Two Different Questions Behind a Spread of PBPK Predictions
A PBPK simulation can produce a spread of predicted exposures for two fundamentally different reasons. The model may use an uncertain population parameter, such as a typical clearance value that is not precisely known. Or it may represent real differences between people, such as variation in body size or enzyme activity. The numerical predictions can look similar in either case, but their interpretation is not: one describes what is not known about a model input, while the other describes how outcomes differ across individuals.
This distinction matters when interpreting a predicted range, comparing a simulated cohort with clinical observations, or deciding whether additional evidence about a parameter would be useful. As established in the earlier material on measurement error, biological variability, and knowledge uncertainty, these sources should be named separately. Here the focus is on how their separation changes the meaning of PBPK outputs.
A parameter can be uncertain without varying from person to person in the model. For example, uncertainty about the population-average intrinsic clearance is not itself a claim that each person has a different population average. Conversely, a model may represent person-to-person differences in clearance around a known population mean. These are distinct features of the model, even when both affect predicted concentrations or AUC.
Let \(\theta\) denote an uncertain population parameter and \(Z\) denote individual-level characteristics. A PBPK output for an individual can be written as \(Y=g(\theta,Z)\). The model’s value of \(\theta\) is shared across individuals in a given parameter scenario; \(Z\) can differ from one individual to another. A distribution assigned to \(\theta\) summarizes knowledge about the parameter, while a distribution for \(Z\) describes the target population.
What the Two Sources Mean for a Simulated Cohort
Suppose a cohort is simulated under one fixed value of \(\theta\). The spread among that cohort’s predictions can reflect interindividual variability in \(Z\), along with any other explicitly modeled individual-level differences. If the simulation is repeated under a different plausible value of \(\theta\), the cohort predictions may shift. That shift represents sensitivity to epistemic uncertainty, not a new set of biological differences between people.
This gives two useful questions to ask of any reported PBPK spread:
- Across whom? Does the spread compare different individuals while holding the population parameter scenario fixed?
- Across what knowledge states? Does it compare predictions under alternative plausible values of an uncertain parameter?
The answers determine whether a range is an individual prediction range, a range of cohort summaries across parameter scenarios, or a combination of both. A single unlabelled range does not tell the reader which one it is.
Worked Example: Uncertainty in a Shared Clearance Parameter
Consider a hypothetical standardized individual for whom the model predicts AUC as \(100/\theta\) mg·h/L. The parameter \(\theta\) is a dimensionless multiplier on a population clearance value. Evidence is insufficient to distinguish two plausible fixed values, \(0.8\) and \(1.2\). For illustration, assign each scenario equal weight. There is no individual variability in this calculation: the same standardized individual is evaluated in both scenarios.
At \(\theta=0.8\), predicted AUC is \(100/0.8=125\) mg·h/L. At \(\theta=1.2\), it is \(100/1.2=83.33\) mg·h/L. The two scenario predictions therefore span 83.33 to 125 mg·h/L.
With the stated equal scenario weights, the average across these two predictions is \((125+83.33)/2=104.2\) mg·h/L. This is a summary of the chosen parameter-uncertainty representation, not the average AUC of a biological population of individuals. The range likewise does not say that individuals in the target population have AUCs spanning those values.
Worked Example: Individual Variation With the Parameter Held Fixed
Now suppose the population parameter is known and held fixed. A hypothetical target population contains two equally common individual profiles, with model-predicted AUCs of 80 and 120 mg·h/L. These different predictions result from individual-level biological characteristics represented by \(Z\); they are not alternative values of the shared population parameter.
The population mean AUC is \((80+120)/2=100\) mg·h/L. Treating these two profiles as the complete two-point population, the population variance is \[ \frac{(80-100)^2+(120-100)^2}{2} =\frac{400+400}{2} =400\;(\text{mg·h/L})^2, \] so the population standard deviation is \(\sqrt{400}=20\) mg·h/L.
Here the 80–120 mg·h/L range describes individual-level variation in the specified hypothetical population under the fixed parameter value. It is not an uncertainty interval for the population mean. If the target population instead consisted of more profiles with different frequencies, its mean and spread would need to be calculated using those frequencies.
Separating the Components in a Combined Prediction
When both sources are represented, let \(E(Y\mid\theta)\) be the mean output across individuals at a fixed parameter value, and let \(\operatorname{Var}(Y\mid\theta)\) be the variance among individuals at that value. If a probability distribution is used to represent uncertainty in \(\theta\), the overall variance under this two-level representation separates into an average within-scenario variance and a variance across scenario means:
The first term summarizes interindividual variability, averaged over the represented parameter uncertainty. The second summarizes how the expected output for a population changes across parameter values. The equation is a bookkeeping decomposition; treating \(\theta\) as a random variable for this calculation does not assert that a fixed population parameter physically changes at random.
Why the Variance Decomposition Holds
The derivation follows by adding and subtracting the conditional mean \(m(\theta)=E(Y\mid\theta)\). Write \(E(Y)\) for the overall mean under the two-level representation. Then
Squaring both sides and taking expectations gives the sum of the two squared terms plus a cross-product. The expected cross-product is zero, because conditional on \(\theta\), the mean of \(Y-m(\theta)\) is zero:
The expectation of the first squared term is \(E_{\theta}[\operatorname{Var}(Y\mid\theta)]\), and the expectation of the second is \(\operatorname{Var}_{\theta}[m(\theta)]\). Adding them yields the stated result. If uncertainty about \(\theta\) is represented only by a range or a set of scenarios without probability weights, this variance decomposition cannot be calculated until weights are specified. The scenarios can still be reported separately without implying a probability distribution.
Worked Example: Decomposing a Combined AUC Spread
For a fully hypothetical calculation, let the individual-level baseline AUC \(Z\) be 80 or 120 mg·h/L with equal frequencies. Let the uncertain shared multiplier \(\theta\) be 0.8 or 1.2, also with equal scenario weights, and let \(Y=\theta Z\). Within each parameter scenario, the two individual profiles remain equally frequent.
At \(\theta=0.8\), outputs are 64 and 96 mg·h/L. The conditional mean is 80 mg·h/L, and the conditional variance is \(\{(64-80)^2+(96-80)^2\}/2=256\;(\text{mg·h/L})^2\).
At \(\theta=1.2\), outputs are 96 and 144 mg·h/L. The conditional mean is 120 mg·h/L, and the conditional variance is \(\{(96-120)^2+(144-120)^2\}/2=576\;(\text{mg·h/L})^2\). The average conditional variance is \((256+576)/2=416\;(\text{mg·h/L})^2\).
The variance across the two equally weighted conditional means, 80 and 120 mg·h/L, is \(\{(80-100)^2+(120-100)^2\}/2=400\;(\text{mg·h/L})^2\). Therefore, the overall variance is \(416+400=816\;(\text{mg·h/L})^2\), and the overall standard deviation is \(\sqrt{816}=28.57\) mg·h/L.
Directly, the four equally weighted outputs are 64, 96, 96, and 144 mg·h/L. Their mean is 100 mg·h/L, and their variance is \(\{1296+16+16+1936\}/4=816\;(\text{mg·h/L})^2\), agreeing with the decomposition. The overall spread combines individual variability and parameter uncertainty; it should not be described as either source alone.
Choosing and Reporting an Analysis
Different summaries answer different questions. A simulation at one fixed, best-supported parameter set can describe modeled interindividual variability conditional on that set, but it does not quantify how uncertain parameter knowledge could move the results. A set of alternative parameter scenarios can show how cohort predictions respond to uncertain inputs, but it does not by itself estimate individual variability. A combined analysis can represent both, provided the two levels are kept identifiable in the design and the report.
| Analysis or summary | What changes? | What the spread can mean |
|---|---|---|
| Individual predictions under one parameter scenario | Individual characteristics | Interindividual variability, conditional on that scenario |
| Cohort means across parameter scenarios | Uncertain population parameter | Parameter-driven uncertainty in the cohort mean |
| Pooled predictions across both levels | Both levels | A combined spread requiring decomposition or clear qualification |
| One prediction at a single parameter set and for one individual | Neither level in the reported result | A conditional point prediction, not a quantified spread |
Before interpreting a prediction distribution, specify the target population, which inputs vary among individuals, which parameters are shared within a scenario, and how alternative parameter values are represented. Report units and say whether each interval or range describes individual outcomes, scenario-dependent cohort summaries, or a combined distribution. If a probability distribution over uncertain parameters is not justified, avoid presenting probability-based summaries of epistemic uncertainty as though their weights were established.
Diagnostics, Pitfalls, and Limits
Several checks can reveal whether the two sources have been mixed. Compare simulations at a fixed parameter scenario: if the individual-level inputs are also fixed, any remaining spread is not interindividual variability generated by those inputs. Next compare cohort summaries across parameter scenarios: changes in those summaries reflect the selected parameter changes, not new individuals. Finally, inspect whether a reported “population range” pools person-level predictions, parameter scenarios, or both. A pooled histogram alone may conceal these distinctions.
- Pitfall: calling every simulated spread biological variability. A wide range obtained by changing a population clearance parameter across runs is not evidence that individual clearances vary by that amount. Hold the parameter scenario fixed when describing the modeled individual distribution.
- Pitfall: treating a parameter-uncertainty range as a population reference range. Alternative values for a shared parameter shift cohort predictions; they do not describe the fraction of people expected to fall within the range.
- Pitfall: reporting a combined standard deviation without qualification. It can combine within-scenario variation and between-scenario uncertainty. State the components or label the summary as combined under the specified representation.
- Pitfall: assigning probabilities to scenarios without justification. Equal weights are an explicit modeling choice, not a default consequence of having two plausible values. If weights are unavailable, present scenario-specific results or a range, not a probability-weighted interpretation.
This separation does not remove uncertainty from a PBPK model, establish that the chosen individual-level inputs represent the target population, or guarantee that all relevant sources have been included. Results depend on the model structure and the chosen representations. In particular, individual variability that is absent from the input model cannot be recovered by labeling parameter uncertainty as variability. Likewise, uncertainty in a shared parameter should not be converted into person-to-person differences merely to produce a wider prediction distribution.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A model predicts AUC for 500 simulated individuals using one shared value of population clearance. Which source does the spread among their predictions represent, if their individual characteristics vary?
- A cohort mean AUC is 90 mg·h/L under one plausible parameter value and 110 mg·h/L under another. What does the difference describe?
- In the combined example, what was the average within-scenario variance, and what was the variance across scenario means?
- Two equally weighted parameter scenarios produce different cohort means, but no probability weights are supported. What is an appropriate way to report their results?
Solutions. (1) The spread represents modeled interindividual variability, conditional on the fixed clearance scenario, assuming the individual-level characteristics are what differ. (2) The 20 mg·h/L difference describes parameter-driven change in the cohort mean across the two scenarios; it is not an individual-level range. (3) The average within-scenario variance was \(416\;(\text{mg·h/L})^2\), and the variance across scenario means was \(400\;(\text{mg·h/L})^2\). (4) Report the two scenario-specific means, or their range of 90–110 mg·h/L, and identify it as a scenario range. Do not imply probability weights or call it an individual prediction range.