Why Separate Concentration Changes from Response Changes?
A combination can change the measured response because one drug changes the concentration of the other, because the response to a given concentration changes, or because both occur. These explanations are not interchangeable. A response that looks unusually large at the administered doses may be fully explained by increased exposure; it is not, by itself, evidence that the drugs interact pharmacodynamically.
The distinction is useful when developing or interpreting combination models. Concentration data inform how dose becomes exposure, while response data inform how exposure becomes effect. Separating those links makes it possible to ask a focused question: if the combination had produced the same relevant concentrations as the drugs given separately, would its response still differ?
These definitions require care about what counts as relevant exposure. Depending on the mechanism, it may be the unbound concentration in plasma, a tissue concentration, an active metabolite, or a concentration at an effect site. A change in total plasma concentration does not necessarily imply a change in unbound target-site exposure; conversely, unchanged plasma concentrations do not rule out altered tissue exposure.
A Model-Based Separation
Let \(C_A(t)\) and \(C_B(t)\) denote the relevant concentration profiles of drugs A and B, and let \(Y(t)\) denote a response. A model separates the dose-to-concentration relationship from the concentration-to-response relationship:
The first link is the PK model; the second is the PD model. A PK interaction changes the first link under combination dosing. A PD interaction changes the second link relative to a defined reference, conditional on the concentrations supplied to it.
For example, suppose the response is a fractional inhibition \(I\), and the single-agent concentration-response functions are \(I_A(C_A)\) and \(I_B(C_B)\). Under a Bliss-independence reference, the predicted remaining response fraction is the product of the two single-agent remaining fractions:
Here \(R_{\mathrm{ref}}\) is a no-PD-interaction prediction at specified concentrations. It is not automatically a no-interaction prediction at specified doses: if combination dosing changes concentrations, the reference must use those changed concentrations to assess the PD component. Other reference models may be appropriate for other mechanisms or endpoints; the choice must be stated before interpreting a deviation as PD interaction.
Worked Examples
Worked Example: A PK Interaction That Mimics PD Synergy
Hypothetical data. At the measurement time, the single-agent concentration-response relationships are \(I_A(C_A)=C_A/(2+C_A)\) and \(I_B(C_B)=C_B/(3+C_B)\), with concentrations in mg/L. Drug A alone gives \(C_A=2\) mg/L; drug B alone gives \(C_B=3\) mg/L. In combination, A remains at 2 mg/L, while B rises to 6 mg/L. The observed combination response is 83.33% inhibition.
At the single-agent concentrations, \(I_A(2)=2/(2+2)=0.5000\) and \(I_B(3)=3/(3+3)=0.5000\). A dose-based comparison that ignores the concentration change would predict, under Bliss independence, \(1-(1-0.5000)(1-0.5000)=0.7500\), or 75.00% inhibition.
But combination dosing raised B concentration to 6 mg/L. At that concentration, \(I_B(6)=6/(3+6)=0.6667\). The exposure-matched reference therefore predicts remaining response \(R_{\mathrm{ref}}=(1-0.5000)(1-0.6667)=0.1667\), or inhibition \(1-0.1667=0.8333\), which is 83.33%.
The observed inhibition matches the exposure-matched prediction. In this example, the apparent excess over the dose-based 75.00% prediction is explained by increased B exposure; there is no additional PD deviation under the chosen reference. The concentration change is consistent with a PK interaction, although identifying its specific cause would require further mechanistic evidence.
Worked Example: Detecting a PD Deviation at Matched Exposure
Hypothetical data. Use the same single-agent functions. In a separate experiment, the combination produces \(C_A=2\) mg/L and \(C_B=3\) mg/L, matching the single-agent concentrations. The measured combination inhibition is 85.00%.
At these matched concentrations, the single-agent inhibitions are each 0.5000. The Bliss reference predicts remaining response \(R_{\mathrm{ref}}=(1-0.5000)(1-0.5000)=0.2500\), equivalent to 75.00% inhibition.
The observed remaining response is \(1-0.8500=0.1500\), which is 0.1000 lower than the reference. Thus, at the measured concentrations and on this response scale, the combination shows 10.00 percentage points more inhibition than the Bliss prediction. Because the relevant concentrations are matched, this difference cannot be explained by a measured concentration increase in this example; it is a PD deviation relative to the stated reference.
That conclusion is conditional. It depends on the concentration measurements representing the effect-driving exposure, on the response being compared at the same time and disease state, and on Bliss independence being an appropriate reference for the scientific question. It does not establish a molecular mechanism or show that the combination is superior in a clinical setting.
Counterfactual Simulations
A counterfactual simulation asks what the model predicts under a deliberately specified alternative. To separate mechanisms, define the scenarios before comparing outputs:
- Reference scenario: use the concentrations predicted for the drugs administered separately and the selected no-PD-interaction reference.
- PK-only scenario: use the concentration profiles predicted or measured under combination dosing, but pass them through the reference PD relationship.
- Full-combination scenario: use combination concentration profiles and the combination PD relationship needed to reproduce the observed response.
The PK-only scenario isolates the response change predicted from altered exposure while holding the PD rule fixed. A difference between the full-combination and PK-only predictions is evidence for an additional PD component within the model. This is a model-based attribution, not an assumption-free causal decomposition.
Worked Example: Decomposing a Response Difference
Hypothetical data. Suppose the reference scenario has \(C_A=2\) mg/L and \(C_B=3\) mg/L. Combination dosing instead gives \(C_A=2\) mg/L and \(C_B=6\) mg/L. The observed combination response is 85.00% inhibition. Use the single-agent functions and Bliss reference from the previous examples.
The reference scenario predicts remaining response \(R_{\mathrm{ref}}=(1-0.5000)(1-0.5000)=0.2500\), or 75.00% inhibition. The PK-only scenario uses the combination concentrations but the unchanged reference PD model: \(R_{\mathrm{PK}}=(1-0.5000)(1-0.6667)=0.1667\), or 83.33% inhibition.
The full-combination response has remaining fraction \(R_{\mathrm{full}}=1-0.8500=0.1500\). On the inhibition scale, the total difference from reference is \(85.00-75.00=10.00\) percentage points. The PK-only prediction accounts for \(83.33-75.00=8.33\) percentage points. The additional difference between the observed response and the PK-only prediction is \(85.00-83.33=1.67\) percentage points.
On the remaining-response scale, the same accounting is \(0.1500-0.2500=(0.1667-0.2500)+(0.1500-0.1667)\), or \(-0.1000=-0.0833-0.0167\), up to rounding. The signs reverse because lower remaining response means greater inhibition. This decomposition is exact on the chosen scale, but the size assigned to each component can depend on the response scale and on which counterfactual is used as the intermediate scenario.
Derivation and Conditions for Interpretation
The separation follows directly from evaluating the PD model twice. Write \(Y_{\mathrm{ref}}\) for the reference response at reference concentrations, \(Y_{\mathrm{PK}}\) for the reference PD response at combination concentrations, and \(Y_{\mathrm{full}}\) for the full combination response. Add and subtract \(Y_{\mathrm{PK}}\):
The first term is the change predicted by replacing reference concentrations with combination concentrations while keeping the PD reference fixed. The second is the further difference between the full combination and the PK-only counterfactual at those combination concentrations. The identity is arithmetic; interpreting the two terms as PK and PD contributions requires that the models and counterfactuals adequately represent the relevant biology.
In particular, the separation assumes that relevant concentrations are measured or predicted adequately, that the concentration-response model is appropriate over the observed range, and that the endpoint, time point, and baseline state are comparable across scenarios. If response depends on delayed effect-site equilibration, active metabolites, or a changing disease state, the model must represent those factors. Matching AUC alone is insufficient when peak concentration, duration above a threshold, or timing relative to a changing state affects response.
Diagnostics, Alternatives, and Limitations
| Question | Useful check | Warning sign |
|---|---|---|
| Did combination dosing change exposure? | Compare concentration-time profiles for both drugs and relevant metabolites under comparable regimens. | A single plasma sample or total concentration is treated as a complete measure of active exposure. |
| Does the PD deviation persist at matched exposure? | Compare observed response with the reference evaluated at the measured relevant concentrations. | The comparison matches doses but ignores concentration differences. |
| Is the model adequate? | Check predictions across time, concentration ranges, and relevant response measures. | Systematic residual patterns or response delays suggest omitted dynamics or an unsuitable exposure metric. |
When both concentration and response data are available, joint PK/PD modeling can use them together and propagate concentration uncertainty into response predictions. A simpler exposure-matched comparison can be useful when the data support it. These approaches differ from comparisons based only on administered dose, which cannot by themselves distinguish altered exposure from altered response. The choice of no-interaction reference is also consequential: a deviation from one reference is not automatically a deviation from every plausible mechanistic model.
Several limitations remain. Concentration-response associations do not alone prove the mechanism that produced them. Unmeasured tissue exposure, protein binding, active metabolites, adherence, or time-varying disease status can make apparently matched groups biologically different. If both drugs alter a shared biomarker or feed back on one another’s disposition, a sequential decomposition may assign effects differently depending on the counterfactual order. Report the reference model, exposure definition, response scale, and scenarios used so the attribution can be evaluated.
Common Pitfall
Check Your Understanding
Answer each question, then compare with the solutions.
- If a combination increases the unbound concentration of drug B and its response matches the reference prediction evaluated at that concentration, what type of interaction is supported?
- Why is matching administered dose not sufficient to establish a PD interaction?
- With reference inhibition 75.00%, PK-only predicted inhibition 83.33%, and observed inhibition 85.00%, calculate the total difference from reference and the additional difference beyond PK-only.
- Name two reasons that matching plasma AUC might fail to match the relevant exposure for response.
Solutions. (1) The data support a PK interaction; they show no additional PD deviation under the selected reference. (2) The same dose can produce different concentration profiles, and therefore different responses, when disposition changes. (3) The total difference is \(85.00-75.00=10.00\) percentage points; the additional difference beyond PK-only is \(85.00-83.33=1.67\) percentage points. (4) Response may depend on peak concentration or exposure timing, or on unbound tissue/effect-site concentrations rather than plasma AUC.