Model Discrepancy in Preclinical-to-Human Extrapolation

Why Include a Discrepancy Term?

A cross-species model can be carefully fitted, its parameters can be uncertain, and its calculations can be correct, yet its human prediction can still be systematically wrong. The model may omit a biological process, use an imperfect scaling relationship, or represent a species difference too simply. Model discrepancy is a term for this residual mismatch between the model's prediction and the biological quantity it is intended to predict. Representing it explicitly prevents a precise model calculation from being mistaken for a complete account of predictive uncertainty.

Discrepancy is not the same as uncertainty in a model parameter. Parameter uncertainty concerns which values of the model's parameters are compatible with the evidence; discrepancy concerns the gap between the model, even at specified parameter values, and biological reality. It is also not automatically between-subject variability or assay error. Those describe different sources of variation and should not be silently combined with discrepancy.

The scale matters. For a positive quantity such as exposure, an additive discrepancy on the natural scale can imply implausible negative values. A log-scale representation is often convenient: if \(B_h\) is a prediction for log exposure, then a discrepancy of \(d\) corresponds to multiplying exposure by \(\exp(d)\). A positive log-scale discrepancy therefore raises the predicted exposure, and a negative one lowers it. The examples below use log exposure and report final intervals in exposure units.

A discrepancy term must also have a defined target. It might describe the mismatch for one human estimand under specified conditions, such as typical exposure at a specified dose, rather than a universal correction for every drug, dose, endpoint, or population. If evidence suggests the mismatch changes with dose or another covariate, a single constant discrepancy may be inadequate; the discrepancy may need a structured dependence on those conditions.

Propagating Discrepancy Into a Human Prediction

Suppose the model-based log-scale prediction \(B_h\) has mean \(\mu_0\) and variance \(v_0\), incorporating the uncertainty sources already included in that prediction. For a simple calculation, assume that \(B_h\) is normally distributed and that the discrepancy is normally distributed with mean \(m_D\) and variance \(\tau_D^2\). If the two are independent, their sum is normal. The resulting mean and variance follow directly from the rules for expectations and variances.

This calculation makes the main consequence transparent: under independence, discrepancy variance adds to the existing predictive variance. A zero-mean discrepancy can widen an interval without shifting its centre on the log scale. A nonzero mean shifts that centre as well. If \(B_h\) and \(D_h\) are dependent, it is not correct simply to add their variances: the covariance term must be included. Independence should be supported by the way the components were constructed, not assumed merely because they have different names.

The normal distribution is a tractable assumption, not a requirement of the idea. If discrepancy is asymmetric, heavy-tailed, or related to conditions such as dose, simulation or another suitable probability calculation can propagate its specified distribution. The essential discipline is to state what the discrepancy represents, what evidence informs it, and which assumptions link it to the human prediction.

What Can Inform Discrepancy?

One possible source is a set of relevant historical predictions paired with suitable reference values. For each case, a log-scale residual can be calculated as the reference value minus the model prediction. The mean residual can inform a systematic shift, and the residual spread can inform discrepancy variability. These summaries are informative only to the extent that the historical cases resemble the intended human prediction in drug properties, biology, endpoint, and conditions.

For illustration, suppose four hypothetical paired cases have log-scale residuals \(0.10\), \(-0.10\), \(0.20\), and \(-0.20\). Their mean is zero. The sample variance is

so the sample standard deviation is \(s=\sqrt{0.03333}=0.1826\) log units. This is a descriptive estimate of residual spread in the hypothetical cases, not a precise or automatically transportable estimate of discrepancy for a new drug. In particular, treating an estimated spread as known omits uncertainty from estimating it; four cases provide little evidence about tail behaviour or how discrepancy varies across conditions.

Judgment or elicitation may be needed when direct calibration evidence is scarce. A useful elicitation specifies a scale and a prediction target: for example, plausible multiplicative departures from a model's human exposure prediction under stated conditions. The resulting distribution should be documented and explored over scientifically defensible alternatives. Assigning a large variance without explaining its basis may make an interval wide, but does not make the discrepancy model well supported.

Checks, Alternatives, and Limitations

Before using a discrepancy distribution, check that the components have clear and non-overlapping meanings. If the model-based predictive variance already includes the same residual mismatch represented by \(D_h\), adding \(\tau_D^2\) again double-counts uncertainty. Conversely, if a model's residual spread is treated only as assay noise when it includes systematic cross-species mismatch, omitting discrepancy can understate predictive uncertainty. The intended uncertainty components should be stated before they are combined.

Also check whether a single discrepancy distribution is plausible for the prediction target. Residuals that change with dose, species, endpoint, or compound class suggest a constant correction may not represent the mismatch well. A normal log-scale discrepancy should be reconsidered if relevant calibration residuals show pronounced asymmetry, extreme values, or distinct patterns. Sparse evidence cannot reliably rule out these problems, so lack of an obvious diagnostic signal is not proof that the assumed distribution is correct.

Discrepancy modelling complements, but does not replace, other ways to represent uncertainty. Parameter uncertainty describes uncertainty within a specified model; population variability describes differences among individuals; assay error describes measurement uncertainty. Structural uncertainty and Bayesian model averaging address uncertainty across candidate model structures. A discrepancy term can capture mismatch that remains even after considering candidate structures, but it does not identify the biological cause of that mismatch. Nor should it become an unexamined catch-all that obscures a fixable model defect.

A key limitation is transportability: residuals from earlier compounds or species may not describe the discrepancy for a new human prediction. Another is confounding: with sparse data, systematic bias, parameter values, and discrepancy can be difficult to distinguish. A broad discrepancy distribution may reflect several unresolved sources at once, while a narrow one may reflect limited evidence rather than genuine precision. Prediction intervals are therefore conditional on both the specified model and the evidence or judgment used to construct the discrepancy distribution.