study suggesting that 7-dayinfusions of amylin-CCK combinations elicited a synergistic increase inweight loss in dietary obese rats (162). A cooperative effect synergy analysiswould also indicate that most dose combinations were synergistic and wouldpresumably also have sufficed to motivate this followup.

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these points are additive for both effect X and effect 0.8X.Isoboles for many more effect levels will also intersect the Xisobole. This cannot be graphed in three dimensions becausesingle (x, y) dose combinations require numerous z-values.This situation is analogous to what are known as multiple-valued mathematical functions, for example y � x0.5. Suchfunctions are one source of four- and higher-dimensionalRiemann surfaces. Thus, Loewe additivity response surfacesare highly complex mathematical structures.

OTHER ADDITIVITIES

A number of other supra-additive synergy models have beenadvanced (30, 32, 42, 74, 76, 92, 148, 150, 174). I brieflyreview four, the first three of which are closely related toLoewe additivity.

The Greco and Minto Models

Greco and colleagues (64, 66) and Minto et al. (108) pro-posed interaction models for drugs whose dose-effect curves fitthe Hill equation, Eq. 9, which was discussed in IndeterminateLoewe Additvity Solutions. Greco and colleagues’ (64, 66)starting point was to accept the assertion by Berenbaum (11,12) that the linear isobole, Eq. 4, is generally valid. They thenused it and a form of Eq. 9 for inhibitory effects to generate aninteraction index that expressed the difference between theobserved combination effects and the linear isobole

1 � a ⁄ �Ax Ex1⁄p� � b ⁄ �Bx Ex1⁄q�� ab ⁄ �Ax Bx Ex�1⁄2p�1⁄2q�� (10)

where Ex is the effect of Ax and of Bx normalized to the0-dose effect ECON [i.e., Ex � E/(ECON E)], p and q arethe slope parameters for drugs A and B, respectively, and �is the interaction index. Note that the two left terms of Eq. 10are similar to Eq. 4. Synergy is indicated by � � 0, andadditivity is indicated by � � 0, i.e., if the right term of Eq. 10drops out. Several modified forms of the original model have beenproposed (16, 24, 74).

The quite different approach by Minto et al. (108) is basedon the dose-effect curves of combinations of fixed-dose ratiosof drugs A and B, i.e., combinations for which a/b is constant.The dose ratios, called �, were expressed with respect to thedrugs’ half-maximal effects, i.e., a/A50 and b/B50. That is,

� � �a ⁄ A50� ⁄ �a ⁄ A50 � b ⁄ B50� , (11)

where � ranges from 0, a mixture of all drug B, to 1, a mixtureof all drug A. Each � value was treated as a new drug whosedose-effect curve is a new Hill equation. Minto et al. (108) thenderived fourth-order polynomial equations to estimate the threeHill equation parameters (i.e., the maximal effect, half-maximaldose, and slope parameter) as functions of �. These polynomialsinvolved five coefficients, �0 �4, but �0 and �1 could beexpressed in terms of � and eliminated. Finally, the type ofinteraction is reflected by values of the remaining three �-coeffi-cients. If all three � � 0, the drugs are additive; if �2 � 0 and�3 � �4 � 0, the drugs synergize; if two or three � � 0, the drugshave complex interactions, for example, interactions includingboth synergy and antagonism.

Both Greco and colleagues (64, 66) and Minto et al. (108)accepted Berenbaum’s assertion (11, 12) of the generality ofEq. 4 and, therefore, that the criterion for additivity is the linearisobole. Unfortunately, as discussed above (Linear Isobolesare a Rarity; also see APPENDIX 3), Berenbaums’s assertion isincorrect, and true Loewe additivity isoboles generated for Hillequation dose-effect curves using the principles of dose equiv-alence and sham addition are to a great extent indeterminate(Indeterminate Loewe Additvity Solutions). Thus, rather thanbeing mathematically valid applications of the axioms ofLoewe additivity, both the Greco and colleagues (64, 66) andMinto et al. (108) algorithms are ad hoc supra-additive synergy

P

Drug 1 (mg/kg)

Drug 2 (mg/kg)

Drug 1 (mg/kg)

Dru

g 2

(mg/

kg)

Drug 2 (mg/kg)

120

80

40

0

0 40 80 120

12080

400

120

80

40

0

1.00

0.67

0.33

0.00

A

B

Fig. 10. Loewe additivity response surface analysis. A: the estimated responsesurface describing the loss of the righting reflex for �30 min (%miceresponding) produced by sham combinations of sodium hexobarbitol withitself (56). A logistical model was used to render the data continuous andestimate the response surface. Because the relative potency of hexobarbitol asdrug 1 and hexobarbitol as drug 2 was a constant 1, the linear response surfacemodel is valid. The data fit the model well (�2 � 0.95). Note that intersectionsof the surface with horizontal planes are linear isoboles for the effect level ofthe height of the plane. B: the estimated linear isobole for 50% efficacy and its95% confidence interval generated from the same data. Note that �78 mg/kghexobarbitol as drug 1 and �84 mg/kg hexobarbitol as dose 2 led to 50%responses. Reprinted with permission.

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metrics. Few of the many discussions of these models mentionthis problem (1, 16, 24, 74, 81, 89, 102, 106, 107, 139, 165).However, it should be noted that this criticism applies only tothe interpretation in terms of synergy, not to the goodness of fitof the Hill equation response surfaces to the experimental data.Thus, both models could be used in general response surfaceanalyses of combination effects, as described in COOPERATIVE

EFFECT SYNERGY (see, for example, Ref. 74).

The Chou Model

Chou (30) and Chou and Talalay (31, 32) described a synergymodel based on the derivation of a generalized equation repre-senting the “unified general theory for the Michaelis-Menten, Hill,Henderson-Hasselbalch, and Scatchard equations” that they de-rived:

fa � 1 ⁄ �1 � �Dm ⁄ D�m� , (12)

where fa is the fractional response, D is dose, Dm is the medianeffective dose, and m is a shape parameter (m � 1 forhyperbolic dose-response curves and m � 1 for sigmoidaldose-response curves). This equation was used to generateequations for the sum of the effects of two or more drugs,which in turn was used in conjunction with Eq. 4, the linearisobologram, to generate an expression for deviations fromLoewe additivity. Lee and Kong (88) recently developed amethod to estimate the confidence intervals of the Chou inter-action index.

There are several problems with Chou and Talalay’s ap-proach for integrative physiology. First, the dose-responsecurves are transformed to fractional or percent maximumresponse, which, as discussed in Linear Isoboles are a Rarity,is not appropriate when the maximal effects differ. Second,although Chou and Talalay do not emphasize it, the method islimited to drugs with constant relative potencies. The calcula-tions do not apply to “mutually nonexclusive drugs,” i.e., thosewhose relative potency is not constant. Therefore, Chou (30)simply posits that “nonexclusivity” indicates synergy (inChou’s words, the author “integrated the nonexclusive condi-tion as an intrinsic contribution to the synergistic effect”).From the perspective of formal theory, this is an indefensiblead hoc maneuver. Third, Greco et al. (64) pointed out a numberof mathematical weaknesses in Chou’s derivations.

Bliss Additivity

Bliss (14) additivity is an axiomatic supra-additive synergymodel based on the principle that drug effects are outcomes ofprobabilistic processes so that zero interaction is equivalent toprobabilistic independence. Thus, combination effects arequantified as joint probabilities according to the familiar rule

P�Ea or Eb� � P�Ea� � P�Eb� � P�Ea and Eb� , (13)

in which P represents the probability of observing the effectindicated (Ea, etc.). The most common criticism of Blissaddtivity is that, except in the case of exponential dose-effectcurves (11, 12, 152), it violates the principle of sham combi-nations; i.e., combinations of a drug with itself lead to synergyor antagonism. Another criticism from the perspective ofintegrative physiology is that interaction seems to be theantithesis of independence.

Bliss additivity is often used in the analysis of synergy inantimicrobiology, toxicology, radiation medicine, and other fields(20, 39, 60, 64, 123, 124, 152, 174, 177). According to Yeh et al.(176), an advantage of Bliss additivity is that it is exactly analo-gous to the definition of epistasis used in molecular biology. Theadditivity algorithm used by the GraphPad statistical package isalso based on Bliss additivity (see http://www.graphpad.com/faq/viewfaq.cfm?faq�991; GraphPad Software, La Jolla, CA).

COOPERATIVE EFFECT SYNERGY

Definition and Advantages

The simplest quantitative concept of synergy is that itreflects an increase in effect over the agents alone. The quan-titative definition is that dose a of drug A and dose b of drugB synergize if E(a � b) � E(a) and E(a � b) � E(b) (recall thatdrugs A and B are drugs with qualitatively similar overteffects). The analogous definition of antagonism is that doses aand b are antagonists if E(a � b) � E(a) or E(a � b) � E(b)(for example, inverse agonists fit this definition). Drug combi-nations with intermediate effects may be termed functionallyneutral. Cooperative effect synergy is also called superior yield,highest single agent, or therapeutic synergy (3, 18, 90, 120, 152a).

Cooperative effect synergy has much to recommend it. Thelack of any addition metric is a major advantage, as it obviatesthe need for an axiomatic mathematical theory for the additionof dose-effect curves and the complications that come alongwith it, as described above. This also enables synergy to beassessed for particular doses without characterization of thedrugs’ dose-effect curves. Thus, simple experiments suffice toidentify interesting results that encourage further mechanisticor translational research, as exemplified in Fig. 1.

Another major advantage of cooperative effect synergy isthat it meets the criteria recently adopted by the US Food andDrug Administration (FDA) for evaluation of combinationtherapies; i.e., “two or more drugs may be combined in a singledosage . . . to enhance the safety or effectiveness of the prin-cipal active component” [Code of Federal Regulations of theUSA, Title 21, Volume 5, Section 300.50(a)(1), April 1, 2011].This is an important shift away from the previous regulatoryguideline that combination therapies demonstrate some form ofsupra-additive synergy (for discussions related to the evolutionof the FDA’s stance, see Refs. 118, 125, 164, and 173). Thischange should influence the designs of both basic discoveryand clinical research.

Response Surface Modeling and Cooperative Effect Synergy

Response surface analysis is a powerful general method toanalyze multivariate data (22, 114) that can be profitably appliedto cooperative action synergy. In drug discovery and other fields,cooperative effect synergy based on response surface analysis isconsidered an excellent strategy for high-throughput drug discov-ery (1, 3, 18, 90, 120, 178).

Response surface analysis is based on modeling the topologyof response magnitudes. This is usually done empirically, i.e.,with no assumptions about the shape of the surface. Second-order polynomial equations, which contain linear, quadratic,and interaction terms, usually suffice to provide good fits to thedata. If several dose combinations are tested, even smallamounts of data (n � 5 or so per dose combination) usually

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http://www.graphpad.com/faq/viewfaq.cfm?faq=991

http://www.graphpad.com/faq/viewfaq.cfm?faq=991


permit rough estimation of the equation parameters. Thisenables predictions of dosages for maximum effects or forparticular effect levels, for example, using algorithms designedto determine “paths of steepest ascent” toward the maximum ordesired response (i.e., lines normal to the fitted surface con-tours directed toward the desired end point). Additional exper-iments targeted to restricted ranges of combinations or com-puter simulations can then be performed to better characterizethe response surface. Such methods have been extended toquite complexly contoured surfaces. Multiple responses can bestudied simultaneously, for example, to optimize “therapeuticwindows,” i.e., identify dose combinations with the largestdifference between therapeutic and undesired responses (90).

An important advantage of response surface analysis ofcooperative effect synergy is that it need not make assumptionsabout the dose-effect curves of the individual drugs. Thus, drugcombinations involving drugs with variable relative potency ordrugs with different maximal effects or one drug that alone hasno measurable effect do not pose modeling problems.

Response surfaces with theoretical bases can also be used.For example, one study of opioid-sedative anesthetic interac-tions fit the data to a compound logistic dose-effect curve (102)and another to a hierarchical function that modeled the ascend-ing neural mechanisms of analgesia and hypnosis (74). Theformer study also exemplifies the predictive use of responsesurface analysis. That is, Manyam et al. (102) first computed anarea on the response surface that identified sedative-hypnoticdose combinations that were optimal for anesthesia and thenused the drugs’ individual clearance times and a stepwiseiterative simulation procedure to identify the dose combina-tions that were predicted to lead to the most rapid recoveryfrom anesthesia.

CONCLUSION

Functional analyses of the effects of combinations of drugsor other manipulations that invoke a notion of additivity aremathematical models that must be developed and validatedmathematically. Unfortunately, this is rarely recognized, muchless done. As a result, much of the theoretical and empiricalsynergy literature propagates fundamental misunderstandingsof Loewe additivity. True Loewe additivity is a formally validaxiomatic mathematical theory, with face validity as a modelof integrative physiological processes. Unfortunately, how-ever, it does not solve the synergy problem. Rather, in mostsituations involving integrative physiology, including applica-tions in energy homeostasis, endocrinology, and anesthesiol-ogy, the shapes of the dose-response curves complicate theapplication of Loewe additivity and prevent clear additivitysolutions. Other supra-additive synergy models are no moresatisfactory (Table 1).

The elusive nature of good quantitative synergy solutionshas long been a concern. In 1953, Loewe (94) wrote, “Al-though, according to this study, the terms synergism andantagonism . . . have no definable place in the treatment ofcombination problems and should be eliminated from the fieldbecause of the menace of confusion, the greater probability isthat they will live on as so many other undefined and undefin-able ‘terms.’” In 1996, the view of Greco et al. (65) was that“good synergy assessment and interpretation will never besimple” and likened the search for such a “proper and easy”

solution to Dorothy’s escape from Oz in that we are waiting“for some wizard to tell us the secret.” And in 2012, Shafer(142), in discussing eight synergy models in anesthesia, con-cluded that, although useful, “all models are wrong.”

Supra-additive synergy approaches also fail to provideunique clues as to mechanism. Yates (175) enumerated ninecharacteristics of a good mathematical model of physiologicalprocesses. Neither Loewe additivity nor the other supra-addi-tivity approaches described here seem to fulfill any of themand, therefore, seem unlikely to be guides to understanding themechanisms of integrative responses.

For these reasons, I conclude that the disadvantages anddifficulties of supra-additive synergy models far outweigh theirbenefits and recommend that they be abandoned.

Nevertheless, synergy is an important physiological andclinical issue worthy of pursuit. I believe that the simpledefinition of synergy as cooperative, i.e., nonantagonistic,action is adequate both for basic discovery research and clin-ical research. The designs and analyses are far simpler thanthose of supra-additive synergy and avoid the many problemsdescribed above. Cooperative effect synergy suffices to iden-tify good candidates for further mechanistic or clinical research(Fig. 1). The definition meets regulatory requirements forestablishing combination therapies. Finally, it has proven pro-ductive in other fields.

APPENDIX 1: LOEWE ADDITIVITY EQUATION FOR LINEARDOSE-EFFECT CURVES

The simplest application of the theory of Loewe additivity is togenerate an additive formula for linear dose-effect curves with zerointercepts and with maxima ignored. In this case, the effects E(a) andE(b) for doses a and b of two drugs (or other manipulations), drugs Aand B, are given by:

E�a� � nAa and E�b� � nBb, (A1.1)

where nA and nB are the slopes of the two lines. The equations can beused to generate equivalent doses of drugs A and B. If E(a) � E(b),then nAa � nBb and

a � �nB ⁄ nA�b. (A1.2)

This equation also indicates that the relative potency of drug A withrespect to drug B is nB/nA. Note that, unlike the situation in APPENDIX 3, 1)

Table 1. Comparison of several synergy metrics

Metric Supra-Additive?Face

Validity?Formal

Validity? Recommended?

Response additivity Yes No No NoLoewe additivity Yes Yes Yes NoGreco model Yes Yes No NoMinto model Yes Yes No NoChou model Yes Yes No NoBliss additivity Yes No Yes NoCooperative effect No Yes N/A Yes

N/A, not applicable. Supra-additive indicates that synergy is assessedagainst an additive or zero-interactive prediction. Face validity is based onwhether the metric distinguishes a drug from itself (the principle of shamcombinations) and other considerations described in the text. Formal validityis indicated if the metric generates additive predictions based on an axiomaticmathematical theory of dose-response curve addition; this is not applicable forcooperative effect synergy because it does not involve addition. Loeweadditivity is not recommended because it is often complex, often leads toindeterminate solutions for many or most drug combinations, and cannot beapplied to some boundary conditions. Please see text for further details.

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Eqs. A1.1 and A1.2 are true at all effect levels, 2) the relative potencyof drugs A and B is derived, not asserted, and 3) nB/nA is constant.Next, consider an effect level X that is given by doses Ax and Bxalone. What combinations of dose a of drug A and dose b of drug Balso give effect X? For any such (a � b) pair, the dose a and the drugA-equivalent dose of drug B must add to Ax because Ax has thedesired effect X. Substituting the formula in Eq. A1.2 for equivalentdoses, this yields

a � �nB ⁄ nA�b � Ax. (A1.3)

Dividing by Ax gives

a/Ax � �nB ⁄ nA�b/Ax � 1. (A1.4)

Finally, again from Eq. A1.2 above, the dose of drug B that isequivalent to Ax is (nB/nA)Bx. Substituting this into Eq. A1.4 gives

a/Ax � �nB ⁄ nA�b/�nB ⁄ nA�Bx � 1, (A1.5)

in which the potency ratio nB/nA cancels, leaving

a/Ax � b/Bx � 1, (A1.6)

the linear-isobole equation (Eq. 4 in the main text). Thus, for theparticular situation of two linear dose-effect curves with zero inter-cepts, the additive prediction for dose combinations yielding a con-stant effect that is generated by the dose-equivalence principle is thelinear isobole equation.

APPENDIX 2: LOEWE ADDITIVITY EQUATION FORRECTANGULAR-HYPERBOLIC DOSE-EFFECT CURVES WITHIDENTICAL MAXIMUM EFFECTS

This proof was presented by Tallarida (156) and Tallarida andRaffa (157). If both drugs A and B have rectangular hyperbolicdose-effect curves, the effects E(a) and E(b) of doses a and b of drugsA and B are

E�a� � EAMax a ⁄ �a � A50� and E�b� � EBMaxb ⁄ �b � B50� , (A2.1)

where EAMax and EBMax are constants describing the maximal effectsof drugs A and B and A50 and B50 are rate constants, or potencies orhalf-maximal effects, of drugs A and B. If the maximal effects areequal, EAMax � EBMax � EMAX. If the effects of doses a and b areequal, then

EMaxa ⁄ �a � A50� � EMaxb ⁄ �b � B50� . (A2.2)

Dividing by EMAX gives

a ⁄ �a � A50� � b ⁄ �b � B50� . (A2.3)

Multiplying by (a � A50) (b � B50) gives

ab � aB50 � ab � bA50, (A2.4)

Subtracting ab from each side and dividing by B50 gives

a � bA50 ⁄ B50, (A2.5)

which is the dose of b that is the dose equivalent to a. Note that, as inthe linear example above, A50/B50 describes the two drugs’ relativepotencies and is constant. Finally, for an effect level X that is given bydoses Ax and Bx alone, what combinations of dose a of drug A anddose b of drug B also give effect X? Again, for any such pair, dose aand the drug A-equivalent dose of drug B must add to Ax because Axhas the desired effect X. Expressing dose b as its drug A-equivalentdose, given in the formula in Eq. A2.5, yields

a � b�A50 � B50� � Ax. (A2.6)

Except for the names of the constants expressing relative potency,this is identical to Eq. A1.3 and again can be transformed to thelinear-isobole equation (i.e., Eqs. 4 and A1.4). Thus, for two hyper-bolic dose-effect curves with equal maxima, the Loewe additive

prediction for dose combinations yielding a constant effect is thelinear-isobole equation.

APPENDIX 3. BERENBAUM’S “GENERAL VALIDATION”OF THE ISOBOLE METHOD BASED ON THESHAM-COMBINATION PRINCIPLE

Berenbaum (12) claims that the following derivation is a generalvalidation of the linear-isobole synergy metric because “Its derivationtook no account, either explicitly or implicitly, of the shapes ofdose-response curves of the agents . . .” In fact, as shown below, thederivation does take account of the shapes of dose-response curves byimplicitly assuming that the two agents have a constant relativepotency. Therefore, its validity is limited to that situation.

The notation is that E(a) and E(b) are the individual effects ofvarious doses a and b of drugs A and B, respectively, and effect levelX is given by doses Ax of A and Bx of B. Berenbaum (12) begins byconsidering a sham dose of drug A, called dose A= that has the sameeffect as Bx, namely X. Because dose Ax is isoeffective with dose Bx,then the sham dose A= must equal the dose BX times the relativepotency of Ax and Bx at effect level X

A� � �Ax ⁄ Bx�Bx. (A3.1)

This is straightforward application of the principle of dose equiv-alence, although Berenbaum (12) did not recognize that as a generalprinciple. Next consider that (a � b) dose combination also yieldeffect X, i.e., doses that have an additive effect according to thelinear-isobole method. Berenbaum (12) wishes to construct an equiv-alent dose a= of drug A that can be substituted for dose b. He writes,“Now, the sham combination is, in fact, a of A plus (Ax/Bx) b of A”(italics added). That is, Berenbaum substitutes dose a= for drug A= anddose b for Bx in Eq. A3.1 to generate the dose a= with an effectequivalent to that of b units of B,

a� � �Ax ⁄ Bx�b. (A3.2)

This substitution is incorrect. There is no reason that (Ax/Bx) b ofA should have the same effect as b. Recall that Ax/Bx is the relativepotency of A and B at effect level X. No information is availableabout the relative magnitudes of A and B that lead to other effectlevels. For example, 0.5 Ax need not have the same effect as 0.5 Bx.Thus, Eq. A3.2 is invalid, and no equivalent dose of drug B in termsof drug A can be generated.

Berenbaum (12) completes the derivation by equating the effect ofthe sham combination with that of Ax

E�a of A � �Ax ⁄ Bx�b of A� � E�Ax� . (A3.3)

Because all the dose terms are in units of A, the doses on each sideof the equation must be equal; thus,

a � �Ax ⁄ Bx�b � Ax. (A3.4)

Finally, dividing by Ax would yield the linear-isobole equation(Eq. 4).

a ⁄ Ax � b/Bx � 1. (A3.5)

ACKNOWLEDGMENTS

I thank Dr. Lori Asarian, Institute for Veterinary Physiology, University ofZurich, Zurich, Switzerland, Dr. Anders Lehman, AstraZeneca, Mölndal,Sweden, Dr. Timothy Moran, Department of Psychiatry, Johns HopkinsUniversity, Baltimore, MD, Dr. Jonathan Roth, Amylin, San Diego, CA, andDr. Gerard P. Smith, Department of Psychiatry, Weill Medical College ofCornell University, New York, NY, for helpful discussions. I also thank thereviewers for their helpful comments.

DISCLOSURES

I have no conflicts of interest, financial or otherwise, to declare.

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AUTHOR CONTRIBUTIONS

N.G. drafted the manuscript.

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