Decomposing Exposure-Mediated and Direct Pharmacodynamic Interactions

Why Decompose a Drug Interaction?

When a perpetrator drug changes the concentration of a victim drug, the resulting response difference can have more than one source. The perpetrator may alter victim-drug exposure through pharmacokinetics; it may also affect response through its own pharmacology or through a pharmacodynamic interaction with the victim drug. A response difference between combination and victim-drug-alone regimens does not, by itself, distinguish these explanations.

The decomposition developed here makes the comparison explicit. It asks what the response contrast would be if victim-drug concentration changed while the perpetrator were absent, and what additional contrast remains when the perpetrator is present at specified concentrations. These are counterfactual, model-based response contrasts. They clarify the estimand; they do not establish a causal mechanism unless the design and assumptions support causal interpretation.

For clarity, the equations below use a scalar concentration at a specified time or a summary exposure measure. The response scale and time point must be held fixed throughout a decomposition. For example, a response in milligrams per deciliter cannot be combined with a response expressed as a log ratio without first defining a common scale.

Define the Total and Exposure-Mediated Contrasts

At a fixed victim-drug regimen \(a\), the total response contrast for adding the perpetrator compares the actual combination concentrations with the concentrations under victim-drug treatment alone:

This contrast includes all consequences of adding the perpetrator represented in the response model: a change in victim-drug concentration, the perpetrator’s own response effect, and any pharmacodynamic interaction. To isolate the response change associated with the altered victim-drug concentration, use a counterfactual comparison in which the perpetrator concentration is held at zero in both terms:

This is an exposure-mediated contrast on the chosen response scale and at the specified reference condition for the perpetrator. It asks how response would differ between the two victim-drug concentrations if the perpetrator were absent. It is not generally equal to a pharmacokinetic change alone: it translates that concentration change through the response relationship for the victim drug.

The remaining contrast after this first subtraction is a controlled direct contrast at the combination’s victim-drug concentration:

Here “direct” means that the victim-drug concentration is held fixed in the two response predictions. This contrast includes the perpetrator’s response effect on its own and any residual interaction conditional on both concentrations. It is therefore not, by itself, a pure measure of pharmacodynamic interaction.

Separate the Perpetrator Effect From Residual Interaction

A further contrast separates the perpetrator’s effect at zero victim-drug concentration from the response departure when both drugs are present. At fixed concentrations \(c_A\) and \(c_B\), define the residual interaction contrast by a difference in differences:

This contrast removes the response at baseline and each drug’s response when given alone. It is zero when the combination response equals the sum of the two single-drug changes from baseline on the chosen scale. A nonzero value indicates departure from that specific additivity reference, at those concentrations and on that response scale.

For \(c_A=C_A^{a,1}\) and \(c_B=C_B^1\), the controlled direct contrast can be written exactly as:

Thus the total contrast has three components: the exposure-mediated contrast, the perpetrator-alone effect, and the residual interaction contrast. This identity is useful because it prevents a nonzero controlled direct contrast from being labelled automatically as an interaction.

Derivation and a Worked Decomposition

The three-part result follows by adding and subtracting response predictions, not by assuming that the response surface is linear. Start with the total contrast and insert the zero-perpetrator prediction at the combination victim concentration, the perpetrator-alone prediction, and the baseline prediction:

The first bracket is \(EM(a)\), the second is the perpetrator-alone effect, and the third is \(I(C_A^{a,1},C_B^1)\). To verify the equality, expand the right side: the terms \(\mu(C_A^{a,1},0)\), \(\mu(0,C_B^1)\), and \(\mu(0,0)\) cancel, leaving precisely the total contrast.

Reference Concentrations, Assumptions, and Interpretation

The decomposition is exact for the response predictions and reference conditions written in its equations. The reference matters. The exposure-mediated contrast above evaluates the victim-drug concentration change with the perpetrator absent. If instead the victim-drug concentration change is evaluated with the perpetrator present, the exposure and direct components can differ. When the response surface contains an interaction, neither allocation is uniquely privileged by algebra; each answers a different counterfactual question.

For a causal interpretation, the counterfactual responses must correspond to well-defined interventions, and observed data must identify or credibly estimate the required predictions. Typical requirements include consistency between assigned interventions and the potential outcomes being modeled, adequate positivity or overlap for the relevant regimens and concentration combinations, and control of confounding of treatment, concentration, and response. In mediation settings, additional assumptions are needed to identify contrasts that intervene on concentration; in particular, mediator-response confounding and confounders affected by treatment require careful handling. A fitted response model alone does not guarantee these conditions.

Diagnostics, Alternatives, and Limitations

Before interpreting a decomposition, check whether its response and concentration predictions are credible in the region being compared. Useful checks include:

• Confirm that the concentration units, response scale, time point, and baseline definition are consistent across all terms.
• Inspect data support for each required concentration combination, especially the counterfactual settings \((C_A^{a,1},0)\) and \((0,C_B^1)\). Predictions far outside observed support rely heavily on model extrapolation.
• Compare observed and model-predicted responses across concentration ranges and regimens. Systematic residual patterns or implausible predictions can make the decomposition misleading even when the algebra is correct.
• Check whether concentrations are adequately measured and whether time-varying profiles have been reduced in a way that preserves the response-relevant information.
• Report uncertainty for the contrasts when estimated from data; uncertainty in several correlated model predictions must be propagated into their differences.

A direct comparison of combination versus victim-drug-alone regimens estimates a total contrast, not its mechanism. The counterfactual decomposition is more informative when the required predictions are identified and supported. A pharmacokinetic analysis can describe how the perpetrator changes victim-drug exposure, while response contrasts translate exposure differences into response units. Neither alone establishes a residual pharmacodynamic interaction.

The residual contrast is also scale-dependent. A departure from additivity on the raw response scale need not be a departure on a transformed or relative scale. Its value depends on the concentrations, endpoint, time, and response scale; it should not be interpreted as a universal property of the drug pair. It is distinct from named drug-combination additivity criteria, which define their own reference models and are treated separately.

Common mistake: “The combination response is 6.5 units higher than the victim-only prediction, so the interaction is 6.5 units.” That comparison is a controlled direct contrast, not the pure residual interaction. In the first worked example it includes a \(2\ R\) perpetrator-alone effect. The residual interaction is \(4.5\ R\), computed by subtracting both single-drug effects and adding back baseline.