## Abstract

Analysis of the interactive effects of combinations of hormones or other manipulations with qualitatively similar individual effects is an important topic in basic and clinical endocrinology as well as other branches of basic and clinical research related to integrative physiology. Functional, as opposed to mechanistic, analyses of interactions rely on the concept of synergy, which can be defined qualitatively as a cooperative action or quantitatively as a supra-additive effect according to some metric for the addition of different dose-effect curves. Unfortunately, dose-effect curve addition is far from straightforward; rather, it requires the development of an axiomatic mathematical theory. I review the mathematical soundness, face validity, and utility of the most frequently used approaches to supra-additive synergy. These criteria highlight serious problems in the two most common synergy approaches, response additivity and Loewe additivity, which is the basis of the isobole and related response surface approaches. I conclude that there is no adequate, generally applicable, supra-additive synergy metric appropriate for endocrinology or any other field of basic and clinical integrative physiology. I recommend that these metrics be abandoned in favor of the simpler definition of synergy as a cooperative, i.e., nonantagonistic, effect. This simple definition avoids mathematical difficulties, is easily applicable, meets regulatory requirements for combination therapy development, and suffices to advance phenomenological basic research to mechanistic studies of interactions and clinical combination therapy research.

- 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 among endocrine and nervous controls is a fundamental characteristic of physiological regulation. Interactions are evident at all levels of organization, from genetic to organismic (34, 67, 78, 85, 104, 160). For example, the secretion of most hormones is under both endocrine and neural control, and many endocrine target cells as well as neurons in a variety of brain sites express receptors to multiple hormones (34, 67, 85, 104). At the organismic level, the coordinated actions of nerves and hormones acting in several tissues are vital to many regulatory functions, including blood glucose homeostasis, fluid and mineral balance, and energy homeostasis, which provides many of the examples in this review (7, 54, 68, 83, 85, 110, 151). That many endocrine and metabolic diseases are treated with combination therapies also reflects the importance of interactions in physiological regulation (10, 29, 85).

Administration of combinations of drugs often produces unexpectedly large responses. This is known as synergy. It is often described as “1 + 1 > 2,” or supra-additive synergy. This is an apt description because, as shown below, addition causes the most complications and misunderstandings in synergy analyses. (Note that I couch the discussion in terms of “drugs” for simpicity; what is meant is any quantifiable manipulation, including doses of hormones, neurotransmitters, neuromodulators, pharmaceutical agents, etc., as well as amounts of activity, hours of food deprivation, etc.).

### What is the Problem?

Despite the apparent simplicity of the concept of supra-additive synergy, translating it into a specific quantitative methodology with formal validity is a thorny problem for which many solutions have been offered (for a sense of the variety of approaches, see Refs. 12, 64, and 121). Furthermore, few investigators appreciate the prerequisites and limitations of supra-additive synergy methodologies. These problems are exacerbated by the different evolution of the concept in different fields, by the relative paucity of theoretical accounts of synergy that are not forbiddingly quantitative, by the failure of most theorists to clearly distinguish their approaches from those of others, by some widely propagated errors in the existing accounts, and by the momentum of the misdirected literature.

Synergy analyses in energy homeostasis research provide an unfortunate example of inappropriate or incomplete synergy analyses. Review of the energy homeostasis literature identified 44 studies, including four of my own, that used the response additivity approach to supra-additive synergy (2, 8, 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 44 studies attempted to establish the validity of this approach, and, as explained in response additivity synergy, it is almost never valid. A simple thought experiment shows why. If a drug has a linear log dose-effect curve, as many do through much of their dynamic ranges, then adding a dose of the drug to itself will 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. This suggests that many response additivity studies that report additivity may in fact have shown synergy.

Eleven energy homeostasis studies that used a linear Loewe additivity approach were identified (9, 13, 50, 84, 91, 131, 134, 136, 137, 161, 169). Again, none of these studies attempted to establish the validity of this approach, and, as explained in *Loewe Additivity Synergy*, it is rarely valid. Other fields have more successfully shunned response additivity but still suffer from misunderstandings of Loewe addivity. For example, although nowhere is the concept of synergy more important and more discussed than in anesthesiology (45, 72, 77, 108, 142, 143), one can find recent examples of uncritical application of linear Loewe additivity there as well (e.g., Refs. 38, 73, and 79).

The gravity of incorrect applications of supra-additive synergy metrics can hardly be overstated. For example, as shown in *Interdeterminate Loewe Additivity Solutions*, uncritical use of linear Loewe additivity can completely reverse the interpretation of combination data, turning what should be antagonism into synergy.

### Why is Understanding Synergy Important?

Improving research practice by understanding synergy is important for at least three reasons. First, if synergy were better understood, researchers would make fewer innocent errors in searching for supra-additive synergy. Second, there would be fewer misguided attempts to understand the physiological bases of what are in fact artifactual synergies. Third, there would be fewer misinformed efforts to translate apparently promising but erroneous basic research synergies into clinical research. These improvements in research practice would lead to substantial savings in human, animal, and financial resources.

### What is the Solution?

My review of supra-additive synergy in response additivity synergy, loewe additivity synergy, and other activities leads me to conclude that there is no metric to detect supra-additive synergy, i.e., 1 + 1 > 2, with adequate formal validity, face validity, and general applicability. Therefore, I recommend adoption of a simpler definition of synergy as a cooperative effect. As I describe in cooperative effect synergy, such synergy is defined simply as a response to a drug combination that is greater than the responses to the drugs individually. Addition is not involved. Figure 1 provides some examples of cooperative effect synergy and its utility in furthering basic and clinical research.

## SUPRA-ADDITIVE SYNERGY IS A FORMAL QUANTITATIVE THEORY

### Mathematical Nature of Synergy

As is often emphasized (19, 49, 65, 142), quantitative synergy analyses are formal mathematical exercises, not wet physiology. There are two main reasons. First, there is not sufficient mechanistic knowledge to enable quantitative predictions of the effects of drug combinations. If, for example, two drugs each bound reversibly to a single receptor with known kinetics and if the drugs' individual effects depended only on the number of receptors bound, then their individual and interactive effects could be computed using the laws of mass action (12, 30, 64, 139, 143, 155). Similarly, interactive effects could be computed if they were shown to result directly from the frequency of action potentials in some accessible population of neurons. However, such quantitative mechanistic knowledge is not yet available in endocrinology or in other branches of integrative physiology. The second reason that synergy is intrinsically mathematical is that physiological measurements cannot be used to test synergy metrics. The theorems derived in Euclidian geometry can be compared with the characteristics of existing objects to determine whether the theory corresponds to our direct experience of physical reality. In contrast, supra-additive synergy metrics lead to mathematical predictions that have no directly observable physiological counterparts. This has the implication that, unlike more mechanistic models, it is difficult or impossible to apply the criteria of construct and predictive validity to synergy. How then to judge synergy metrics? I suggest the criteria of *1*) face validity, *2*) mathematical soundness, and *3*) utility with respect to furthering basic and clinical research.

Although my approach to synergy is quantitative, and many important points can be made only mathematically, the mathematical content of the article is minimal. I present mathematical derivations in the appendices in this article, and I illustrate many points graphically.

### Mathematical Notation

The mathematical notation is as follows: a and b denote particular amounts or doses of “*drugs*” *A* and *B*, which might be hormones or any other quantifiable manipulations; E_{A} and E_{B} are the dose effect functions of *drugs A* and *B*, respectively, although the subscripts are omitted in unambiguous situations, situations; E(a) or E_{A}(a) and E(b) or E_{B}(b) are the effects of dose a of A and b of B when applied individually, and E(a + b) is the effect of a and b applied simultaneously. E_{ADD}(a + b) is the theoretical additive effect of the a + b combination. For clarity, 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 *A* and *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 metric, the additive or zero-interactive effect of a drug combination is the sum of the individual effects,

The problem with the method is that it usually violates the “principle of sham combinations,” which was introduced by Loewe and Muischnek (93, 94, 96) but independently conceived by others (e.g., see Ref. 69). The principle is simply that a drug cannot synergize with itself (or, equivalently, it must add to itself according to the additivity metric). This seems self-evident. Response additivity violates this principle, because for any drug with a curvilinear dose-effect curve sham combinations indicate synergy in concave-up parts of the dose-effect curve (i.e., 2nd derivative of the dose effect equation > 0) and indicate infra-additivity in concave-down parts (2nd derivative < 0). Figure 2 depicts this graphically. Only if both drugs have linear dose-effect curves with zero intercepts is response additivity synergy valid with respect to this principle. Of course, this is very rarely the case. Rather, most dose-effect curves in pharmacology and physiology are curvilinear (35, 37, 105, 130). Thus, response additivity synergy is generally invalid if one accepts the principle of sham combinations.

Synergy is sometimes claimed if combinations of subthreshold doses of two drugs yield significant effects. This form of response additivity is also invalid. The dose-effect curve of any drug with a threshold is empirically concave-up in the region of the threshold because all subthreshold effects are indistinguishable from zero and thus lie on a horizontal line. Thus, response additivity in the threshold region violates the principle of sham combinations. This is shown schematically in Fig. 2. The mistaken belief that two effects must differ if one is significant and 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 emphasizes the critical point that, in the context of synergy analyses, addition refers not simply to the addition of two numbers but to the addition of dose response curves. Thus, supra-additive synergy requires the development of a theory of addition of dose-effect curves.

It is also worth noting that factorial analysis of variance of the individual and combination effects of particular doses of *drugs A* and *B* [i.e., E_{A}(a), E_{B}(b) and E(a + b), respectively] is inappropriate for assessing synergy (28). The problem is that, in factorial analysis of variance, deviations from additivity are indicated by significant interaction effects, i.e., E_{A}(a) + E_{B}(b) ≠ E(a + b). This is the equivalent of testing *Eq. 1* or simple response additivity; i.e., it is not a valid approach to synergy if one accepts the principle of sham combinations.

Finally, it is important to note that not all theorists agree on the necessity of fulfilling the principle of sham combinations. This is the case, for example, in a recent response additvity approach developed for the detection of synergy in antimicrobial effects (20, 92, 159) as well as in the Bliss additivity approach (please see *Bliss Aditivity*).

## LOEWE ADDITIVITY SYNERGY

### Theory and Validity

Originally, Loewe additivity referred to a graphic way to analyze 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 the principles underlying the method and recognize that they could lead to a variety of nonlinear forms of drug additivities. The principles are the sham combination principle and the drug dose equivalence principle.

Grabovsky and Tallarida (63), Tallarida (153–156), and Tallarida and Raffa (157) articulated these principles into an axiomatic mathematical model. Loewe additivtiy is sometimes called the isobolographic method, dose additivity, or, in the toxicology literature, concentration additivity (76, 119, 126). It is also the basis of comparison in most response surface synergy metrics. Despite the method's popularity, its prerequisites and shortcomings are not widely understood. Therefore, I describe them here in detail.

The principles of sham combinations and drug dose equivalence combine to produce additive predictions as follows: *dose a* of *drug A* is equivalent to *dose b*_{a} of *drug B* if *a* and *b*_{a} have equal effects (principle of dose equivalence), and *b*_{a} can be added to any other *dose b* of *drug B* to give the additive effect of the (a + b) combination (principle of sham combinations). Dose equivalence can be computed on the basis of transforming either from the dose-effect curve of *A* to that of *B* (A → B) or from *B* to *A* (B → A). That is,
_{B} is measured on the dose-effect curve of *drug B* and E_{A} is measured on the dose-effect curve of *drug A*. This is shown schematically in Fig. 3.

The method is usually used to derive pairs of *dose a* of *drug A* 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

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

If the dose-effect curves of *drugs A* and *B* are characterized mathematically, then formulas for equivalent dose of one drug in terms of the other and additive predictions can be sometimes be derived algebraically, as discussed in *Linear Isoboles are a Rarity* and *Curvilinear Isoboles*. However, if the form of the dose-effect curve is complex, a solution may not be possible, as discussed in *Indeterminate Loewe Additivity Solutions* and *Boundary Conditions*.

Loewe addition has good face validity. It suggests that, at the level of intracellular postreceptor effects or postsynaptic neuronal processing, the representations of diverse stimuli that synergize are transformed, at least in part, into a single representation. This is exactly how we understand the integrative actions of intracellular and interneuronal signaling. It is a logical generalization of the classical Sherringtonian principle of neural signal processing (25, 146) that if two stimuli elicit the same reflex, they must converge into a single final common pathway. From an information-processing perspective, the input signals generated by the two stimuli lose their individual identities at the point of convergence; i.e., the output could be driven by either stimulus, which corresponds to the principle of drug dose equivalence. In addition, the integrated output signal can be increased identically by appropriate increases in either input signal, which corresponds to the principle of sham combinations. A response in the Sherringtonian context is an abstraction referring to an elemental neural reflex, such as the iconic scratch reflex. However, it can be extended to any elemental neuroendocrine response, for example, the secretion of insulin by the pancreatic β-cells or the momentary rate of licking liquid food by a rat. By extension, more integrated responses that are driven in part by the elemental responses, 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 straight lines. The simplest case is if both drugs have linear dose-effect curves with zero *y*-intercepts and without maxima in the region studied. The isobol for effect level X is the follwing line
*doses Ax* and *Bx* alone each lead to effect level X). The derivation of *Eq. 4* from the principles of drug dose equivalence and sham combinations in this simple case is given in appendix 1. As mentioned above, synergy (or supra-additivity) occurs if combinations of *doses a* and *b* that fall on the isobole produce an effect greater than X or, alternatively, if combinations of *doses a* and *b* that lie below the isobol produce effect X.

Isoboles are also straight lines for dose-effect curves that approximate rectangular hyperbolas with equal maxima. Rectangular hyperbolas have the form
_{A}(a) is the effect of *dose a* of *drug A*, E_{AMax} is its maximal effect, and A_{50} is its rate or potency constant, which is equal to dose producing the half-maximal effect. If *drugs A* and *B* have rectangular hyperbolic dose-effect curves with equal maxima (i.e., E_{AMax} = E_{BMax}), then the isobole is again linear and described by *Eq. 4*, as shown in *Appendix 2* (156, 157). Figure 5 shows typical rectangular hyperbolic dose-effect curves. 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 the same maximal effects substantially reduces the generality of the linear isobole.

It is possible, of course, to equate different maximum effects by expressing data as percent maximal effect, but this is not advisable in situations where transformation changes the relationship of the variables change to the underlying physiologically relevant variables. For example, if one drug's maximum weight loss effect is 10 kg and another's is 20 kg, then 1% maximum response for the former represents only half as much weight loss in kilograms as 1% maximum response for the latter. Synergy analyses in percent transforms in such situations seems senseless.

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

Additivity is often defined as the ability of doses of two drugs to substitute for each other in proportion to their potencies to produce the half-maximal effect (recall that potencies are the doses producing the half-maximal effects A_{50} and B_{50}); that is, for predicted additivity, the *dose b* of *drug B* to add with a particular *dose a* of *drug A* is
*Eq. 4*. Thus, this formulation is not generally valid. Rather, it is valid only for dose-effect curves for which the principles of dose equivalence and 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 are curvilinear (19, 63, 99, 100, 155, 156, 172). For example, rectangular hyperbolic dose-effect curves with different maxima produce curvilinear isoboles (please see *Curvilinear Isoboles*). *Equation 4*, and therefore linear isoboles, results only when 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* of *drug A* and *b* of *drug B* for which the (a + b) combination leads to the same effect. If the relative potency of *drugs A* and *B* is variable, then the isobole is not linear.

Relative potency is most conveniently tested using log dose-effect curves. Drugs with parallel log dose-effect curves have constant relative potency. Therefore, if the slopes of two log dose-effect curves are not significantly different, it is appropriate to analyze synergy with linear isoboles (or the resulting response surfaces; please *Loewe Additvity Response Surfaces*). This criterion is rarely, if ever, applied. Furthermore, parallel log dose-effect curves seem to be the exception rather than the rule; for example, I have found no such instances in interaction studies in energy homeostasis research. Even rather small deviations from constant relative potency can lead to curvilinear isobolograms (please see *Curvilinear Isoboles*) and worse, indeterminate solutions (please see *Indeterminate Loewe Additvity Solutions*). Without consideration of the individual dose-effect curves, deviations from the linear predictions indicate only that the two drugs are not identical and are not both agonists of a single receptor (139, 155). They do not indicate that they are supra-additive or synergistic.

The misunderstanding of the applicability of linear isoboles stems in large part from influential reviews by Berenbaum (11, 12) that included derivations supposedly establishing that *Eq. 4* is true regardless of the form of the drugs' dose-effect curves. This derivation is incorrect (please see appendix 3)*.* The proof is in fact limited to dose-effect curves whose relative potency is constant. Either this was not recognized or its implications were not understood, and the linear-isobole (and linear response surface method described in the next section) quickly became the gold standard for synergy studies (21, 55).

Although he did not compute additive effects mathematically, Loewe (94) understood and stated clearly that linear isoboles are valid only if the dose-effect curves of the drugs considered have constant relative potency and that this is rarely the case (“. . . the overwhelming probability is that the isobole deviates from the endpoint diagonal with either SW- or NE-convexity” and that “. . . heterodynamic isoboles usually have two isoboles for the same endpoint, one deviating to NE and the other to SW from the endpoint diagonal;” the endpoint diagonal is the linear isobole; NE and SW are compass directions relative to the *y*-axis, which is 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 log dose-effect curves generate curvilinear isoboles. This appears to 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 curves W and Z in Fig. 5. For such curves, Grabovsky and Tallarida (63) derived the following equivalent dose formula

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

### Indeterminate Loewe Additivity Solutions

Perhaps the greatest disadvantage of Loewe additivity is that often no clear prediction can be generated. Such indeterminancy occurs when the additive prediction based on using the B-equivalent dose of A differs from that based on the A-equivalent dose of B so that drug-dose combinations between the two solutions are supra-additive according to one transformation and infra-additive according to the other.

Indeterminant Loewe additivity solutions occur frequently. Tallarida (155, 156) gives an example of indeterminate Loewe additivity based on the Hill equation (or Hill-Langmuir equation). The Hill equation is a three-parameter equation that results in sigmoid functions that provide good fits to a wide range of pharmacological and physiological data. It is considered the basis of quantitative pharmacology (33, 58, 62). The form of the equation is
*drug A*, E_{AMax} is the maximum effect, A_{50} 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 the rectangular-hyperbola equation, *Eq. 5*). Examples are shown in Fig. 6.

If the dose-effect curves of two drugs are given by Hill equations with different values of *p*, solving for the isobole leads to two different curvilinear solutions, depending on whether one uses the A → B or B → A transformation (i.e., whether one uses the B-equivalent dose of A or the A-equivalent dose of B). This is shown in Fig. 7. Figure 8 shows graphically how this can result from combinations of a drug with a linear dose-effect curve and a drug with a rectangular hyperbolic dose-effect curve.

The interpretation different outcomes of A → B or B → A transformations is problematic. Tallarida (155) argues that, because each solution represents additivity, the region contained between them is a region of additivity. But each solution represents additivity for only one of the two equally valid transformations and represents infra- or supra-additivity for the other. Thus, it seems more reasonable to interpret both solutions and the area between them as indeterminate outcomes. This counterintuitive result can be described as a situation in which E_{A}(a) = E_{B}(b_{a}) and E_{B}(b) = E_{A}(a_{b}) but E_{A}(a + a_{b}) ≠ E_{B}(b_{a} + b). Synergy is then restricted to the area below both solutions and antagonism to the area above both.

Loewe addition frequently leads to such indeterminate regions. Furthermore, the region of indeterminancy is often large. 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 → A transformations of the data resulted in indeterminate areas that filled ∼25–75% of the total dose space [i.e., the area bounded by the 4 points (0, 0), (Ax, 0), (0, Bx), and (Ax, Bx)]. Lorenzo and Sánchez-Marín (97) concluded that the gravity and frequency of this problem suffice to invalidate Loewe additivity as a general solution to synergy.

Combinations of two drugs with linear log dose-effect curves that have different slopes also lead to indeterminate outcomes. The energy homeostasis literature exemplifies this. The 11 Loewe additivity energy homeostasis studies that I know of all used a linear 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 log dose-effect curves of cholecystokinin (CCK) alone and amylin alone differed by ∼25%. Correct calculation of Loewe additivity indicated that this difference is sufficient to change the interpretation of some of the CCK-amylin combination effects from synergy, as these authors concluded, to antagonism. The individual 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 correct analyses would also change the interpretation of these studies.

### Boundary Conditions

Boundary conditions pose additional problems for Loewe additivity. If dose equivalence is computed graphically, there is no exact transform for subthreshold doses. A reasonable solution is to use linear interpolation between zero and the smallest dose with a measureable effect. If dose equivalence is computed mathematically, below-threshold doses are not a problem.

More difficult is when doses of *drug A* lead to effects exceeding the maximum effect of *drug B* (i.e., *drug B* is a partial agonist). Such doses of *drug A* cannot be transformed to equivalent doses of *drug B* either graphically or mathematically, leaving the upper border of Loewe additivity undefined. The problem of different maxima is common. One example is provided by leptin. As shown in Fig. 1, *C* and *E*, exogenous leptin alone often has only modest effects on food intake and body weight in rats but apparently large interactive effects when combined with CCK or amylin. As discussed in *Linear Isoboles are a Rarity*, normalizing to equate maxima, for example, by percent maximum transformations, is an inadequate solution. Another possibility is to develop an alternative supra-additivity metric. Howard and Webster (76), for example, recently proposed a “generalized concentration addition” method that was based on the inverse functions of the drugs' dose-effect curves {function f^{−1}(x) is the inverse of function f(x) if f^{−1}[f(x)] = x}. Although this clever mathematical ploy avoids the partial agonist problem, it abandons the axiomatic foundation of Loewe additivity.

A conceptually similar situation is when in one drug is completely inactive. Much of the synergy literature in clinical anesthesiology is designed to meet just this problem (e.g., Refs. 45, 74, 77, 102, 108, 142, and 143). The basic approach of contemporary anesthesiology is to capitalize on the potent synergistic effects of combinations of a hypnotic drug with an analgesic drug that alone has little or no hypnotic potency; i.e., it is a partial agonist (e.g., Refs. 74 and 102). Loewe additivity cannot handle these situations because there are no equivalent doses. In response, anesthesiologists have developed a variety of alternative additivity metrics (45, 74, 77, 102, 108, 142, 143), frequently not emphasizing that these are no longer true Loewe additivity analyses. Tallarida (153) suggested a solution based on an ad hoc formula to generate pseudoequivalent doses. Another is to abandon supra-additive synergy, as described more in *Response Surface Modeling and Cooperative Effect Synergy*.

### Loewe Additivity Response Surfaces

The response surface approach extends the analysis of synergism from a single-effect level to a range of effect levels. In simple cases, the surface is a three-dimensional contour describing the Loewe additive effects of continuous ranges of doses derived mathematically from the principles of sham addition and dose equivalence in a manner analogous to the derivation of isoboles (153). Doses of *drugs A* and *B* are plotted on the *x*- and *z*-axes, and the predicted E(a + b) is plotted on the *y*-axis. The intersection of the response surface with a horizontal plane at height *z* = X is isobole for effect level X. If the dose-effect curves of *drugs A* and *B* are linear with zero *z*-intercepts, rectangular hyperbolas with equal maxima, or parallel logits or probits, these intersections will be linear. 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 points above the surface and infra-additivity by points below the surface. Alternatively, the observed data can be fit to a surface using second- or higher-order polynomials (57, 64, 108, 153). The observed contour is then compared with the additive predictions. Dose combinations under areas of the observed contour that lie above the additive prediction are synergistic.

Response surfaces are more complex, and indeed are not surfaces in the usual sense at all, if the drugs' dose-effect curves lead to indeterminate solutions, as discussed in *Indeterminate Loewe Additvity Solutions*. In such cases, single-dose combinations will approach as a limit the additive solutions of many effect levels, not just one. This can be envisioned with reference to Fig. 7, which shows the additive solutions for effect X, which is obtained with *dose 60* of *drug A* alone and *dose 22* of *drug B* alone. If one now imagines the additive solutions for effect 0.8X given by dose 50 of *drug A* alone and *dose 18* of *drug B* alone, it should be clear that the lower additivity solution for X and the upper additivity solution for 0.8X will intersect; i.e., the dose combinations represented by these points are additive for both effect X and effect 0.8X. Isoboles for many more effect levels will also intersect the X isobole. This cannot be graphed in three dimensions because single (*x*, *y*) dose combinations require numerous *z*-values. This situation is analogous to what are known as multiple-valued mathematical functions, for example *y* = *x*^{0.5}. Such functions are one source of four- and higher-dimensional Riemann surfaces. Thus, Loewe additivity response surfaces are highly complex mathematical structures.

## OTHER ADDITIVITIES

A number of other supra-additive synergy models have been advanced (30, 32, 42, 74, 76, 92, 148, 150, 174). I briefly review four, the first three of which are closely related to Loewe additivity.

### The Greco and Minto Models

Greco and colleagues (64, 66) and Minto et al. (108) proposed interaction models for drugs whose dose-effect curves fit the Hill equation, *Eq. 9*, which was discussed in *Indeterminate Loewe 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 then used it and a form of *Eq. 9* for inhibitory effects to generate an interaction index that expressed the difference between the observed combination effects and the linear isobole
_{CON} [i.e., Ex = E/(E_{CON} − E)], p and q are the slope parameters for *drugs A* and *B*, respectively, and α is the interaction index. Note that the two left terms of *Eq. 10* are similar to *Eq. 4*. Synergy is indicated by α > 0, and additivity is indicated by α = 0, i.e., if the right term of *Eq. 10* drops out. Several modified forms of the original model have been proposed (16, 24, 74).

The quite different approach by Minto et al. (108) is based on the dose-effect curves of combinations of fixed-dose ratios of *drugs A* and *B*, i.e., combinations for which a/b is constant. The dose ratios, called θ, were expressed with respect to the drugs' half-maximal effects, i.e., a/A_{50} and b/B_{50}. That is,
*drug B*, to 1, a mixture of all *drug A*. Each θ value was treated as a new drug whose dose-effect curve is a new Hill equation. Minto et al. (108) then derived fourth-order polynomial equations to estimate the three Hill equation parameters (i.e., the maximal effect, half-maximal dose, and slope parameter) as functions of θ. These polynomials involved five coefficients, β_{0} − β_{4}, but β_{0} and β_{1} could be expressed in terms of θ and eliminated. Finally, the type of interaction is reflected by values of the remaining three β-coefficients. If all three β = 0, the drugs are additive; if β_{2} > 0 and β_{3} = β_{4} = 0, the drugs synergize; if two or three β > 0, the drugs have complex interactions, for example, interactions including both synergy and antagonism.

Both Greco and colleagues (64, 66) and Minto et al. (108) accepted Berenbaum's assertion (11, 12) of the generality of *Eq. 4* and, therefore, that the criterion for additivity is the linear isobole. Unfortunately, as discussed above (*Linear Isoboles are a Rarity*; also see appendix 3), Berenbaums's assertion is incorrect, and true Loewe additivity isoboles generated for Hill equation dose-effect curves using the principles of dose equivalence and sham addition are to a great extent indeterminate (*Indeterminate Loewe Additvity Solutions*). Thus, rather than being mathematically valid applications of the axioms of Loewe additivity, both the Greco and colleagues (64, 66) and Minto et al. (108) algorithms are ad hoc supra-additive synergy metrics. Few of the many discussions of these models mention this problem (1, 16, 24, 74, 81, 89, 102, 106, 107, 139, 165). However, it should be noted that this criticism applies only to the interpretation in terms of synergy, not to the goodness of fit of the Hill equation response surfaces to the experimental data. Thus, both models could be used in general response surface analyses 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 synergy model based on the derivation of a generalized equation representing the “unified general theory for the Michaelis-Menten, Hill, Henderson-Hasselbalch, and Scatchard equations” that they derived:
_{a} is the fractional response, D is dose, D_{m} is the median effective dose, and m is a shape parameter (m < 1 for hyperbolic dose-response curves and m ≥ 1 for sigmoidal dose-response curves). This equation was used to generate equations for the sum of the effects of two or more drugs, which in turn was used in conjunction with *Eq. 4*, the linear isobologram, to generate an expression for deviations from Loewe additivity. Lee and Kong (88) recently developed a method to estimate the confidence intervals of the Chou interaction index.

There are several problems with Chou and Talalay's approach for integrative physiology. First, the dose-response curves are transformed to fractional or percent maximum response, 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 is limited to drugs with constant relative potencies. The calculations do not apply to “mutually nonexclusive drugs,” i.e., those whose relative potency is not constant. Therefore, Chou (30) simply posits that “nonexclusivity” indicates synergy (in Chou's words, the author “integrated the nonexclusive condition as an intrinsic contribution to the synergistic effect”). From the perspective of formal theory, this is an indefensible ad hoc maneuver. Third, Greco et al. (64) pointed out a number of mathematical weaknesses in Chou's derivations.

### Bliss Additivity

Bliss (14) additivity is an axiomatic supra-additive synergy model based on the principle that drug effects are outcomes of probabilistic processes so that zero interaction is equivalent to probabilistic independence. Thus, combination effects are quantified as joint probabilities according to the familiar rule
_{a}, etc.). The most common criticism of Bliss addtivity is that, except in the case of exponential dose-effect curves (11, 12, 152), it violates the principle of sham combinations; i.e., combinations of a drug with itself lead to synergy or antagonism. Another criticism from the perspective of integrative physiology is that interaction seems to be the antithesis of independence.

Bliss additivity is often used in the analysis of synergy in antimicrobiology, 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 analogous to the definition of epistasis used in molecular biology. The additivity algorithm used by the GraphPad statistical package is also 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 it reflects an increase in effect over the agents alone. The quantitative definition is that *dose a* of *drug A* and *dose b* of *drug B* synergize if E(a + b) > E(a) and E(a + b) > E(b) (recall that *drugs A* and *B* are drugs with qualitatively similar overt effects). The analogous definition of antagonism is that *doses a* and *b* are antagonists if E(a + b) < E(a) or E(a + b) < E(b) (for example, inverse agonists fit this definition). Drug combinations with intermediate effects may be termed functionally neutral. 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. The lack of any addition metric is a major advantage, as it obviates the need for an axiomatic mathematical theory for the addition of dose-effect curves and the complications that come along with it, as described above. This also enables synergy to be assessed for particular doses without characterization of the drugs' dose-effect curves. Thus, simple experiments suffice to identify interesting results that encourage further mechanistic or translational research, as exemplified in Fig. 1.

Another major advantage of cooperative effect synergy is that it meets the criteria recently adopted by the US Food and Drug Administration (FDA) for evaluation of combination therapies; i.e., “two or more drugs may be combined in a single dosage . . . to enhance the safety or effectiveness of the principal active component” [Code of Federal Regulations of the USA, Title 21, Volume 5, Section 300.50(a)(1), April 1, 2011]. This is an important shift away from the previous regulatory guideline that combination therapies demonstrate some form of supra-additive synergy (for discussions related to the evolution of the FDA's stance, see Refs. 118, 125, 164, and 173). This change should influence the designs of both basic discovery and clinical research.

### Response Surface Modeling and Cooperative Effect Synergy

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

Response surface analysis is based on modeling the topology of 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 the data. If several dose combinations are tested, even small amounts of data (*n* = 5 or so per dose combination) usually permit rough estimation of the equation parameters. This enables predictions of dosages for maximum effects or for particular effect levels, for example, using algorithms designed to determine “paths of steepest ascent” toward the maximum or desired response (i.e., lines normal to the fitted surface contours directed toward the desired end point). Additional experiments targeted to restricted ranges of combinations or computer simulations can then be performed to better characterize the response surface. Such methods have been extended to quite complexly contoured surfaces. Multiple responses can be studied simultaneously, for example, to optimize “therapeutic windows,” i.e., identify dose combinations with the largest difference between therapeutic and undesired responses (90).

An important advantage of response surface analysis of cooperative effect synergy is that it need not make assumptions about the dose-effect curves of the individual drugs. Thus, drug combinations involving drugs with variable relative potency or drugs with different maximal effects or one drug that alone has no measurable effect do not pose modeling problems.

Response surfaces with theoretical bases can also be used. For example, one study of opioid-sedative anesthetic interactions fit the data to a compound logistic dose-effect curve (102) and another to a hierarchical function that modeled the ascending neural mechanisms of analgesia and hypnosis (74). The former study also exemplifies the predictive use of response surface analysis. That is, Manyam et al. (102) first computed an area on the response surface that identified sedative-hypnotic dose combinations that were optimal for anesthesia and then used the drugs' individual clearance times and a stepwise iterative simulation procedure to identify the dose combinations that were predicted to lead to the most rapid recovery from anesthesia.

## CONCLUSION

Functional analyses of the effects of combinations of drugs or other manipulations that invoke a notion of additivity are mathematical models that must be developed and validated mathematically. Unfortunately, this is rarely recognized, much less done. As a result, much of the theoretical and empirical synergy literature propagates fundamental misunderstandings of Loewe additivity. True Loewe additivity is a formally valid axiomatic mathematical theory, with face validity as a model of integrative physiological processes. Unfortunately, however, it does not solve the synergy problem. Rather, in most situations involving integrative physiology, including applications in energy homeostasis, endocrinology, and anesthesiology, the shapes of the dose-response curves complicate the application of Loewe additivity and prevent clear additivity solutions. Other supra-additive synergy models are no more satisfactory (Table 1).

The elusive nature of good quantitative synergy solutions has long been a concern. In 1953, Loewe (94) wrote, “Although, according to this study, the terms synergism and antagonism . . . have no definable place in the treatment of combination problems and should be eliminated from the field because of the menace of confusion, the greater probability is that they will live on as so many other undefined and undefinable ‘terms.’” In 1996, the view of Greco et al. (65) was that “good synergy assessment and interpretation will never be simple” 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, concluded that, although useful, “all models are wrong.”

Supra-additive synergy approaches also fail to provide unique clues as to mechanism. Yates (175) enumerated nine characteristics of a good mathematical model of physiological processes. Neither Loewe additivity nor the other supra-additivity approaches described here seem to fulfill any of them and, therefore, seem unlikely to be guides to understanding the mechanisms of integrative responses.

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

Nevertheless, synergy is an important physiological and clinical issue worthy of pursuit. I believe that the simple definition of synergy as cooperative, i.e., nonantagonistic, action is adequate both for basic discovery research and clinical research. The designs and analyses are far simpler than those of supra-additive synergy and avoid the many problems described above. Cooperative effect synergy suffices to identify good candidates for further mechanistic or clinical research (Fig. 1). The definition meets regulatory requirements for establishing combination therapies. Finally, it has proven productive in other fields.

## DISCLOSURES

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

## AUTHOR CONTRIBUTIONS

N.G. drafted the manuscript.

## ACKNOWLEDGMENTS

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

## APPENDIX 1: LOEWE ADDITIVITY EQUATION FOR LINEAR DOSE-EFFECT CURVES

The simplest application of the theory of Loewe additivity is to generate an additive formula for linear dose-effect curves with zero intercepts and with maxima ignored. In this case, the effects E(a) and E(b) for *doses a* and *b* of two drugs (or other manipulations), *drugs A* and *B*, are given by:
_{A} and n_{B} are the slopes of the two lines. The equations can be used to generate equivalent doses of *drugs A* and *B*. If E(a) = E(b), then n_{A}a = n_{B}b and

This equation also indicates that the relative potency of *drug A* with respect to *drug B* is n_{B}/n_{A}. Note that, unlike the situation in appendix 3, *1*) *Eqs. A1.1* and *A1.2* are true at all effect levels, *2*) the relative potency of *drugs A* and *B* is derived, not asserted, and *3*) n_{B}/n_{A} is constant. Next, consider an effect level X that is given by doses Ax and Bx alone. What combinations of *dose a* of *drug A* and *dose b* of *drug B* also give effect X? For any such (a + b) pair, the *dose a* and the *drug A*-equivalent dose of *drug B* must add to Ax because Ax has the desired effect X. Substituting the formula in *Eq. A1.2* for equivalent doses, this yields

Dividing by Ax gives

Finally, again from *Eq. A1.2* above, the dose of *drug B* that is equivalent to Ax is (n_{B}/n_{A})Bx. Substituting this into *Eq. A1.4* gives
_{B}/n_{A} cancels, leaving
*Eq. 4* in the main text). Thus, for the particular situation of two linear dose-effect curves with zero intercepts, the additive prediction for dose combinations yielding a constant effect that is generated by the dose-equivalence principle is the linear isobole equation.

## APPENDIX 2: LOEWE ADDITIVITY EQUATION FOR RECTANGULAR-HYPERBOLIC DOSE-EFFECT CURVES WITH IDENTICAL MAXIMUM EFFECTS

This proof was presented by Tallarida (156) and Tallarida and Raffa (157). If both *drugs A* and *B* have rectangular hyperbolic dose-effect curves, the effects E(a) and E(b) of *doses a* and *b* of *drugs A* and *B* are
_{AMax} and E_{BMax} are constants describing the maximal effects of *drugs A* and *B* and A_{50} and B_{50} are rate constants, or potencies or half-maximal effects, of *drugs A* and *B*. If the maximal effects are equal, E_{AMax} = E_{BMax} = E_{MAX}. If the effects of *doses a* and *b* are equal, then

Dividing by E_{MAX} gives

Multiplying by (a + A_{50}) (b + B_{50}) gives

Subtracting ab from each side and dividing by B_{50} gives
_{50}/B_{50} describes the two drugs' relative potencies and is constant. Finally, for an effect level X that is given by doses Ax and Bx alone, what combinations of *dose a* of *drug A* and *dose b* of *drug B* also give effect X? Again, for any such pair, *dose a* and the *drug A*-equivalent dose of *drug B* must add to Ax because Ax has the desired effect X. Expressing *dose b* as its *drug A*-equivalent dose, given in the formula in *Eq. A2.5*, yields

Except for the names of the constants expressing relative potency, this is identical to *Eq. A1.3* and again can be transformed to the linear-isobole equation (i.e., *Eqs. 4* and *A1.4*). Thus, for two hyperbolic dose-effect curves with equal maxima, the Loewe additive prediction for dose combinations yielding a constant effect is the linear-isobole equation.

## APPENDIX 3. BERENBAUM'S “GENERAL VALIDATION” OF THE ISOBOLE METHOD BASED ON THE SHAM-COMBINATION PRINCIPLE

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

The notation is that E(a) and E(b) are the individual effects of various *doses a* and *b* of *drugs A* and *B*, respectively, and effect level X is given by doses Ax of A and Bx of B. Berenbaum (12) begins by considering a sham dose of *drug A*, called *dose A′* that has the same effect as Bx, namely X. Because dose Ax is isoeffective with dose Bx, then the sham *dose A′* must equal the dose B_{X} times the relative potency of Ax and Bx at effect level X

This is straightforward application of the principle of dose equivalence, although Berenbaum (12) did not recognize that as a general principle. Next consider that (a + b) dose combination also yield effect X, i.e., doses that have an additive effect according to the linear-isobole method. Berenbaum (12) wishes to construct an equivalent *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′* and *dose b* for Bx in *Eq. A3.1* to generate the *dose a′* with an effect equivalent to that of b units of B,

This substitution is incorrect. There is no reason that *(Ax/Bx) b of A* should have the same effect as b. Recall that Ax/Bx is the relative potency of A and B at effect level X. No information is available about the relative magnitudes of A and B that lead to other effect levels. 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 terms of *drug A* can be generated.

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

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

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

- Copyright © 2013 the American Physiological Society

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