Propagating Physiological Scaling Uncertainty

Why Physiological Scaling Uncertainty Matters

A physiology-based prediction of human hepatic clearance can combine measurements made in vitro with estimates of human liver size, hepatic blood flow, enzyme abundance, and drug binding. Each input is uncertain, and its effect on predicted clearance depends on how it enters the model. In particular, a percentage uncertainty in blood flow does not generally have the same effect as the same percentage uncertainty in enzyme abundance: the difference depends on hepatic extraction.

The aim here is to quantify that propagation for a specified hepatic clearance model. The result is conditional on the model and the uncertainty assigned to its inputs; it does not establish that the model structure itself is correct. The earlier material on decomposing uncertainty in cross-species predictions distinguishes sources of predictive uncertainty; here the focus is the effect of uncertain physiological scaling inputs within a clearance calculation.

Set Up the Clearance Model

Consider a well-stirred liver model. Let \(Q_h\) be hepatic blood flow in L/h, \(f_{u,p}\) the unbound fraction in plasma, and \(CL_{int,u,h}\) the unbound intrinsic hepatic clearance in L/h. The predicted hepatic clearance is

For the scaling calculation, suppose unbound intrinsic clearance is represented as

Here \(I\) is the measured apparent in vitro intrinsic clearance per milligram of microsomal protein, \(MPPGL\) is microsomal protein per gram of liver, \(W_L\) is liver mass, and \(R_E\) is a dimensionless enzyme-abundance scaling ratio relative to the system used to obtain \(I\). The in vitro unbound fraction \(f_{u,inc}\) corrects the apparent in vitro clearance for binding in the incubation. Units must be consistent: if \(I\) is L/h/mg and \(MPPGL\) is mg/g, then multiplying by \(W_L\) in g gives L/h; the abundance ratio and binding fractions are dimensionless.

Define \(Z=f_{u,p}CL_{int,u,h}\), the product of plasma unbound fraction and unbound intrinsic clearance. Then \(CL_h=Q_hZ/(Q_h+Z)\). The hepatic extraction ratio in this model is \(E_h=Z/(Q_h+Z)\), so \(CL_h=Q_hE_h\) and \(1-E_h=Q_h/(Q_h+Z)\). Both \(E_h\) and \(1-E_h\) lie between zero and one.

Local Sensitivities and Uncertainty

A useful first approximation is to examine how a small proportional change in each input changes clearance. The elasticity of clearance with respect to an input is its proportional change in clearance divided by the input’s proportional change. Differentiating the log of the well-stirred equation gives

Because \(Z\) is proportional to \(W_L\), \(R_E\), and \(f_{u,p}\), and inversely proportional to \(f_{u,inc}\), the corresponding elasticities are \(1-E_h\) for liver mass, enzyme abundance, and plasma unbound fraction, and \(-(1-E_h)\) for the in vitro unbound fraction. The negative sign means that a higher \(f_{u,inc}\) reduces the binding-corrected intrinsic clearance in this model.

For small uncertainties, a first-order approximation on the log scale propagates input variances through these sensitivities. If the inputs are treated as independent and their relative standard deviations are modest, the approximate variance of log clearance is

where \(CV_X=SD(X)/E(X)\) is the coefficient of variation of input \(X\). This expression uses the approximation that the standard deviation of \(\log(X)\) is close to \(CV_X\), which is most reasonable for modest relative uncertainty. It also assumes the stated inputs are independent. In applications where input distributions are materially skewed, bounded, or dependent, this approximation may not adequately represent predictive uncertainty.

Why Extraction Changes the Uncertainty Contributions

The sensitivity result follows directly by differentiating \(CL_h=Q_hZ/(Q_h+Z)\). For flow,

Similarly, differentiating with respect to \(Z\) gives

When extraction is low, \(Z\) is small relative to flow, so \(E_h\) is small and clearance is more sensitive to intrinsic-clearance inputs than to flow. When extraction is high, the clearance approaches flow: uncertainty in flow matters more, while proportional changes in intrinsic clearance have a smaller effect on total clearance.

Binding Fractions, Diagnostics, and Alternatives

Binding fractions deserve particular care because they are bounded and can enter in opposite directions. In the specified model, \(f_{u,p}\) increases \(Z\), whereas \(f_{u,inc}\) appears in the denominator of intrinsic clearance. A mistake in the direction of the in vitro correction can therefore reverse the predicted response. Also check the measured quantity: if an in vitro value is already unbound intrinsic clearance, dividing by \(f_{u,inc}\) again would double-correct it.

Useful diagnostics include checking units at every scaling step, confirming that each input is on the scale assumed by the formula, and recomputing predictions at plausible low and high input values. Compare the local approximation with such scenarios: large asymmetry or a major change in extraction ratio signals that a first-order approximation may be inadequate. Inputs near zero, very broad uncertainty, or fractions close to their bounds can also make a normal approximation on the raw scale implausible.

A local sensitivity calculation is efficient and shows which inputs matter near the nominal prediction. Deterministic scenario analysis is transparent about chosen combinations of inputs but does not assign probabilities to them. A full distribution-based calculation can represent nonlinear responses and bounded input distributions more directly, but requires defensible distributions and dependence assumptions. Independence should not be used merely for convenience: physiological quantities and assay-derived scaling inputs may share sources of information. If dependencies matter, use a joint representation rather than interpreting the independent-input result as complete.