Worked Analysis of an Exposure-Mediated Drug Interaction

Why Separate Exposure Change From Response Change?

A perpetrator drug can alter the pharmacokinetics of a victim drug, changing the victim’s exposure at a fixed dose. It may also be associated with a pharmacodynamic change that remains after accounting for exposure. These pathways answer different questions: the total effect compares responses under the combination and victim drug alone at the same nominal victim dose, while an exposure-adjusted contrast compares responses at a common victim exposure.

The distinction matters because a response difference at the same nominal dose does not, by itself, show a pharmacodynamic interaction. If the perpetrator increases victim exposure, the response may change even when the response-versus-exposure relationship is otherwise unchanged. The analysis below uses hypothetical data to estimate the exposure shift, fit a response model, and decompose a total contrast.

Definition: Let \(P=0\) denote absence of the perpetrator and \(P=1\) its presence. Let \(D\) be the nominal victim-drug dose, \(C\) the measured victim exposure, and \(Y\) a continuous response. A total contrast at dose \(d\) compares \(Y\) under \(P=1,D=d\) with \(Y\) under \(P=0,D=d\). An exposure-adjusted contrast compares the response under \(P=1\) with that under \(P=0\) at a specified common value of \(C\).

Models and Conditions for Interpretation

An exposure model describes how victim exposure varies with victim dose and perpetrator status. One useful starting point is a log-linear model:

\log(C_{ij})=\alpha+\lambda\log(D_i)+\delta P_{ij}+\varepsilon_{ij}.

Here \(i\) indexes dose and \(j\) indexes observations. The coefficient \(\delta\) represents the log exposure ratio associated with perpetrator presence at a fixed victim dose, provided the model’s dose relationship is appropriate. Exponentiating gives the exposure ratio, \(\exp(\delta)\). For this interpretation, exposure must be measured comparably across conditions, the dose range must support the assumed dose relationship, and important dose-by-perpetrator differences must not be hidden by the model.

A simple response model for a continuous outcome is:

Y_{ij}=\beta_0+\beta_C C_{ij}+\beta_P P_{ij}+\epsilon_{ij}.

In this model, \(\beta_C\) is the response change per exposure unit, and \(\beta_P\) is the exposure-adjusted perpetrator contrast, conditional on the model and the observed exposure range. The model assumes a linear response-exposure relationship over that range, an additive perpetrator effect, and errors with mean zero conditional on the predictors. For standard least-squares uncertainty calculations, errors are also assumed independent with constant variance; repeated observations on the same individual require an appropriate within-individual covariance model.

The additive model can be checked against a model that permits the perpetrator contrast to vary with exposure:

Y_{ij}=\beta_0+\beta_C C_{ij}+\beta_P P_{ij}+\beta_{CP}C_{ij}P_{ij}+\epsilon_{ij}.

With this extension, the exposure-adjusted contrast at exposure \(c\) is \(\beta_P+\beta_{CP}c\). A meaningful comparison requires overlap: the data should include or support both perpetrator conditions at the exposure values used for the contrast. Extrapolating a fitted line far beyond observed exposures can produce a precise-looking but unsupported estimate.

Worked Analysis of Hypothetical Data

Consider a hypothetical study of a victim drug given at 1, 2, or 3 mg, with or without a perpetrator. Victim exposure is measured as area under the concentration-time curve (AUC), in mg·h/L. The continuous response is in arbitrary response units. The values below are illustrative and are not from a real study.

Worked Example: Estimating the Perpetrator-Associated Exposure Shift

At each nominal dose, suppose the observed mean victim AUC is 1, 2, and 3 mg·h/L without the perpetrator, and 1.5, 3, and 4.5 mg·h/L with it. Fit the simple exposure relationship \(C=D\exp(\delta P)\), where \(D\) is the nominal dose in mg.

At each dose, the ratio of exposure with versus without the perpetrator is the same:

\frac{1.5}{1}=\frac{3}{2}=\frac{4.5}{3}=1.5.

Thus the estimated exposure ratio is 1.5, corresponding to a 50% increase in victim AUC at a fixed dose. On the log scale, the perpetrator coefficient is:

\hat{\delta}=\log(1.5)=0.4055.

This result is consistent with the assumed proportional exposure change: the AUC scales with dose, and the perpetrator multiplies AUC by the same factor at each dose. If the ratios had varied substantially by dose, a single \(\delta\) would not adequately describe the data; a dose-by-perpetrator term or another exposure model would need consideration.

Now suppose the response observations are as follows. Two observations are shown for each dose-perpetrator condition; these replicate values are arranged symmetrically around their cell mean to make the model calculations transparent.

Victim dose (mg)	Perpetrator \(P\)	Victim AUC \(C\) (mg·h/L)	Responses \(Y\) (units)
1	0	1	11, 13
2	0	2	13, 15
3	0	3	15, 17
1	1	1.5	15, 17
2	1	3	18, 20
3	1	4.5	21, 23

Worked Example: Fitting the Exposure-Adjusted Response Model

Fit \(Y=\beta_0+\beta_C C+\beta_P P+\epsilon\). The cell means are 12, 14, and 16 units without the perpetrator at AUC values 1, 2, and 3; with the perpetrator, they are 16, 19, and 22 units at AUC values 1.5, 3, and 4.5.

Using the three means without the perpetrator, the change in response per AUC unit is:

\hat{\beta}_C=\frac{16-12}{3-1}=2 \quad\text{response units per mg·h/L}.

The fitted no-perpetrator line therefore has intercept \(12-(2)(1)=10\) response units. At the common observed exposure of 3 mg·h/L, the mean response is 16 units without the perpetrator and 19 units with it. The exposure-adjusted contrast is \(19-16=3\) response units, so \(\hat{\beta}_P=3\). Together, these give:

\hat{Y}=10+2C+3P.

The table also permits a check of the additive assumption. Without the perpetrator, the slope from AUC 1 to 3 is \((16-12)/(3-1)=2\). With it, the slope from AUC 1.5 to 4.5 is \((22-16)/(4.5-1.5)=2\). The equal slopes are consistent with no exposure-by-perpetrator interaction in this hypothetical dataset. This is a useful descriptive check, not proof that the model is correct.

Deriving the Total and Exposure-Adjusted Contrasts

The decomposition follows directly from adding and subtracting the response that would occur without the perpetrator at the exposure observed with it. Let \(C_0\) be the victim exposure at a given nominal dose without the perpetrator and \(C_1\) the exposure at that same dose with the perpetrator. Under the additive linear response model:

\begin{aligned} Y(C_1,1)-Y(C_0,0) &=\{Y(C_1,1)-Y(C_1,0)\}+\{Y(C_1,0)-Y(C_0,0)\}\\ &=\beta_P+\beta_C(C_1-C_0). \end{aligned}

The first bracket is the exposure-adjusted perpetrator contrast evaluated at \(C_1\); in the additive model it equals \(\beta_P\) at any exposure. The second bracket is the response change associated with moving from \(C_0\) to \(C_1\) along the no-perpetrator response curve. This algebraic decomposition depends on the specified response model. It does not, by itself, prove that either component is a causal effect.

Worked Example: Comparing Total and Exposure-Adjusted Effects

At a nominal victim dose of 2 mg, AUC is 2 mg·h/L without the perpetrator and 3 mg·h/L with it. The observed mean responses are 14 and 19 units, respectively. The total contrast at the same nominal dose is:

19-14=5\text{ response units}.

The fitted response model attributes \(2(3-2)=2\) response units to the exposure increase along the no-perpetrator response curve. The remaining exposure-adjusted contrast is 3 units. Equivalently, at the common exposure of 3 mg·h/L, the model predicts 19 units with the perpetrator and \(10+2(3)=16\) units without it, giving \(19-16=3\) units. The decomposition checks:

\underbrace{5}_{\text{total contrast}} = \underbrace{2}_{\text{exposure-associated component}} + \underbrace{3}_{\text{exposure-adjusted component}}.

These are response-unit contrasts for this hypothetical model, not percentages and not universal properties of the drug pair. In particular, the adjusted contrast is conditional on the response model and is not automatically a causal direct effect.

Diagnostics, Alternatives, and Limitations

Before interpreting the decomposition, inspect whether the assumed models are credible. For the exposure model, plot victim AUC against dose separately by perpetrator status. A changing exposure ratio across doses, curvature, or influential observations can undermine a single multiplicative shift. For the response model, inspect residuals against fitted values and exposure, and compare observed cell means with model predictions. Curvature suggests that a linear exposure term may be inadequate; systematic differences between the two perpetrator-specific slopes suggest that \(\beta_{CP}\) may be needed. Large residuals, unequal spread, or dependence among repeated observations call for a revised error or covariance model.

A response model using a nonlinear exposure-response function, such as an Emax model, may be more appropriate when the observed exposure range includes saturation. In that case, the decomposition should be calculated from model predictions rather than by applying the linear formula \(\beta_C(C_1-C_0)\). Another alternative is joint modeling of exposure and response, which can account for uncertainty and dependence in both processes. A two-stage analysis that first estimates exposure and then treats it as known in the response model can understate uncertainty if exposure is measured imprecisely; uncertainty propagation should be considered.

The exposure-adjusted contrast also requires care in causal interpretation. It compares conditions at a common measured exposure, but exposure is itself affected by the perpetrator. If factors that influence both exposure and response are not adequately handled, or if the relevant exposure ranges do not overlap, a conditional model contrast need not represent a causal direct effect. The total contrast and the adjusted contrast therefore answer distinct questions and should be reported with their estimands, model assumptions, and uncertainty made explicit.

Common pitfall: “The response is 5 units higher with the perpetrator at the same victim dose, so the direct pharmacodynamic interaction is 5 units.” This reasoning ignores that victim exposure also increased. In the worked data, the total contrast is 5 units, but 2 units are associated with the exposure shift under the fitted model; the exposure-adjusted contrast is 3 units. Neither contrast should be given a causal interpretation without the additional assumptions required for that interpretation.

Check Your Understanding

Use the hypothetical fitted model and distinguish the total contrast from the exposure-adjusted contrast.

If victim AUC rises from 2 to 3 mg·h/L at a fixed dose, what exposure ratio and percentage increase does that represent?
Under the fitted response model, what is the predicted response without the perpetrator at AUC 2.5 mg·h/L?
At the 2 mg dose in the worked data, identify the total contrast and the exposure-adjusted contrast.
What pattern in a response-versus-exposure plot would make the additive response model questionable?
Why does adjusting for victim exposure not automatically make the perpetrator coefficient a causal direct effect?

Solutions. (1) The ratio is \(3/2=1.5\), a 50% increase. (2) \(10+2(2.5)=15\) response units. (3) The total contrast is \(19-14=5\) response units; the exposure-adjusted contrast is \(3\) response units, with \(2\) units associated with the exposure shift under the model. (4) Curvature or systematically different slopes by perpetrator status would challenge the linear additive specification. (5) Exposure is affected by the perpetrator, and confounding of exposure and response or lack of exposure overlap can make the conditional contrast differ from a causal direct effect.