Why the Source of Uncertainty Matters
A PBPK input can be uncertain for different reasons. An assay may produce slightly different readings when the same sample is measured repeatedly. Real individuals may also differ in a physiological or drug-related characteristic. Separately, a value may be poorly known because evidence is sparse or methods disagree. These are not interchangeable explanations: they refer to different features of the data and have different implications for how an input should be interpreted.
For example, repeated measurements of the same plasma sample can reveal assay repeatability, but they do not describe the range of plasma binding across a patient population. Conversely, a wide spread of measurements across patients does not by itself show that the assay is inaccurate. And when two sources give different values for a tissue partition input, that disagreement does not establish that patients vary by the same amount.
The key question is therefore not just “How uncertain is this input?” It is also “What is uncertain, and why?” A PBPK model may target a typical individual, a defined population, or a particular patient. The relevance of biological variability depends on that target. Measurement error and knowledge uncertainty, in contrast, concern how well the biological quantities or model inputs are known.
A Measurement Model for Repeated Observations
A simple model makes the distinction precise. Suppose \(Y_{ij}\) is the \(j\)th assay result for individual \(i\). Write
Here, \(\mu\) is the population mean of the biological quantity, \(B_i\) is individual \(i\)'s deviation from that mean, and \(E_{ij}\) is the measurement error for that assay result. Assume, for this model, that the \(B_i\) values have mean zero and variance \(\sigma_B^2\); the \(E_{ij}\) values have mean zero and variance \(\sigma_E^2\); and measurement errors are independent of each other and of the biological deviations. The model does not require the errors to be normally distributed for the variance result below, although normality may be adopted for some inferential procedures.
Under these assumptions, the variation in results across individuals and measurements has two components. Conditional on an individual, the only variation across replicate assays is measurement error. Across individuals, results vary both because individuals differ biologically and because their results contain measurement error.
The first result follows from the law of total variance. Let \(I\) denote the individual measured. Within an individual, \(\operatorname{Var}(Y_{ij}\mid I)=\sigma_E^2\), while the conditional mean is \(\operatorname{E}(Y_{ij}\mid I)=\mu+B_i\). Therefore,
For the average of \(r\) replicate results, the individual component \(B_i\) is shared by every replicate, but the independent error terms are averaged. The variance of their average is \(\sigma_E^2/r\), giving \(\operatorname{Var}(\bar{Y}_i)=\sigma_B^2+\sigma_E^2/r\). This argument also shows why repeated assays on one person cannot estimate between-person variation: they all share the same \(B_i\).
The model represents random error, not every possible defect in a measurement process. A calibration error that shifts all readings in one direction is systematic bias; repeating the same biased procedure may give highly consistent results without correcting the bias. Drift over time, sample degradation, or a method that measures a different quantity than intended can also undermine interpretation.
Worked Example: Estimating Repeatability for a Binding Assay
In a hypothetical laboratory exercise, the same plasma sample is assayed five times for the unbound fraction, \(f_{u,p}\). The results are 0.897, 0.903, 0.900, 0.906, and 0.894. Assume the sample is stable during the run and the replicates are independent measurements of the same underlying sample value. Calculate the mean and sample standard deviation.
The mean is
The sum of squared deviations from the mean is \(0.000009+0.000009+0+0.000036+0.000036=0.000090\). Thus,
In this hypothetical run, the replicate results have a sample standard deviation of about 0.00474 in unbound-fraction units. This describes repeatability for this sample under these assay conditions. It is not an estimate of biological variation in \(f_{u,p}\) across patients, and it does not rule out a shared calibration bias.
Separating Assay Error from Between-Subject Variation
To learn about biological variation, measurements must include multiple individuals. Replicates within each individual are useful because they provide information about assay error, which can otherwise inflate the apparent spread between individuals. The simple random-effects model provides a way to make this separation when the same assay protocol is applied consistently and the assumptions are reasonable.
Worked Example: Comparing Donors After Replicate Assays
In a hypothetical study, \(f_{u,p}\) is measured twice for each of four donors. The results are: donor 1, 0.076 and 0.084; donor 2, 0.096 and 0.104; donor 3, 0.116 and 0.124; donor 4, 0.136 and 0.144. Assume donors are sampled from a relevant population, replicate errors are independent and unbiased, and the assay procedure is the same across donors.
The donor means are 0.080, 0.100, 0.120, and 0.140. Their mean is \(\bar{x}=0.110\). The sample variance across donor means is
The within-donor deviations from each donor mean are \(-0.004\) and \(0.004\) for every donor. The sum of their squares across the four donors is \(8(0.004^2)=0.000128\). With \(4(2-1)=4\) within-donor degrees of freedom, the pooled within-donor variance is
With two replicates per donor, the replicate-error contribution to the variance of donor means is estimated as \(0.000032/2=0.000016\). Subtracting it gives an estimated between-donor variance:
The corresponding estimated between-donor standard deviation is \(\sqrt{0.0006507}=0.02551\). This is an estimate of biological spread under the model, not a precise population constant: only four donors were measured, so the estimate itself is uncertain. The calculation also depends on the assumptions that the donors are relevant to the target population and that the assay errors behave similarly across donors. If the variance subtraction were negative, that would not imply negative biological variance; it would indicate that sampling noise or the model assumptions prevent a positive estimate from these data.
A further source of uncertainty arises because the population values, assay properties, and model inputs are not known perfectly. For instance, a PBPK model may require a tissue-to-blood partition coefficient, but available evidence may be sparse or method-dependent. The modeler must distinguish uncertainty about which input value is appropriate from genuine variation in that coefficient across individuals.
Worked Example: Conflicting Estimates of a Tissue Input
Suppose two hypothetical experimental reports give a tissue partition coefficient \(K_p\) of 2.2 and 2.8, respectively. They used different methods, and neither provides evidence that the difference reflects variation among patients. For illustration, consider a fixed blood concentration of 1.5 mg/L and the simple relation \(C_{\text{tissue}}=K_p C_{\text{blood}}\).
Using the first estimate gives \(C_{\text{tissue}}=2.2(1.5)=3.30\) mg/L. Using the second gives \(C_{\text{tissue}}=2.8(1.5)=4.20\) mg/L. The two input choices therefore produce a 0.90 mg/L difference in this calculated tissue concentration.
That difference illustrates how uncertainty in the evidence for \(K_p\) affects a model calculation. It does not establish a 2.2-to-2.8 range of patient-to-patient values, nor does it establish that either report is unbiased. Choosing a value or representing the uncertainty requires additional judgment about the methods and evidence; simply averaging the two values would not, by itself, resolve their disagreement.
Classification, Diagnostics, and Common Pitfalls
When an input looks variable or poorly determined, classify the source by asking what was actually observed and what inference is being made. Replicate assays of a stable sample primarily inform measurement repeatability. Measurements across individuals can inform biological variation, but the observed spread may also include assay error, differences in sampling conditions, and population sampling uncertainty. Disagreement among studies may reflect methodological differences, population differences, systematic bias, or limited knowledge; the disagreement alone does not identify which explanation is correct.
| Source | What varies or is uncertain? | Useful evidence | What it does not establish by itself |
|---|---|---|---|
| Random measurement error | Repeated readings of the same target under the same procedure | Replicate assays, repeatability checks, calibration records | Variation among individuals or absence of systematic bias |
| Biological variability | The actual biological quantity across individuals or occasions | Repeated measurements across appropriately sampled individuals | That observed differences are free of assay error or sampling effects |
| Knowledge uncertainty | Which value or description is best supported by limited evidence | Method comparisons, validation data, replicated studies, evidence review | A distribution of actual values across individuals |
Useful diagnostics include examining replicate differences for trends with run order, checking standards and controls for shifts, and comparing within-individual variation with between-individual variation. A large difference between runs may point to drift or method changes rather than random assay noise. A wide patient spread that is also present in replicate means may be consistent with biological heterogeneity, but it still requires scrutiny for differences in sample handling or measurement conditions. If reported values come from unlike methods, treating their spread as a single biological standard deviation is not justified without further evidence.
A common mistake is to put every uncertain PBPK input into a patient-level variability distribution. The reasoning might be: “The exact value is unknown, so different patients must have different values.” That conclusion does not follow. Limited knowledge about a parameter is not evidence that the parameter differs among individuals. The correction is to identify whether data support actual individual-level variation, measurement error, uncertainty about a shared or typical value, or some combination. Keep the interpretation tied to the evidence and the model’s target.
The main alternatives answer different questions. Repeated measurement of one sample is appropriate for assessing repeatability but cannot characterize population variability. Cross-sectional measurements in many individuals can describe observed population spread but need replication or other assay information to separate that spread from measurement error. Comparing independent studies can reveal unresolved knowledge uncertainty, but it is not a substitute for a representative individual-level sample. In practice, a PBPK analysis may need to acknowledge more than one source; this classification is a prerequisite for choosing how each source is handled.
Check Your Understanding
Classify the main source of uncertainty in each situation and check the solution reasoning below.
- A laboratory measures one stable liver microsome preparation six times and obtains slightly different intrinsic-clearance readings. What source is primarily being assessed?
- A study measures the same physiological input in many people using a validated, consistent protocol and finds substantial differences. What source may this spread reflect, and what other component should still be considered?
- Two studies report different values for a tissue partition coefficient, but neither reports individual-level data. Does their disagreement demonstrate interindividual variability?
- Why can repeated assays on one person not, by themselves, estimate between-person biological variation?
Solutions: (1) The primary target is random measurement error or assay repeatability, assuming the preparation is stable and the procedure is consistent; systematic bias is not ruled out. (2) The differences may reflect biological variability across people, but measurement error and sampling or protocol differences should also be considered. (3) No. The disagreement indicates unresolved differences in evidence or methods; it does not establish patient-to-patient variation. (4) All replicate results share that person’s underlying biological value, so replication informs within-person measurement error but supplies no comparison with other individuals.