Why Input Distributions Matter
A PBPK prediction is affected not only by the central values and spread assigned to uncertain inputs, but also by how probability is distributed across their range. Two input distributions can share the same mean and variance yet assign different probability to extreme values. When the PBPK model responds nonlinearly, those differences can produce meaningfully different prediction quantiles or probabilities of crossing a decision threshold.
The aim of a robustness comparison is not to select whichever distribution produces the most reassuring result. It is to determine whether conclusions persist across plausible input-distribution assumptions, and to make clear which predictions depend on assumptions about the tails. The assumptions being compared should be scientifically defensible and stated explicitly.
Let \(X\) denote an uncertain input, or a vector of uncertain inputs, and let \(Y=g(X)\) denote a PBPK output, such as concentration at a specified time, AUC, or Cmax. Under candidate input distribution \(j\), write the distribution of the output as \(F_{Y,j}\). Its \(p\)th quantile is \(q_{p,j}=\inf\{y:F_{Y,j}(y)\ge p\}\). A tail probability at a threshold \(c\) is \(P_j(Y>c)\). These are different summaries: a comparison of quantiles does not by itself establish equality of threshold probabilities, or vice versa.
Set Up a Fair Comparison
First define the output, the population or scenario being predicted, and the summaries that matter. For example, a comparison might focus on the 95th percentile of a concentration prediction and the probability that concentration exceeds a specified threshold at a particular time. Then specify the candidate input distributions and why each is plausible. Alternatives might represent different scientifically supported shapes, such as a bounded distribution versus a positive skewed distribution, rather than arbitrary mathematical choices.
When a comparison is intended to isolate distribution shape, match relevant summaries such as input means and variances. Depending on the parameter and scientific evidence, it may be more appropriate to match a median and geometric standard deviation, or a central quantile range. State which summaries are matched and why. Matching a mean and variance does not imply matching medians, quantiles, skewness, support, or tail probabilities. If the alternatives differ in these summaries, explain why that difference is intentional; otherwise a prediction difference cannot be attributed to distribution shape alone.
The usual propagation procedure is to draw inputs from each candidate distribution, evaluate the same PBPK model, and summarize the resulting predictions. Earlier material on Monte Carlo convergence and prediction quantiles is relevant here: simulation uncertainty in each estimated summary must be small enough to distinguish it from differences between assumptions. Using common random numbers can help compare alternatives when inputs can be coupled meaningfully, but it does not make the alternatives equivalent or remove the need to assess Monte Carlo error.
Quantiles and Tail Sensitivity
A useful justification for examining output quantiles directly follows when the model output is a continuous, strictly increasing function of a single input on an interval containing the support of that input. If \(Y=g(X)\), then for \(y\) in the range of \(g\), \(F_Y(y)=P(g(X)\le y)=P(X\le g^{-1}(y))=F_X(g^{-1}(y))\); below that range \(F_Y(y)=0\), and above it \(F_Y(y)=1\), with boundary values determined by limits. Under these conditions, \(q_p(Y)=g(q_p(X))\). Thus a change in an input's upper-tail quantile transfers directly through the response function. With multiple inputs or non-monotone responses, this simple identity need not hold, so the output distribution should be propagated through the model rather than inferred from one input quantile.
Worked Example: Comparing Two Candidate 99th Percentiles
In a hypothetical sensitivity calculation, two candidate input specifications produce a positive output \(Y\) whose 99th-percentile formulas are \(q_{0.99,A}=10\exp(0.1103\ln 99)\) and \(q_{0.99,B}=10\exp(0.2\times2.326)\), in mg·h/L. These formulas are assumed to follow from the respective input distributions and response mappings in this illustrative calculation. They are not asserted to be matched-summary alternatives.
Candidate A. Using \(\ln(99)=4.59512\), \(0.1103\ln(99)=0.506842\), so \(q_{0.99,A}=10\exp(0.506842)\approx16.6004\) mg·h/L.
Candidate B. The exponent is \(0.2\times2.326=0.4652\), so \(q_{0.99,B}=10\exp(0.4652)\approx15.9233\) mg·h/L.
The absolute difference is \(16.6004-15.9233=0.6771\) mg·h/L. The relative difference, using Candidate B as the reference, is approximately \(0.6771/15.9233=0.0425\), or 4.25%. Whether that difference matters depends on the context and the decision criterion. Because these specifications have not been shown to match relevant input summaries, this calculation alone does not isolate a tail-shape effect; it is a comparison of the stated candidate predictions, not evidence that the assumptions are otherwise equivalent.
The preceding example illustrates why a reported quantile difference needs context. A small difference in a high percentile can be important if a decision threshold lies between the predictions; a larger difference might be immaterial for a decision far from either prediction. Robustness is best judged against a prespecified, scientifically meaningful tolerance or decision criterion, not by looking only at whether two numerical summaries are exactly equal.
Matched Moments Do Not Guarantee Matched Tails
A simple example demonstrates that matching the mean and variance can leave upper-tail predictions quite different. Suppose a standardized uncertain input \(X\) has either of two plausible distributions. Under Candidate A, it takes values \(-1\) and \(1\), each with probability \(0.5\). Under Candidate B, it is uniform on \([-\sqrt{3},\sqrt{3}]\). Both have mean zero and variance one: Candidate A has \(E(X)=0\) and \(E(X^2)=1\); Candidate B has mean zero and variance \((2\sqrt{3})^2/12=1\).
Worked Example: Same Mean and Variance, Different Upper Quantiles
Suppose the hypothetical PBPK output is \(Y=10\exp(0.2X)\) mg/L, an increasing function of \(X\). Compare the 99th percentiles under the two input distributions just described.
Under Candidate A, the distribution function reaches 0.5 at \(-1\) and 1 at \(1\), so its 99th percentile is \(q_{0.99}(X)=1\). Therefore \(q_{0.99,A}(Y)=10\exp(0.2\times1)=10\exp(0.2)\approx12.2140\) mg/L.
Under Candidate B, the 99th percentile of a uniform distribution on \([-\sqrt{3},\sqrt{3}]\) is \(-\sqrt{3}+0.99(2\sqrt{3})=0.98\sqrt{3}\approx1.6974\). Therefore \(q_{0.99,B}(Y)=10\exp(0.2\times1.6974)\approx14.043\) mg/L.
The difference is about \(14.043-12.214=1.829\) mg/L, despite the matched input means and variances. For a threshold of \(13.5\) mg/L, \(Y>13.5\) is equivalent to \(X>\ln(1.35)/0.2\approx1.501\). Candidate A assigns probability zero to this event, since its largest value is 1. Candidate B assigns \((\sqrt{3}-1.501)/(2\sqrt{3})\approx0.0667\), or about 6.67%. The alternative tail assumptions therefore lead to different threshold conclusions in this hypothetical case.
This example uses a two-point distribution to make the arithmetic transparent; it is not a claim that a two-point distribution is appropriate for a particular PBPK input. The scientific question is whether each candidate support and shape is plausible for the parameter under study. A distribution that is mathematically convenient but inconsistent with biological constraints is not a useful robustness alternative.
Compare the Prediction Summaries That Matter
For each plausible input specification, compare the same output summaries in the same units. A central summary, such as the mean or median, can reveal shifts in typical predictions. Upper and lower quantiles reveal changes in the predictive spread and tails. If a decision is based on crossing a threshold, calculate the corresponding exceedance or non-exceedance probability as well. A conclusion about a threshold should not be inferred only from a central estimate.
The choice of matching strategy depends on the scientific parameter. For a positive parameter, matching the arithmetic mean and variance may not preserve a scientifically meaningful relative spread; matching a median and geometric standard deviation may be more suitable. For a bounded parameter, candidate distributions should respect its bounds. For correlated inputs, changing marginal distributions while silently changing their dependence structure can confound the comparison. State whether dependence is held fixed, changed intentionally, or not represented.
A compact comparison table helps make the basis of interpretation visible. The entries should distinguish input summaries from output summaries; matching input moments does not mean that output moments or quantiles are expected to match.
| Comparison element | What to report | Interpretive purpose |
|---|---|---|
| Input assumptions | Distribution, support, and matched or differing summaries | Shows what changed between candidates |
| Central predictions | Mean or median, with units | Shows whether typical predictions shift |
| Tail predictions | Relevant quantiles, with units | Shows whether extreme predictions depend on assumptions |
| Decision summaries | Threshold probability or decision classification | Shows whether the practical conclusion changes |
| Simulation precision | Monte Carlo uncertainty for estimated summaries | Helps distinguish simulation noise from assumption sensitivity |
Diagnostics, Pitfalls, and Alternatives
A distributional comparison can be misleading if its inputs are not checked. Inspect the range and tail behavior implied by each distribution, confirm that sampled values respect biological or physical constraints, and verify that the same sampling and PBPK evaluation procedures are used across candidates. For each output summary, check whether Monte Carlo uncertainty is small relative to the difference being interpreted. If the estimated difference is similar in size to its simulation uncertainty, more simulation or a more cautious conclusion may be needed; it is not evidence that the assumptions produce the same prediction.
Other warning signs include a candidate distribution that puts substantial probability on implausible inputs, different parameter dependence across alternatives without explanation, and conclusions based on only one summary when the decision concerns another. A large discrepancy in a high percentile deserves attention, but does not by itself identify which distribution is better supported. Scientific evidence, not the more favorable prediction, must guide that choice.
- Wrong reasoning: “The input means and variances match, so the 99th-percentile predictions must match.” Correction: Means and variances do not determine tail probabilities or quantiles. Compare the propagated summaries relevant to the decision.
- Wrong reasoning: “One alternative produces a higher upper percentile, so it is the conservative and correct distribution.” Correction: A more extreme prediction is not automatically more plausible. Assess support and tail assumptions using scientific evidence.
- Wrong reasoning: “The two estimated percentiles differ, so the distributional assumptions caused the full observed difference.” Correction: Check that other assumptions were held constant and that Monte Carlo uncertainty is small enough to interpret the difference.
An alternative to comparing several distributions is to use one selected distribution and report uncertainty under that specification. That can be appropriate when evidence supports a particular input model, but it does not show robustness to alternatives. Conversely, comparing plausible alternatives is informative only if the alternatives are justified and the comparison is designed so its results can be interpreted. Neither approach removes the need to state the assumptions.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Two candidate distributions for a PBPK input have the same mean and variance. Does that establish that their 95th percentiles are equal? Explain.
- A decision depends on whether an output exceeds \(c\). Which summary should be compared directly across input assumptions?
- For \(Y=10\exp(0.2X)\), an increasing function, Candidate A has \(q_{0.95}(X)=0.5\) and Candidate B has \(q_{0.95}(X)=1\). Calculate both output 95th percentiles and their difference.
- What should be checked if two simulated output quantiles differ by less than the Monte Carlo uncertainty in their estimates?
- Why can matching a positive input's arithmetic mean and variance be less informative than matching its median and geometric standard deviation?
Solutions. (1) No. Equal means and variances do not determine the distribution shape, support, or tail probabilities, so the 95th percentiles may differ. (2) Compare \(P(Y>c)\) directly under each candidate distribution, in addition to any other summaries needed. (3) Candidate A gives \(10\exp(0.2\times0.5)=10\exp(0.1)\approx11.05\); Candidate B gives \(10\exp(0.2)\approx12.21\). The difference is about \(12.21-11.05=1.16\) in the output's units. (4) The apparent difference may be simulation noise. Increase simulation precision as needed and avoid claiming a meaningful assumption effect until the estimates are sufficiently precise. (5) For a positive parameter, relative variability and multiplicative spread may be more scientifically interpretable than absolute deviations around an arithmetic mean; matching median and geometric standard deviation can better align those features, depending on the parameter and evidence.