Transferring PBPK Predictions Into Population PK Priors

Why a PBPK Prediction Needs a Prior Distribution

A physiologically based pharmacokinetic (PBPK) model can predict a clearance or volume under specified population and clinical conditions. A population PK analysis, however, estimates parameters from data and represents uncertainty about those parameters. Transferring a PBPK result therefore requires more than copying a point prediction: the target parameter, scale, and uncertainty represented by the PBPK output must be made explicit.

Here the target is a population typical value, such as the typical clearance \(CL_{\text{pop}}\) or typical volume \(V_{\text{pop}}\) under a stated condition. A prior distribution describes uncertainty about that target before the PK data being analyzed are used. It is not the same as the distribution of individual patient clearances or volumes. The alignment of population and clinical conditions must also be established before transfer; otherwise the prior may describe a different target.

A useful starting point for a positive parameter is a normal prior on its logarithm. For example, if the numerical value of clearance is expressed in L/h, write \(Z_{CL}=\log(CL_{\text{pop}})\) and specify \(Z_{CL}\sim N(\mu,\sigma^2)\). Then \(CL_{\text{pop}}\) is lognormally distributed and remains positive. All logarithms below apply to numerical values expressed in the stated units.

Turning PBPK Uncertainty Into a Prior

The transfer is most defensible when the PBPK output includes a probability distribution or an uncertainty interval for the relevant population typical parameter. If the PBPK model gives a central estimate and a central interval, the interval can be translated to a log-scale standard deviation when its endpoints are positive and are intended to represent symmetric uncertainty on the log scale. For a central 90% interval \([L,U]\), the normal quantile is \(z_{0.95}=1.64485\). If the interval is log-symmetric, then \(\mu=(\log L+\log U)/2\) and \(\sigma=(\log U-\log L)/(2\times1.64485)\).

This conversion depends on what the interval means. It must represent uncertainty about the PBPK-predicted population parameter, not a prediction interval for individual patients. It must also be clear whether the reported center is a median or arithmetic mean. A median directly determines \(\mu\); an arithmetic mean does not unless the spread is also known.

If an output instead gives an arithmetic mean and coefficient of variation (CV), those quantities must be converted rather than treated as the lognormal location and spread. For a lognormal variable, the mean is \(\exp(\mu+\sigma^2/2)\), while the variance is \(\exp(2\mu+\sigma^2)(\exp(\sigma^2)-1)\). Dividing the variance by the squared mean gives \(CV^2=\exp(\sigma^2)-1\). Thus \(\sigma^2=\log(1+CV^2)\), and \(\mu=\log(M)-\sigma^2/2\), where \(M\) is the arithmetic mean.

Prior Strength and a Simple Update

A transferred prior should not be more precise than its supporting information warrants. If PBPK parameter uncertainty, alternative model structures, or uncertain physiological inputs are omitted, the resulting interval may be too narrow. One transparent way to examine the consequence of prior strength is to compare a prior with a hypothetical likelihood on the same log scale. For independent normal prior and likelihood distributions, their precisions (inverse variances) add. This calculation illustrates how information is weighted; it does not establish that the chosen prior is scientifically justified.

The formula follows by multiplying the prior density and likelihood and collecting terms in \(Z\). The exponent, apart from a constant, is \(-\frac12[(Z-m_0)^2/v_0+(Y-Z)^2/v_y]\). Expanding the squares gives a coefficient of \(Z^2\) equal to \(1/v_0+1/v_y\), and the coefficient of \(Z\) is \(-2(m_0/v_0+Y/v_y)\). Completing the square gives the stated variance and mean.

Checks, Alternatives, and Failure Modes

Before using a transferred prior, check that its target matches the population PK parameter: the same analyte, parameter definition, population, dose context, and clinical conditions should be intended. Confirm units and whether the PBPK center is a median, arithmetic mean, or another summary. Check that the uncertainty interval is about the population typical parameter and that its stated coverage level and construction are understood. A prior predictive check—simulating parameter values from the prior and asking whether they are scientifically plausible—can reveal an implausible center or spread before fitting.

Clearance and volume may share PBPK inputs or physiological assumptions. If the model supplies paired uncertainty draws, inspect whether those draws indicate dependence rather than automatically assigning independent priors. Independent marginal priors can imply combinations of clearance and volume that the PBPK model regards as unlikely. The dependence should be represented only when it is supported and the PK estimation framework can use it.

Several failure modes are especially important. A patient-level prediction interval is not automatically a prior interval for a population typical value: it may mostly describe physiological variability. A point estimate without uncertainty does not justify a narrow prior. A rounded interval should not be converted into many apparently precise decimal places. If the lognormal approximation is poor, or the PBPK uncertainty distribution is notably asymmetric or multimodal, preserve a more suitable distribution rather than forcing a single normal distribution on the log scale.

A useful sensitivity analysis repeats the population PK estimation under defensible alternatives, such as a wider transferred prior or a weakly informative prior. Large changes in estimates or conclusions indicate that the analysis relies materially on the prior. That is not automatically an error, but it should be visible and discussed. A prior is an expression of uncertainty and information, not a guarantee that the PBPK model is correct.