Emax Response Surfaces for Two-Drug Combinations

From Single-Drug Curves to a Combination Surface

A single-drug Emax curve describes how response changes as one concentration increases. A two-drug response surface extends that relationship across two concentration axes: each point on the surface predicts a response for one concentration of drug 1 and one of drug 2. The surface is useful only when its reference behavior is explicit. In particular, a claim of interaction depends on what response would be expected in the absence of interaction and on the scale used to define that expectation.

Begin with an effect-additive reference. This says that each drug contributes its own Emax effect and that the two effects sum on the response scale. An additional term can then describe departures from that reference. A different model can describe a drug changing the apparent potency of the other drug. These are distinct ways to represent an interaction; they should not be treated as interchangeable parameters in one generic “interaction” model.

The additivity assumption is a reference, not a biological law. For example, a response might have a physiological ceiling, or one drug might alter the potency of the other. In those settings the effect-additive surface may be inadequate even if each single-drug curve fits well. The model also presumes that concentrations are the relevant exposures at the effect site; if the drugs alter one another’s pharmacokinetics, measured or modeled concentrations should reflect that change before a direct pharmacodynamic interaction is inferred.

Adding a Departure From Effect Additivity

One direct way to represent a departure from the additive reference is to add a saturable interaction contribution. Define \(h_1\) and \(h_2\) as concentration fractions for the interaction term. Their potency parameters, \(K_{12,1}\) and \(K_{12,2}\), need not equal the single-drug EC50 parameters. The parameter \(\gamma_{12}\) is the maximum interaction contribution on the response scale.

The product form makes a simple boundary condition explicit: a combination-specific contribution cannot appear when either drug is absent. It also creates a smooth, bounded interaction contribution rather than one that grows without limit. It does not guarantee that the total response is bounded: the two single-drug terms still approach their separate maxima, and the interaction term approaches \(\gamma_{12}\). If a strict physiological ceiling is required, a different surface formulation may be needed.

Representing a Potency Shift

A drug can alter the apparent potency of another without changing its maximum effect. One way to express this is to let drug 2 modify drug 1’s EC50 through a concentration-dependent shift. Let \(q_2=C_2/(K_2+C_2)\), where \(K_2\) controls how quickly the shift develops. The apparent EC50 for drug 1 is \(EC_{50,1}\exp(\beta_{12}q_2)\). The exponential keeps that EC50 positive for any real value of \(\beta_{12}\).

If both drugs can shift one another’s potency, a corresponding term can be specified for drug 2, with its own direction and parameters. The two shifts need not be equal. The model’s baseline behavior should be checked: setting either concentration to zero must recover the intended single-drug curve for the other drug. A potency-shift model can yield a nonzero difference from the effect-additive prediction during co-exposure even though it has no separate \(\gamma_{12}\) term. That difference arises through altered potency rather than an added maximum interaction effect.

Derivation, Assumptions, and Diagnostics

The Emax contribution follows from a simple saturating relationship. For one drug, write \(f(C)=E_{\max}C/(EC_{50}+C)\). At zero concentration, \(f(0)=0\). At \(C=EC_{50}\), \(f(EC_{50})=E_{\max}/2\). Dividing by \(C\) in the numerator and denominator shows that, as concentration grows, the fraction approaches one:

Consequently, the concentration-response contribution approaches \(E_{\max}\). The additive surface follows by summing the two single-drug changes from the same baseline. For the interaction-augmented form, each fraction \(C_j/(K_{12,j}+C_j)\) is zero at zero concentration and approaches one as \(C_j\) grows. Their product therefore vanishes on either single-drug axis and approaches one when both concentrations grow, establishing the boundary and limiting interpretations of \(\gamma_{12}\).

These interpretations rely on the model assumptions, not just the algebra. The basic Emax form fixes the Hill coefficient at one; it assumes a monotone saturating contribution and stable Emax and EC50 values across the design. The additive reference assumes response changes sum on the modeled scale. The interaction product assumes a smooth, monotone, saturable joint departure. A potency-shift form assumes that the modifier acts through the apparent EC50 as specified, rather than through Emax or another mechanism. If the endpoint has a floor, ceiling, delayed response, or nonmonotone behavior, these assumptions may be inappropriate.

Useful checks include plotting observed responses and model predictions over the full two-concentration grid, including the single-drug axes. A surface that predicts a combination effect at \(C_1=0\) from its interaction term violates the product model’s boundary condition. Systematic residual patterns along one axis may indicate an inadequate single-drug curve; patterns concentrated in co-exposure regions may indicate that the chosen interaction structure is too restrictive. Check predictions at high concentrations as well: an implausible total response can reveal that an additive, unbounded surface is unsuitable for a constrained endpoint.

Choosing and Using a Surface

The effect-additive surface is a useful reference when the scientific question is whether the observed combination response departs from the sum of single-drug response changes. The \(\gamma_{12}\) extension makes that departure explicit and separates its limiting magnitude from the concentration scales controlling its onset. The potency-shift model instead asks whether one drug changes the concentration required for the other to produce a given fractional effect. It can preserve the shifted drug’s Emax while allowing its apparent EC50 to vary with the partner concentration.

These approaches are not equivalent to Bliss, Loewe, or highest-single-agent additivity, nor are their interaction parameters interchangeable with interaction contrasts on a continuous response scale. Before interpreting a coefficient, state the response scale, baseline, reference surface, and parameterization. A positive \(\gamma_{12}\) means an upward departure from the particular effect-additive reference; a positive \(\beta_{12}\) means a rightward shift in drug 1’s apparent EC50. Neither sign has a model-independent meaning.

A common mistake is to call \(\gamma_{12}\) “the combination Emax” or to interpret a positive value as proof of synergy. The correction is that \(\gamma_{12}\) is the limiting additional response relative to a specified additive surface, while the total high-concentration prediction also includes \(E_0\) and both single-drug Emax contributions. Likewise, \(\beta_{12}\) is a log-scale potency-shift parameter: the limiting fold change in drug 1’s EC50 is \(\exp(\beta_{12})\), not \(\beta_{12}\) itself. Parameter meaning always follows from the equation and its reference behavior.

Finally, a mathematically valid surface is not automatically a clinically plausible one. Check the concentration units, baseline, limiting responses, axis behavior, and predictions over the range relevant to use. If the response is bounded, if a drug changes the other drug’s exposure, or if the observed surface departs from the selected interaction shape, the model may need a different link, exposure input, or response structure. Interpretation should remain conditional on the chosen model and the observed concentration range.