Why Separate Exposure From Potency?
When a drug is given with a second drug, a change in response may arise because the concentration of the first drug changes, because its concentration-effect relationship changes, or because both happen. These explanations have different implications. A pharmacokinetic interaction changes exposure; a pharmacodynamic interaction changes the response associated with a given concentration. If concentration and response are considered only as a combined treatment contrast, these mechanisms can be mistaken for one another.
A crossover study helps by measuring each participant under both conditions: the index drug alone and the index drug with the interacting drug. The within-participant comparison controls stable individual differences, but it does not automatically resolve the mechanisms. Concentrations and a continuous pharmacodynamic endpoint must be measured in a way that supports separate estimates, and the analysis must account for pairing.
For clarity, let \(C\) be the index-drug concentration and let \(E(C)\) be the baseline-adjusted effect, with larger values representing more of the effect of interest. A simple saturable model is the Emax model with Hill coefficient fixed at 1:
Here \(E_{\max}\) is the maximum modeled effect and \(EC_{50}\) determines the concentration scale of the response. This model is a useful teaching and analysis tool when the response increases monotonically and approaches a plateau. It is not a universal description of pharmacology; the data must support its shape and assumptions.
Design and Assumptions for a Paired Analysis
In a two-period crossover, each participant receives both conditions, with the order assigned by the study design. For the within-participant comparison to be interpretable, the periods should have adequate washout, the treatment sequence should not create material carryover, and the endpoint should be measured under comparable conditions. Period effects, changes in disease state, adherence, and time-dependent co-medications can still matter. Randomizing sequence helps balance systematic order effects; it does not substitute for checking them.
The PK comparison typically uses a prespecified exposure measure, such as AUC over a defined interval. AUC should be measured consistently in both periods, and the conditions should differ in the intended interacting drug rather than in unrelated features of dosing or sampling. To compare concentration-effect relationships, concentration and effect must be temporally aligned. If the response lags behind plasma concentration, a direct plot of response against the same-time concentration can create hysteresis and an apparent potency shift. Sampling around the relevant response time, or using an appropriate effect-compartment model, may be needed.
A basic potency comparison also assumes that a shared \(E_{\max}\) is a reasonable approximation and that the concentration range in each condition is informative about the curve. If observations cover only low concentrations, \(E_{\max}\) may be poorly identified; if they cover only a narrow range, \(EC_{50}\) may be weakly estimated. A direct effect of the interacting drug on the endpoint, unrelated to the index-drug concentration, must be absent or separately represented. Otherwise, the fitted curve may incorrectly attribute that direct effect to altered potency.
Pairing is reflected in the analysis. For exposure, calculate each participant’s ratio of combination-condition AUC to alone-condition AUC, then analyze its logarithm. For concentration-effect modeling, a paired or repeated-measures model can allow participant-specific baseline or curve parameters while estimating the condition contrast. Treating all observations as independent discards the crossover structure and generally gives an inappropriate uncertainty estimate.
Estimating the Exposure Change
Because exposure ratios are positive and often vary multiplicatively, the standard comparison is on the log scale. For participant \(i\), define \(R_i=AUC_{i,\mathrm{combination}}/AUC_{i,\mathrm{alone}}\). The mean log ratio is exponentiated to obtain the geometric mean ratio. A confidence interval can be formed on the log scale using a paired \(t\) interval, provided the log ratios are reasonably compatible with the model assumptions; exponentiating its endpoints returns the interval in ratio units.
Worked Example: Paired AUC Comparison
In a hypothetical six-participant crossover, the index-drug AUC values under the alone and combination conditions, respectively, are 100 and 120, 110 and 143, 90 and 126, 120 and 180, 105 and 168, and 95 and 161.5 mg·h/L. The paired ratios are therefore \(1.20, 1.30, 1.40, 1.50, 1.60,\) and \(1.70\).
Their natural logarithms are approximately \(0.1823, 0.2624, 0.3365, 0.4055, 0.4700,\) and \(0.5306\). The mean log ratio is \(0.3646\), and its sample standard deviation is \(0.1302\). Thus the geometric mean ratio is:
The combination condition has an estimated 1.44-fold AUC, or about 44.0% greater AUC, than the alone condition. For a 95% paired \(t\) interval, \(n=6\), \(df=5\), and \(t_{0.975,5}=2.571\). The standard error of the mean log ratio is \(0.1302/\sqrt{6}=0.05315\). The interval on the log scale is:
Exponentiating both endpoints gives a 95% confidence interval of approximately \([1.256,\ 1.651]\) for the geometric mean AUC ratio. Under the paired log-scale model, this interval describes uncertainty in the population geometric mean ratio; it is not a probability statement about the fixed parameter. The interval is above 1, indicating evidence of increased exposure in this hypothetical sample. With only six participants, the estimate is imprecise, and the log-ratio distribution cannot be assessed reliably.
Separating Exposure and Potency in the Effect Model
A concentration-effect comparison asks whether participants have different expected effects at the same index-drug concentration in the two conditions. A simple paired model can use a shared \(E_{\max}\) and estimate separate \(EC_{50}\) values, while allowing for participant-level differences when the data support them. A larger \(EC_{50}\) means that more concentration is required to achieve a given fraction of \(E_{\max}\), so it represents a rightward shift and lower potency under this model.
Worked Example: Interpreting a Potency Shift
Suppose a hypothetical concentration-effect analysis estimates \(E_{\max}=100\) effect units in both conditions. Under index drug alone, \(EC_{50}=4\) mg/L; with the interacting drug, \(EC_{50}=8\) mg/L. To see what the estimates imply at matched concentrations, evaluate the Emax equation at \(C=4\) mg/L.
For the alone condition, \(E=100(4)/(4+4)=50.0\) effect units. For the combination condition, \(E=100(4)/(8+4)=33.33\) effect units. At the same concentration, the model therefore predicts 16.67 fewer effect units in the combination condition.
The estimated \(EC_{50}\) ratio is \(8/4=2.00\). In this model, the combination condition requires twice the concentration to reach half of the shared maximum effect. This is evidence of a potency shift only if the curve estimates are supported by the data, concentration and effect are properly aligned, and direct effects or time-related differences have not been misattributed to potency.
The distinction can be expressed through a useful algebraic result. Suppose the combination changes concentration by a multiplicative factor \(r\) at a fixed regimen and changes \(EC_{50}\) by a factor \(q\), while \(E_{\max}\) remains the same. The combination effect at the regimen is:
The last equality follows by dividing the numerator and denominator of the middle expression by \(q\). Thus the two changes act through the net concentration-to-potency factor \(r/q\). If \(r=q\), the model predicts no net shift in effect at that regimen; if \(r>q\), the net factor is above 1; and if \(r<q\), it is below 1. This result depends on the shared-\(E_{\max}\), Hill-1 model. It does not imply that every interaction can be reduced to one ratio.
Quantifying the Combined Clinical Consequence
Worked Example: Predicting the Net Effect at a Regimen
Continue with the hypothetical potency estimates above. Suppose the index-drug regimen produces a representative concentration of 4 mg/L alone, while the exposure interaction raises that concentration by a factor \(r=1.50\) under coadministration. The combination concentration is \(1.50(4)=6\) mg/L. The potency ratio is \(q=8/4=2.00\).
At the alone regimen, the predicted effect is \(100(4)/(4+4)=50.0\) units. With coadministration, use both the increased concentration and the shifted \(EC_{50}\): \(E=100(6)/(8+6)=42.86\) units. The combined interaction therefore predicts a decrease of \(50.0-42.86=7.14\) effect units at this regimen, despite the 50% increase in concentration.
As a check, the net factor is \(r/q=1.50/2.00=0.750\), so the interaction is equivalent in this model to reducing the concentration from 4 to \(0.750(4)=3\) mg/L on the alone-condition curve. That curve predicts \(100(3)/(4+3)=42.86\) units, matching the direct calculation. If the potency shift were ignored, the predicted effect at 6 mg/L would instead be \(100(6)/(4+6)=60.0\) units. That exposure-only prediction has the wrong direction relative to the combined model.
This is a model-based consequence for the selected regimen and endpoint, not proof of a clinical outcome. Whether a 7.14-unit change matters depends on the endpoint’s clinical interpretation, uncertainty in the PK and PD estimates, and the relationship between this endpoint and patient benefit or harm.
Diagnostics, Alternatives, and Limitations
Before interpreting the estimates, inspect the data and model behavior. For the exposure analysis, check individual paired values, missing periods, implausible AUCs, and whether log ratios are strongly skewed or dominated by one participant. For the PD analysis, plot effect against concentration by condition and participant, inspect residual patterns, and check whether both conditions cover enough of the concentration range to identify the curve. A condition-specific residual trend, systematic time pattern, or apparent hysteresis suggests that the static concentration-effect model may be inadequate.
Important failure modes include carryover, period effects, nonadherence, inaccurate concentration measurements, and an unmodeled direct effect of the interacting drug. A particularly serious problem occurs when the combination condition has concentrations only in a range not observed under the alone condition: comparing the curves then relies on extrapolation rather than a direct matched-concentration contrast. Uncertainty in the estimated exposure ratio, \(EC_{50}\) ratio, and predicted effect should also be carried into the final consequence estimate. Reporting point estimates alone can make a fragile result appear definitive.
A simpler alternative is to compare response by treatment condition without conditioning on concentration. That estimates a regimen-level contrast in the studied design, but it cannot separate exposure from potency. An exposure-only explanation uses measured PK change and assumes the concentration-effect curve is unchanged; it is appropriate only if that assumption is supported. A joint PK/PD model can estimate exposure and effect parameters together and propagate their uncertainty, but it requires stronger modeling choices and adequate data. The paired summaries and concentration-effect comparison taught here are easier to inspect, but can be less efficient and may not adequately represent nonlinear individual differences.
The main limitation is that the decomposition is conditional on the chosen exposure measure, effect model, endpoint, and sampling schedule. AUC may not capture the exposure feature driving response; peak concentration, trough, or time above a threshold may matter instead. A change in \(E_{\max}\), a different Hill coefficient, delayed response, active metabolites, or a direct effect of the interacting drug can invalidate the simple \(r/q\) summary. In those cases, the model should be expanded only when the design and data can identify the added parameters.
A Practical Analysis Sequence
Confirm the two conditions, sequence, washout, timing, adherence, and availability of paired observations.
Calculate participant-level exposure ratios, analyze log ratios, and report the exponentiated estimate with an interval and units.
Compare concentration-effect relationships at matched concentrations, accounting for pairing and checking whether the chosen model fits.
Apply the exposure and potency changes together at the regimen of interest, then report the predicted endpoint change and its assumptions.
Common Mistake
Wrong reasoning: “AUC increased by 50%, so the drug effect must increase by 50%.” This confuses exposure with effect and assumes a linear, unchanged concentration-effect relationship. In a saturable model, a concentration increase does not translate proportionally into effect, and a potency shift can offset or reverse the exposure change.
Correction: Estimate the exposure change and the conditional concentration-effect change separately. Then calculate the effect under each condition at the regimen of interest. State the endpoint units and acknowledge uncertainty; do not interpret a PK ratio as a clinical effect ratio.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If the geometric mean AUC ratio is 0.80, what does that say about exposure in the combination condition relative to the alone condition?
- Under a shared-\(E_{\max}\) model, \(EC_{50}\) rises from 5 to 10 mg/L. At \(C=5\) mg/L, calculate the predicted effect as a fraction of \(E_{\max}\) in each condition.
- If exposure changes by \(r=1.20\) and potency changes by \(q=1.20\), what net factor \(r/q\) does the Hill-1 model imply?
- Name two reasons a same-time plasma concentration may not be suitable for estimating a concentration-effect relationship.
- Why can a treatment-versus-treatment response contrast fail to identify whether an interaction is pharmacokinetic or pharmacodynamic?
Solutions. (1) The combination condition has an estimated geometric mean AUC 20% lower than the alone condition, since \(0.80-1=-0.20\). (2) Alone, \(E/E_{\max}=5/(5+5)=0.500\); combination, \(E/E_{\max}=5/(10+5)=0.333\). At the same concentration, the shifted curve predicts a smaller fraction of the maximum. (3) \(r/q=1.20/1.20=1.00\), so the model predicts no net concentration-to-potency shift at a fixed regimen, assuming the shared-\(E_{\max}\), Hill-1 model. (4) The response may lag plasma concentration, creating hysteresis, or the relevant effect-site concentration may differ from plasma concentration; poorly timed sampling is another concern. (5) That contrast combines all ways the conditions affect response. Without concentration information and a conditional concentration-effect analysis, exposure changes and potency changes are not separately identified.