Eliciting Priors From Biological Knowledge

From Biological Knowledge to a Prior

A QSP model may contain parameters that are difficult to estimate from the available clinical data. Turnover rates, binding affinities, and baseline concentrations may nevertheless be informed by laboratory studies, related populations, physiological constraints, or expert judgment. The calibration task is to represent that knowledge as a probability distribution over parameter values, rather than to insert a point estimate and treat it as certain.

A prior distribution is conditional on how the parameter and model are defined. A binding-affinity estimate, for example, is useful only if its assay conditions and biological context are sufficiently relevant to the modeled interaction. A reported half-life is not itself a turnover rate, and a baseline concentration from one compartment is not automatically informative about another. Elicitation therefore begins with the scientific meaning and units of the parameter, then makes the evidence and uncertainty explicit.

A prior is not a claim that every parameter value is equally plausible, nor is an expert’s best guess by itself a complete prior. A useful elicitation records what a range means: for example, whether it is a hard physical bound, a central 90% plausible range, or a rough range whose endpoints are uncertain. Those interpretations lead to different distributions.

Specify the Parameter and the Uncertainty Statement

Before choosing a distribution, write down the parameter’s definition, units, and biological context. Confirm whether it is positive, bounded, or allowed to be zero. Then identify the form of the available knowledge: a central estimate, quantiles, a plausible interval, a bound, or a qualitative statement. This step prevents an apparently precise prior from being built on an ambiguous range.

For a positive parameter with uncertainty that is naturally multiplicative, a lognormal distribution is often a useful choice. In the convention used here, \(\log(\theta)\) is normal with mean \(\mu_{\log}\) and variance \(\sigma_{\log}^2\). The median of \(\theta\) is \(\exp(\mu_{\log})\); its arithmetic mean is \(\exp(\mu_{\log}+\sigma_{\log}^2/2)\). These are different quantities, so an elicited “typical” or “central” value should be identified as a median or a mean.

Here \(Q_a(\theta)\) is the \(a\)th quantile and \(z_a\) is the \(a\)th quantile of a standard normal distribution. If experts provide a central \(100(1-\alpha)\%\) interval \([L,U]\), with endpoints intended as equal-tailed quantiles, then \(z=z_{1-\alpha/2}\) and

This calculation matches the two stated quantiles under the lognormal assumption. It does not establish that the distribution is biologically correct. In particular, the interval must be interpreted as a probability interval, not silently treated as a hard bound. For this symmetric-on-the-log-scale construction, the median is \(\sqrt{LU}\). If the supplied central estimate is not close to that value, the estimate and interval do not all fit this simple specification; revisit the elicited information or choose a more suitable model.

Use Literature Estimates Without Treating Them as Certainties

A literature value can inform a prior, but it should be assessed for comparability before its reported precision is used. Relevant questions include whether the study measured the same molecular interaction, used a comparable assay and temperature, and reported the same affinity definition and units. Differences in construct, species, tissue, or experimental conditions may make the published estimate only indirectly informative. A prior should also allow uncertainty beyond the standard error of one experiment when transfer to the modeled context is uncertain.

For positive affinity parameters, eliciting a central estimate and a plausible multiplicative range can be a practical approach. The prior’s spread should reflect both measurement uncertainty and the uncertainty involved in applying that evidence to the QSP model. The following example treats its chosen range as an elicited probability interval; it does not claim that every literature estimate should be converted in this way.

Represent Expert Judgments Transparently

Expert judgment can be valuable when direct evidence is sparse, but it should be elicited as concrete probability statements rather than as a request for a single “reasonable” number. Ask for a median or another specified quantile, and ask for a range the expert expects to contain a stated proportion of plausible values. Clarify the biological quantity, units, context, and time scale first. If experts disagree, preserve that disagreement for discussion rather than concealing it inside an unexplained compromise.

Assumptions, Checks, and Alternative Choices

Every elicited prior rests on assumptions. In the examples above, positive support is appropriate for the parameters; the lognormal family is judged adequate; and the stated ranges are interpreted as central probability intervals. The turnover example additionally assumes first-order loss and a valid conversion from half-life to rate. The affinity example assumes the available evidence is relevant enough to inform the modeled interaction. If these assumptions are doubtful, the numerical fit can be exact while the prior remains scientifically unsuitable.

A short audit should check the following before using the prior in calibration:

• Definition and units: Confirm that the prior is on the exact parameter in the model, with compatible units and time scales.
• Support: Check that the distribution does not assign values to impossible regions, such as negative concentrations or rates.
• Meaning of the range: Record whether endpoints are hard bounds, quantiles, or informal judgments. Do not turn a vague range into a 95% interval without agreement.
• Location and spread: Verify whether the reported central value is a mean, median, or another quantile, and whether it agrees with the selected distribution.
• Biological plausibility: Inspect representative values from the distribution, especially its tails, against known physiological and experimental constraints.
• Evidence provenance: Record study context, assay conditions, expert rationale, and any adjustment made for differences from the modeled setting.

The distribution family is a modeling choice, not a consequence of having positive parameters. A lognormal prior is useful when uncertainty is naturally multiplicative and a roughly symmetric range on the log scale is reasonable. A gamma distribution also supports positive values and can be parameterized to match a chosen mean and variance; it may be preferable when its shape better represents the elicited uncertainty. A uniform distribution over hard bounds is straightforward when all values inside the bounds are genuinely treated alike, but it is not a neutral default: it concentrates probability according to the chosen parameter scale and gives no extra weight to a biologically favored central value.

A diagnostic warning arises when several elicited summaries cannot all be matched by the chosen family. For example, if the stated median is not \(\sqrt{LU}\) for a proposed central interval \([L,U]\), then a lognormal prior with symmetric equal-tailed quantiles cannot match all three statements exactly. Do not quietly alter the expert’s statements or report a misleadingly precise fit. Revisit which information is most reliable, elicit additional quantiles, or select a distribution capable of representing the intended asymmetry.

Why Careful Elicitation Matters

In a mechanistic model, parameters can affect multiple states and time courses at once. A poorly justified prior may therefore steer calibration toward implausible biological regimes, or exclude values that the available evidence does not actually rule out. A well-documented prior makes the scientific judgment visible: another modeler can see what evidence was used, how uncertainty was quantified, and which assumptions would need reconsideration if the biological context changes.

The main pitfall is to confuse numerical precision with strength of evidence. A value reported to several decimal places may be the output of an assay or fitting procedure, not a claim that the parameter is known that precisely in the current model context. The correction is to separate the reported estimate from the uncertainty relevant to the QSP application, state the prior’s meaning explicitly, and check its implied values against biological knowledge.