Why the Type of Uncertainty Matters
A physiological input in a PBPK model may be represented by one number, such as an organ volume of 150 mL. That number is useful as a reference value, but it does not say how precisely the volume was measured, how much estimates differ across studies, or how much the organ volume varies among people. Those are different features of the evidence. Treating them as one generic “range” can lead to a model input that represents the wrong source of variation.
For example, repeated measurements of one liver estimate how imprecisely the measurement procedure records that liver. Measurements from several people describe biological differences among those people, as well as any measurement error. A set of reported study means describes differences between studies, which may reflect population, protocol, measurement method, or sampling. The numerical spread could look similar in all three cases, but its interpretation is not interchangeable.
These categories can coexist. A study may measure a biologically variable quantity with an imprecise method, and its estimate may differ from other studies. The first task is therefore not to select a distribution. It is to identify what a reported number represents, what source of spread is available, and what quantity the PBPK application needs.
What a Range or Distribution Should Represent
A point estimate is a convenient reference value, not a complete description of uncertainty or variability. A range can summarize observed values or mark a plausible interval, but the record must state which. A minimum and maximum from a small sample are not automatically population limits. Likewise, a range that describes plausible measurement error should not be interpreted as the range of organ volumes across people.
A distribution provides more detail by assigning relative weight to different values. Its form should fit both the evidence and the physiological quantity. A normal distribution can be a reasonable approximation for a positive quantity when its spread is modest relative to its mean and negative values are negligible. A lognormal distribution is often considered for positive, right-skewed quantities. A proportion constrained between zero and one may be represented by a beta distribution. These are modeling choices, not findings guaranteed by the name of the distribution; their parameters and suitability still need justification.
When evidence provides only lower and upper bounds, a uniform distribution assigns equal weight throughout those bounds. That is an additional assumption, not a neutral way to encode “we do not know.” If some values are more plausible than others, a uniform distribution may misrepresent the evidence. A range can be preferable to a distribution when the data do not support a credible probability shape.
Separating Variation in a Simple Measurement Model
A basic model illustrates why repeated readings and measurements across individuals answer different questions. Let \(X_i\) be the underlying physiological value for individual \(i\), and let \(Y_{ij}\) be the \(j\)th recorded measurement of that individual. Suppose
where \(\varepsilon_{ij}\) is measurement error with mean zero, variance \(\sigma^2_{\varepsilon}\), and independence from \(X_i\). Then the variance of a recorded value is
The equality follows by expanding \(\operatorname{Var}(X_i+\varepsilon_{ij})\): it equals the two variances plus twice their covariance, and independence makes that covariance zero. Thus the spread of readings across people combines biological variability and measurement error under these assumptions. Repeated readings on the same individual can help characterize measurement error because \(X_i\) is held fixed in the model; readings across different individuals include both sources.
This simple decomposition does not automatically separate between-study variation. Studies can differ in their target populations, protocols, and measurement bases, so the variation among study estimates is not necessarily a single random error component. Any decomposition that treats study effects as random requires assumptions about how those studies were selected and why they differ. For a parameter record, it is often more defensible to describe the study-to-study spread and its likely sources than to label it pure measurement error.
Worked Example: Repeatability of One Organ Measurement
Suppose, hypothetically, that the same organ in one reference individual is measured three times by the same procedure, giving 118, 122, and 120 mL. Summarize the readings and identify what their spread can represent.
The mean reading is
The sample standard deviation is
The readings average 120 mL, with a sample standard deviation of 2.000 mL. Because all three readings concern the same individual, this spread is informative about repeatability under the stated procedure, subject to assumptions such as unchanged physiology and comparable measurement conditions. It does not estimate biological variability across individuals. A suitable record might therefore list “reference reading: 120 mL; repeatability SD: 2.000 mL; three repeated measurements in one individual,” rather than presenting 118–122 mL as the population range.
Between-Study Differences Are Not Individual Variability
Study-level summaries can be compared descriptively, provided they refer to sufficiently comparable quantities. Before summarizing them, check that the organ, measurement basis, units, target population, and relevant covariates are stated. Even with those details aligned, differences among study means do not directly give the distribution of values among individuals. The earlier material on evaluating sources of reference physiology data is useful for assessing what each reported estimate actually represents.
Worked Example: Summarizing Study-Level Estimates
Imagine three hypothetical studies reporting reference kidney volumes on the same measurement basis, with estimates of 1.80, 2.10, and 2.40 L. Treat these as study-level estimates, not measurements from three individuals. Calculate their mean, sample standard deviation, and observed range.
The mean is
The sample standard deviation is
The observed study-estimate range is 1.80–2.40 L. A transparent summary is therefore “mean of three study estimates: 2.10 L; sample SD across study estimates: 0.3000 L; observed range: 1.80–2.40 L.” The SD describes these three estimates, not individual kidney-volume variation and not measurement repeatability. With only three studies, it is also a limited description of between-study differences, not a well-established distribution for future study estimates.
Representing Biological Variability
When the intended input should describe variation among people, the data need to represent individuals from a defined target population, or otherwise provide a defensible population distribution. The distribution should be conditional on the covariates relevant to the application. For example, a broad range across adults of different body sizes should not be presented as unexplained variation for a single reference adult if the model instead requires a value conditional on body size.
A sample mean and sample standard deviation can summarize observed individual values, but their meaning depends on the sampling and measurement design. A coefficient of variation, \(100s/\bar{x}\), expresses the sample SD as a percentage of the sample mean and can help compare relative spreads for positive quantities with different scales. It does not remove the need to state the units and population context.
Worked Example: Describing Individual Variation in Organ Volume
Suppose, hypothetically, that five individuals from a narrowly defined reference group have measured spleen volumes of 140, 150, 160, 170, and 180 mL. Assume for this illustration that measurement error is small relative to individual differences. Calculate the sample mean, sample SD, and coefficient of variation, then state one possible distributional summary.
The sample mean is
The sample standard deviation is
The coefficient of variation is
If a normal approximation were reasonable for this group, one illustrative model would be \(X\sim N(160,250)\), with volume in mL; the second parameter is the variance, so the SD is \(\sqrt{250}\approx15.81\) mL. Its approximate central 95% interval is \(160\pm1.96(15.81)\), or 129.0–191.0 mL. This interval is a model-based description of individual variability under the normal approximation, not a confidence interval for the population mean. The sample has only five individuals, so it cannot establish that the normal model or its estimated parameters are reliable. The observed sample range is 140–180 mL and should not be confused with the model-based interval.
Recording Inputs and Avoiding Misleading Combinations
A useful physiological parameter record makes uncertainty interpretable without forcing every input into a distribution. At minimum, include the parameter name and definition, units, reference population and covariates, point estimate, type of uncertainty or variability, range or distribution and its parameters, measurement basis, and source or rationale. If the evidence is sparse, say so. An explicitly qualified range is often more honest than a probability distribution with unsupported precision.
Also consider whether inputs are dependent or constrained. Organ volumes, tissue fractions, and flows may be related through shared body size, composition, or physiological constraints. Sampling each input independently from its own marginal distribution can produce combinations that are implausible or inconsistent. Where dependence or a total constraint is relevant, record it and use a joint representation or an appropriate conditional parameterization rather than pretending the inputs vary independently.
A practical recording sequence is:
State the organ or tissue, units, measurement basis, and the population or covariate conditions.
Classify it as measurement uncertainty, between-study variation, biological variability, or a clearly described combination.
Report an observed or plausible range, summary statistics, or a distribution with stated parameters and assumptions.
Confirm that the spread matches the intended population and that related inputs are not represented as independent when they are constrained or associated.
Common Pitfalls and Diagnostics
- Calling every spread “biological variability.” A spread across repeated readings of one individual primarily informs measurement repeatability; a spread across study means describes study differences. Label the evidence by what was measured.
- Using a confidence interval as an individual range. An interval for a population mean concerns uncertainty in that mean, not the range containing most individuals. State whether the interval refers to an estimate, an individual distribution, or observed values.
- Treating observed minima and maxima as physiological limits. Sample extremes depend on sample size and selection. Call them observed sample limits unless there is evidence that they are genuine bounds.
- Assigning a distribution because software needs one. Check whether the chosen distribution respects positivity or bounds and whether its shape is supported. If evidence is inadequate, report the limitation instead of implying certainty.
- Ignoring covariates or dependence. A spread may reflect mixed body sizes or other population differences. Inspect the population definition and ask whether independent sampling could create implausible combinations.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Repeated readings of one organ in one individual vary by 3 mL. What kind of spread do they primarily describe, and what do they not establish?
- Three studies report means of 1.5, 1.8, and 2.1 L. What is their mean, and does their spread directly estimate variation among individuals?
- A sample of individuals has mean organ volume 200 mL and SD 20 mL. What is its coefficient of variation, and what does it summarize?
- Why might a uniform distribution between two reported values be an unjustified representation of uncertainty?
- What should be checked before independently sampling organ volumes and tissue fractions?
Solutions. (1) The readings primarily describe measurement repeatability under that procedure, assuming physiology and conditions stayed stable; they do not establish between-person biological variability. (2) The mean is \((1.5+1.8+2.1)/3=1.8\) L. The spread is among study estimates and does not directly estimate individual variation. (3) The coefficient of variation is \(100(20/200)=10\%\); it summarizes the sample SD relative to its mean for that sampled population. (4) A uniform distribution assumes every value within the bounds is equally plausible, which the bounds alone do not demonstrate. (5) Check whether the inputs share covariates or are constrained by composition or totals; if so, independent sampling may produce implausible combinations.