Why Elicit an Unmeasured Input?
A PBPK model can require an input for which direct measurements are unavailable, sparse, or poorly matched to the population and conditions of interest. Expert judgment can help characterize what is plausible and how uncertain the input is. It does not turn a judgment into a measurement: the result is a documented representation of current knowledge, conditional on how the question was framed.
The first task is to specify the quantity. An expert cannot give a useful range for “tissue binding” until the tissue, species, population, experimental or physiological conditions, units, and exact parameter definition are clear. If those details are not fixed, two different answers may both be reasonable but describe different quantities. Also state whether the uncertainty concerns knowledge about one model input or variation among individuals. Those are distinct sources of uncertainty and should not be combined without a defined purpose.
An elicitation should begin with existing evidence, including indirect measurements and physiological or physicochemical constraints. The expert’s role is to interpret what that evidence implies for the specific input, identify gaps, and express uncertainty. A numerical response is meaningful only when the expert understands whether the question asks for a typical value, a physical limit, or a probability statement.
Frame the Quantity and the Questions
Before asking for numbers, write a short target specification. It should state the parameter name and definition, unit, tissue or compartment, target species and population, reference conditions, and any fixed model assumptions that affect its interpretation. If the intended input is a value for a representative adult, say so; do not leave an expert to answer for a different population or for variability across individuals.
Then elicit judgments that have explicit meanings. For a positive parameter, useful prompts can ask for a median and selected percentiles, such as the 5th, 50th, and 95th percentiles. For a bounded fraction, ask for a specified central probability interval and, if appropriate, the probability of being below or above a stated threshold. Request the reasoning and evidence supporting each judgment, not just the number.
When experts are uncertain about a percentile, asking for a range of plausible values for that percentile can be more informative than demanding false precision. Clarify whether “plausible range” means an interval expected to contain a specified proportion of values, a range judged possible but not assigned a probability, or an absolute limit. These interpretations are not interchangeable.
Define the input, units, population, conditions, and whether the target is knowledge uncertainty or individual variability.
Present suitable measurements, indirect evidence, and established physiological or physicochemical constraints without implying more certainty than they support.
Use clearly labeled quantities such as a median, percentile, or central interval; define “plausible” before recording a numerical answer.
Ask the expert to explain the bounds and probabilities in their own words, then check ordering, units, constraints, and compatibility with any proposed distribution.
Record the elicitation date, expert role, questions, evidence considered, assumptions, judgments, rationale, and any unresolved disagreement or ambiguity.
From Elicited Percentiles to a Distribution
An elicited range does not uniquely determine a probability distribution. A distribution can be fitted only after making additional assumptions, such as its family and the meaning of the reported endpoints. For example, one might treat two values as the 5th and 95th percentiles and use a lognormal distribution for a positive parameter. That is a modeling choice to check and document, not an automatic consequence of hearing two numbers.
For a lognormal model, suppose \(X>0\) and \(\ln X\sim N(\mu,\sigma^2)\). The \(q\)th percentile of \(X\) is \(\exp(\mu+z_q\sigma)\), where \(z_q\) is the \(q\)th standard-normal quantile. If the elicited endpoints \(L\) and \(U\) are intended as the 5th and 95th percentiles, respectively, then \(z_{0.95}=1.644853626\) and \(z_{0.05}=-1.644853626\). Taking logarithms gives two equations:
Subtracting the first equation from the second cancels \(\mu\), giving the spread parameter. Adding them gives the location parameter. This also provides a direct check: the fitted median is the geometric midpoint of the two endpoints.
Worked Example: Fitting a Lognormal Range for a Positive Input
Suppose an expert judges that a hypothetical positive PBPK input has a 90% central interval from \(0.400\) to \(3.60\) in the input’s stated units. The expert confirms that this means the 5th and 95th percentiles, not absolute limits. Assume, for this calculation, that a lognormal distribution is an adequate representation of the elicited uncertainty.
Using \(L=0.400\), \(U=3.60\), and \(2(1.644853626)=3.289707252\):
The location parameter is
Thus the elicited judgments imply \(\ln X\sim N(0.1823,0.6679^2)\) under the stated lognormal assumption. The median is \(\exp(\mu)=\exp(0.1823)\approx 1.200\) input units. This is a fitted representation of the specified judgments, not evidence that the true input must follow a lognormal distribution.
Check Whether the Judgments Agree
Percentiles must be ordered: if \(q_1<q_2\), then the \(q_1\)th percentile cannot exceed the \(q_2\)th percentile. More subtly, a proposed distribution may not be able to match all elicited percentiles at once. If an expert supplies a median and two other quantiles, compare the observed spacing with the spacing implied by the proposed family. Do not silently discard a judgment or force a fit when the mismatch is substantial.
Worked Example: Testing Elicited Quantiles Against a Lognormal Model
For a hypothetical positive tissue input, an expert gives a 10th percentile of \(2.00\), a median of \(5.00\), and a 90th percentile of \(8.00\) units. Check whether these three judgments can be represented exactly by one lognormal distribution.
For a lognormal distribution, the median is \(\exp(\mu)\), so the median judgment gives \(\mu=\ln(5.00)=1.609438\). The 10th and 90th percentiles are equally far from \(\mu\) on the log scale, because \(z_{0.10}=-1.2815516\) and \(z_{0.90}=1.2815516\). The lower judgment implies
The upper judgment implies
These estimates of \(\sigma\) differ considerably. Therefore, no single lognormal distribution can match all three judgments exactly. The appropriate response is to revisit the questions: the expert may have given rounded or approximate values, may have interpreted one range differently, or may judge the distribution to be asymmetric on the log scale. Record the clarification. If the judgments remain as stated, consider a different representation rather than claiming an exact lognormal fit.
Consistency checks should also include units, positivity or other support restrictions, and plausibility in light of the target conditions. If a fraction is elicited above 1, for example, first determine whether the quantity was misunderstood or the stated units differ; do not simply truncate the value without recording that decision.
Bounds Need a Shared Interpretation
A hard bound is a limit outside which the parameter is impossible under the model definition or specified conditions. A probability interval describes where a stated proportion of uncertainty is believed to lie; values outside it remain possible. A range described only as “plausible” does not reveal which meaning the expert intended.
Worked Example: Resolving an Ambiguous Range for a Fraction
An expert gives \(0.15\) to \(0.55\) as a plausible range for a hypothetical tissue fraction, and the model definition requires \(0\leq X\leq 1\). Initially, it is unclear whether the range is a pair of hard limits or a probability interval. In a follow-up, the expert confirms that it is intended as a 90% central, equal-tail interval.
Under that interpretation, the elicited statements are \(P(X<0.15)=0.05\) and \(P(X>0.55)=0.05\), so \(P(0.15\leq X\leq 0.55)=0.90\). The support constraint remains \(0\leq X\leq 1\); the interval does not rule out values between 0 and 0.15 or between 0.55 and 1. In contrast, treating \(0.15\) and \(0.55\) as hard limits would assign zero probability outside that narrower range. Those are different uncertainty models, so the clarification changes the model input representation materially.
The equal-tail interval still does not determine the distribution’s shape within the interval or how probability is distributed in the tails. More judgments or an explicitly justified distributional assumption would be needed. The elicitation record should preserve both the expert’s intended interpretation and the modeler’s remaining assumptions.
Documentation, Alternatives, and Limitations
A useful record allows another modeler to reconstruct what was asked and why the answer was encoded as it was. Record the exact target definition and units; the expert’s relevant expertise; the evidence and constraints shown; the wording of each question; whether numbers are exact, rounded, or approximate; the intended meanings of ranges; the fitted distribution and fitting method, if any; and reasons for accepting or revising judgments. Note unresolved ambiguities rather than concealing them in a precise-looking parameter file.
Elicitation is one source of information, not a substitute for other approaches. Direct measurements that match the target conditions are generally more informative about that target than unsupported judgment, although measurements also have limitations. Indirect data may help constrain an input but require assumptions connecting the measured quantity to the model parameter. Physiological constraints can exclude impossible values but often do not identify a narrow distribution. A modeler may also use a default distribution, but its provenance and relevance to the target should be made explicit. These alternatives can inform an elicitation; none makes its assumptions disappear.
Elicited judgments are sensitive to question framing, expert knowledge, rounding, and anchoring on values shown during the discussion. A single expert may overlook evidence or misinterpret a percentile. Multiple experts can provide useful perspectives, but disagreement should not be hidden by reporting a pooled number without explaining the method. The aim here is to obtain a defensible, transparent representation of uncertainty, not to make subjective judgments look like a large measurement study.
An important limitation is that an elicited distribution describes the uncertainty represented by the elicited judgments and assumptions. It does not by itself establish how uncertain a PBPK prediction will be, nor does it determine how input uncertainty should be propagated through a model. Those require separate analyses. Likewise, a range for knowledge uncertainty should not be interpreted as a population distribution of person-to-person values unless the elicitation explicitly targets that variability.
Common Pitfall
Correction: Ask whether \(a\) and \(b\) are physical limits, approximate plausible values, or specified quantiles. Even after that meaning is clear, the range alone does not determine a distribution within it. Any distributional form must be supported by additional judgments or stated modeling assumptions.
A practical diagnostic is to ask the expert to restate the range in probability language: “What proportion of your uncertainty do you expect to fall inside these limits?” If the answer is unclear, the range is not yet suitable for distribution fitting. Also verify that every percentile lies within the parameter’s support and that a proposed distribution reproduces the elicited judgments to an acceptable degree.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- An expert gives a “likely range” for a positive input. What clarification is needed before fitting a distribution?
- For a lognormal input, an expert’s 5th and 95th percentiles are \(1.00\) and \(9.00\) units. What is the implied median if those judgments are fit exactly?
- A bounded fraction has an elicited 90% central equal-tail interval of \(0.20\) to \(0.60\). What probabilities are implied below and above the interval?
- Why is it a problem if a proposed distribution matches the elicited median and lower percentile but not the upper percentile?
- Name two items that should be retained in an elicitation record besides the final distribution parameters.
Solutions. (1) Clarify whether the endpoints are hard limits, approximate plausible values, or specified quantiles, and ask what probability statement is intended. (2) For an exact lognormal fit to the 5th and 95th percentiles, the median is their geometric mean: \(\sqrt{1.00\times9.00}=3.00\) units. (3) Equal-tail means 5% below \(0.20\) and 5% above \(0.60\), with 90% between them; the interval does not make those endpoints hard limits. (4) The fit would misrepresent one elicited judgment, so the family, interpretation, or judgments need review and the mismatch must not be hidden. (5) Examples include the exact target definition and units, elicitation questions, evidence shown, range interpretation, expert rationale, and unresolved ambiguity.