Why PK Misspecification Matters for PD
A PD model may be written in terms of a concentration that is not observed at every response measurement. The PK model supplies that concentration or exposure. If the PK structure is wrong—for example, it omits a distribution phase or absorption delay—the resulting exposure predictions can be systematically wrong even when the PK parameters have been estimated precisely. A PD model fitted to those predictions can then attribute PK error to PD parameters.
Sequential fitting first estimates PK parameters from PK data and then treats the resulting exposures as fixed inputs to the PD fit. Joint fitting estimates PK and PD parameters together from their likelihood contributions. Joint estimation can use PD information to alter PK estimates, but it does not make an incorrect PK structure correct. Both approaches can produce biased estimates when the candidate model cannot represent the data-generating process.
Let \(C_{\mathrm{true}}(t)\) denote the concentration relevant to the response at time \(t\), and let \(C_{\mathrm{fit}}(t)\) be the concentration supplied by the fitted PK model. These symbols distinguish the conceptual generating exposure from a model prediction; in practice, \(C_{\mathrm{true}}(t)\) is usually not directly known. A PD model might relate response \(Y(t)\) to concentration through a function \(g\), such as \(Y(t)=g(C(t),\boldsymbol{\beta})+\epsilon(t)\), where \(\boldsymbol{\beta}\) denotes PD parameters and \(\epsilon(t)\) is response error.
How Exposure Error Transfers to a PD Parameter
A simple linear relationship makes the transfer of bias explicit. Suppose the true mean response is \(E_0+\beta C_{\mathrm{true}}\), where \(E_0\) is baseline response and \(\beta\) is the change in response per concentration unit. Suppose the fitted PK model supplies \(C_{\mathrm{fit}}=a+bC_{\mathrm{true}}\), with \(b\ne0\). If the PD fit includes an intercept, the same mean response can be written as
Thus, under this exact affine exposure relation, the fitted PD slope is \(\beta/b\), and the intercept absorbs the additive offset. This is a model-specific result, not a universal correction: with nonlinear PD functions, time-varying exposure error, no free intercept, or random effects, the bias need not follow this formula. It does show why a precise PD fit can still estimate the wrong parameter when its exposure input is biased.
Worked Example: Multiplicative Exposure Underprediction
Consider hypothetical observations with concentrations in mg/L. Assume the true mean response is \(50+4C_{\mathrm{true}}\) response units, while a misspecified PK model supplies \(C_{\mathrm{fit}}=0.8C_{\mathrm{true}}\). There is no response noise in this illustration, and the PD fit includes an intercept.
| \(C_{\mathrm{true}}\) (mg/L) | \(C_{\mathrm{fit}}\) (mg/L) | Mean response (units) |
|---|---|---|
| 1 | 0.8 | 54 |
| 2 | 1.6 | 58 |
| 4 | 3.2 | 66 |
The true slope is \(4\) response units per mg/L. Using the fitted concentrations, the slope between the first two points is \((58-54)/(1.6-0.8)=4/0.8=5\) response units per mg/L. The other pairs give the same slope. The fitted intercept is \(54-5(0.8)=50\) response units. Equivalently, the affine-bias result gives \(\beta/b=4/0.8=5\). The fitted slope is larger because each unit of supplied concentration represents only \(0.8\) mg/L on the true scale.
This example assumes a constant multiplicative error and a correctly specified linear PD relationship. If the PK error instead changes over time or concentration, the PD slope may not simply be rescaled; the PD fit may also show systematic residual structure.
What Sequential and Joint Fitting Target
In a sequential fit, the PK stage estimates parameters from PK observations, then supplies fitted concentrations to the PD stage. Conditional on those supplied concentrations, the PD stage estimates its parameters as if the inputs were fixed. This procedure relies on more than accurate optimization: it requires the supplied exposure trajectory to be adequate for the PD relationship. Accounting for uncertainty in fitted PK parameters can address uncertainty transfer, but it cannot by itself correct a systematically wrong exposure trajectory.
In a joint fit, the PK and PD likelihood contributions are optimized together. Under a correctly specified model and appropriate likelihood assumptions, the joint likelihood can use both data sources to estimate shared parameters. Under misspecification, however, it instead finds a best compromise within the candidate family. In general, the limiting estimate targets the parameter values that optimize the expected log likelihood under the true data-generating process, not necessarily the scientifically true values. If the PK structure is wrong, the joint fit may alter PK parameters to improve the PD fit, alter PD parameters to accommodate the PK fit, or do both.
Worked Comparison of Exposure-Ordered Residuals
Residuals can help reveal that a PD fit is not capturing its mean pattern. For a response observation \(Y_i\) and fitted response \(\hat{Y}_i\), define the residual as \(r_i=Y_i-\hat{Y}_i\). A residual pattern may be consistent with exposure error, but it is not unique evidence of it: PD-model misspecification, covariates, timing errors, or other data features can also produce structure.
Worked Example: Curvature in a Response-versus-Exposure Fit
Suppose a hypothetical linear PD model is fitted to four paired values of model-supplied exposure and response. Exposure is in mg/L and response is in response units. The table reports rounded observations; the displayed regression summaries and residuals are calculated from the corresponding unrounded values.
| Exposure, descending order (mg/L) | Observed response (units) |
|---|---|
| 5.963 | 73.839 |
| 3.175 | 60.572 |
| 2.704 | 59.092 |
| 0.858 | 51.497 |
For these unrounded hypothetical data, \(\bar{x}=3.175\), \(\bar{y}=61.25\), \(S_{xx}=\sum_i(x_i-\bar{x})^2=13.3675\), and \(S_{xy}=\sum_i(x_i-\bar{x})(y_i-\bar{y})=58.725\). The ordinary least-squares slope and intercept are
The fitted line is therefore approximately \(\hat{Y}=47.3019+4.3931x\), with slope units of response units per mg/L. Subtracting the fitted response from each observed response, in descending exposure order, gives residuals of approximately \(0.339,-0.678,-0.088,\) and \(0.426\) response units. The signs are positive at the high and low ends and negative at the two middle exposures, a pattern compatible with curvature rather than random scatter around a straight-line mean.
This small example does not establish that PK misspecification caused the curvature. It establishes only that this linear response-versus-supplied-exposure fit leaves a structured residual pattern in these data. A misspecified concentration-time shape could contribute if observations at particular exposures or times are systematically misrepresented, but a nonlinear PD relationship could produce a similar pattern. The next step is to inspect the exposure and response predictions in their time and design context rather than assign a unique cause from four residuals.
Why a Joint Fit May Move the Estimates
The following simplified calculation illustrates the compromise a joint fit can make. It is not a full nonlinear PK/PD analysis. Let \(q\) be a scale parameter in a misspecified PK family: changing \(q\) changes predicted exposure magnitude, but cannot repair an incorrect concentration-time shape. Suppose, locally, independent PK and PD information can be summarized by estimates \(0.80\) and \(1.00\), with standard errors \(0.10\) and \(0.20\), respectively. Assume both likelihood contributions are approximately quadratic, use the same \(q\), and that the PD information is interpretable for \(q\) because the relevant PD parameters are otherwise known or identified.
Worked Example: A Quadratic Joint-Likelihood Compromise
Under the stated approximation, the joint objective to minimize, up to a constant factor, is
Differentiating and setting the derivative to zero gives
The joint estimate \(0.84\) lies between the PK-only value \(0.80\) and the PD-informed value \(1.00\). The PK contribution has greater weight because its assumed variance is smaller. This movement can improve the combined likelihood, yet the model's concentration-time shape remains misspecified. The calculation therefore illustrates reallocation of fit, not a guarantee that joint estimation recovers the true exposure or an unbiased PD parameter.
Assumptions, Comparisons, and Failure Modes
Interpretation depends on which assumptions hold. A sequential PD estimate is most straightforward when the PK model describes the relevant concentration profile, the stage-one estimates are suitable for the target population, and the PD model is correctly specified conditional on exposure. A joint estimate additionally depends on a coherent joint likelihood and a parameterization capable of representing the important PK and PD features. Both analyses require attention to dependence, random effects, observation error, and design; the simplified calculations above deliberately omit those complexities.
| Feature | Sequential fit | Joint fit |
|---|---|---|
| PK parameters during PD estimation | Fixed at stage-one estimates | Estimated with PK and PD contributions together |
| Response to PD evidence about exposure | Cannot revise the supplied PK estimates | May shift shared PK parameters, if the model allows it |
| Effect of wrong PK structure | Biased exposure can be absorbed by PD parameters | Mismatch can be redistributed across fitted parameters |
| Structural correction | Not achieved by treating estimates as fixed | Not achieved merely by fitting jointly |
A useful warning sign is a coherent residual pattern against fitted exposure, time, dose, or another relevant predictor, especially when the pattern corresponds to a plausible omitted feature of the concentration profile. Another warning is marked movement in PK or PD estimates when the PD likelihood is included. Such movement may reflect useful information, but under misspecification it may also reflect compensation. A good overall likelihood or a visually close response fit is not proof that the individual PK and PD components are scientifically adequate.
A common mistake is to argue, “The joint fit uses more data, so its PD estimate must be less biased.” More information can improve precision when the model is adequate; it does not guarantee accuracy when a structural assumption is false. The correction is to separate two questions: whether the fitting strategy uses information efficiently, and whether the candidate model represents the relevant processes. Compare the strategies under a plausible model, inspect exposure and response patterns, and consider whether a scientifically justified structural alternative is needed. A more flexible model is not automatically preferable if its parameters cannot be identified by the available design.
The central limitation is that the true exposure is typically unobserved between PK samples. Residual patterns therefore provide clues, not direct proof, of exposure bias. Sparse or poorly timed samples may leave several PK structures compatible with the observations, while a flexible PD model may conceal some consequences of PK error. A strategy comparison is meaningful only with the structural assumptions stated clearly; neither a sequential nor a joint estimate should be interpreted independently of those assumptions.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If the true PD slope is \(6\) response units per mg/L and the supplied exposure is \(0.75\) times the true exposure, what slope is expected from an intercept-containing linear PD fit under the exact multiplicative-bias assumptions?
- Why does a joint fit not guarantee removal of PK structural bias?
- Residuals are positive at low and high exposures and negative in the middle. What can be concluded, and what cannot be concluded from that pattern alone?
- In the quadratic illustration, why does the joint estimate lie closer to the PK-only estimate than to the PD-informed estimate?
Solutions. (1) The fitted slope is \(\beta/b=6/0.75=8\) response units per mg/L; the result assumes exact multiplicative exposure bias and a freely fitted intercept. (2) The joint fit optimizes within its candidate model family. If that family cannot represent the true concentration-time behavior, joint optimization can shift parameters to compromise but cannot add the missing structure. (3) The pattern suggests the fitted linear mean may be inadequate, but it does not prove PK misspecification; nonlinear PD behavior and other factors can create similar residuals. (4) The PK information has variance \(0.10^2=0.01\), compared with \(0.20^2=0.04\) for the PD-informed information, so its inverse-variance weight is larger and the combined estimate, \(0.84\), is nearer \(0.80\).