Propagating Uncertainty Under Parameter Dependence

Why Parameter Dependence Matters

A PBPK uncertainty analysis samples uncertain inputs and propagates each sampled parameter set through the model. The parameter distributions alone do not fully describe those sets: the joint distribution also specifies whether parameters tend to be high or low together. Sampling each marginal distribution independently therefore makes a substantive assumption. If the parameters are dependent, independent sampling can overstate or understate the spread of a predicted exposure.

This tutorial isolates that effect using a local affine approximation to a PBPK output. The approximation keeps the arithmetic transparent: for an affine output, the effect of dependence on variance is exact. A real PBPK model may respond nonlinearly, so this calculation is not a substitute for propagating joint draws through the full model.

We will work with two positive PBPK inputs: a tissue-to-plasma partition coefficient \(K_p\) and an intrinsic-clearance parameter \(CL_{\mathrm{int}}\). Define their centered log deviations from reference values \(K_{p,0}\) and \(CL_{\mathrm{int},0}\) by \(U=\ln(K_p/K_{p,0})\) and \(V=\ln(CL_{\mathrm{int}}/CL_{\mathrm{int},0})\). Log-scale sampling represents positive parameters without assigning probability to negative values. Suppose \(U\) and \(V\) have means zero, standard deviations \(\sigma_U\) and \(\sigma_V\), and correlation \(\rho\).

For illustration, take the local output approximation

Here \(Y\) is a hypothetical exposure endpoint, and the coefficients are local sensitivities on the log-parameter scale: a one-unit increase in either log deviation changes the approximated output by \(10\ \text{mg}\cdot\text{h}/\text{L}\). This affine relation is deliberately a simplified surrogate, not a universal PBPK equation.

How Dependence Enters Output Uncertainty

For any two random variables with finite variances and constants \(a\), \(b\), and \(c\), expanding the squared deviation of \(Y=c+aU+bV\) gives

The middle line follows by multiplying out the square: the two squared terms have expectations \(a^2\sigma_U^2\) and \(b^2\sigma_V^2\), while the cross-product has expectation \(2ab\operatorname{Cov}(U,V)\). Independence implies zero covariance, so the cross-term vanishes. Dependence changes the variance through this term; it does not change the marginal variances in this calculation.

When \(a\) and \(b\) have the same sign, positive correlation increases the variance relative to independent sampling, while negative correlation decreases it. If the sensitivities have opposite signs, the direction reverses. Thus, “positive correlation increases uncertainty” is not a general rule without considering how each parameter affects the output.

Generating Joint Parameter Draws

A simulation must preserve the intended dependence within each row of a parameter set. For two jointly normal log deviations, one convenient construction starts with independent standard normal random variables \(Z_1\) and \(Z_2\), each distributed as \(N(0,1)\). Set

This construction gives \(E[U]=E[V]=0\), \(\operatorname{Var}(U)=\sigma_U^2\), and \(\operatorname{Var}(V)=\sigma_V^2\). Its covariance is \(\sigma_U\sigma_V\rho\): the \(Z_1\) contribution to \(V\) is shared with \(U\), while the independent \(Z_2\) contribution adds variation without adding covariance. The square root is real for \(-1\leq\rho\leq1\).

Diagnostics, Alternatives, and Limits

Before interpreting a propagated distribution, check that the dependence model is supported and that the sampled pairs preserve it. Useful checks include:

• Inspect paired draws. A scatterplot of \(U\) against \(V\) can reveal the direction, strength, curvature, or clusters in the simulated relationship. A single correlation value will not describe every possible dependence pattern.
• Check the joint model. Confirm that the correlation or covariance comes from evidence or a defensible elicitation, rather than a convenient assumption. For multiple parameters, the proposed covariance matrix must be positive semidefinite; otherwise no joint distribution can have that matrix.
• Verify the sampling implementation. Compare simulated marginal means, standard deviations, and correlations with their intended values. Check that each row passed to the PBPK model is a joint draw, rather than a mixture of values drawn from different rows.
• Check plausibility and support. The log-scale construction ensures positive parameter values. It does not ensure that every combination is biologically plausible; an assumed joint distribution can still assign weight to implausible combinations.

Independent Monte Carlo sampling is appropriate when independence is justified, or when used explicitly as a comparison scenario. Joint sampling is preferable when evidence or scientific knowledge supports dependence. A covariance-based calculation such as the one derived here is an efficient check for an affine output, but it summarizes only second moments. Directly propagating joint samples through a PBPK model also reflects its actual output calculation, though its results still depend on the chosen joint input distribution. Marginal ranges alone cannot determine dependence: many different joint distributions share the same marginals.

The worked variance formula is exact for an affine output with finite input variances. The normal intervals and the specific sampling construction additionally assume a bivariate-normal distribution on the log scale. Correlation alone does not establish that distribution, and equal correlations can coexist with different joint shapes. For a nonlinear PBPK response, dependence can affect more than output variance; that requires a separate analysis of the response to uncertain inputs. Finally, epistemic uncertainty in parameters should not be confused with interindividual variability: a joint parameter distribution used to express uncertainty does not, by itself, represent a population of individuals.