Why Plausibility Audits Matter
A PBPK uncertainty analysis can be numerically tidy and still be misleading. A reported range may be mistaken for a standard deviation, a sampling distribution may generate impossible values, or a prediction interval may combine uncertainty about the model with differences among people. Even a good fit to an observed concentration profile may be achieved through parameter values that conflict with independent physiological or physicochemical evidence.
These errors do not all have the same remedy. A range requires interpretation before it can inform a distribution; a physical bound must be respected by the sampling method; variability and uncertainty must be represented as distinct sources; and a well-fitting parameter set must still pass a biological plausibility check. The aim of an audit is not to reject uncertainty analysis, but to make clear what the sampled quantities mean and whether the resulting parameter sets are defensible.
A Range Is Not Automatically a Standard Deviation
A reported range can refer to different things: the smallest and largest measurements in a sample, an expert-judged plausible interval, a central quantile interval, or limits imposed by physics. These meanings are not interchangeable. In particular, the endpoints alone do not generally identify a standard deviation. Converting a range into a distribution requires an explicit interpretation and assumptions about the distribution and how the endpoints were obtained.
If a report states that the interval from \(L\) to \(U\) is a central 95% interval for a normally distributed parameter, then the interval has a specific quantile interpretation. If it instead gives observed minimum and maximum values, the same calculation is not justified. For a positive parameter modeled as lognormal, a stated central 95% interval can be translated on the log scale, provided the lognormal assumption is appropriate. That is a conditional calculation, not a general rule for every reported range.
Worked Example: Interpreting a Stated 95% Range
Suppose a hypothetical source reports that a positive tissue partition coefficient has a central 95% interval of 4 to 16, and the analysis represents it as lognormal. Find the log-scale standard deviation. Contrast this with what could be concluded if 4 and 16 were merely observed extremes.
For a lognormal variable \(K\), \(\ln K\) is normally distributed. A central 95% interval places the endpoints at the 2.5th and 97.5th percentiles, corresponding to approximately \(-1.96\) and \(1.96\) standard deviations from the log-scale mean. Thus,
The geometric midpoint of the endpoints is \(\sqrt{4 \times 16}=8\), so this interval is centered at 8 on the log scale. Under the stated model, the corresponding log-scale mean is \(\ln(8)\). The result \(\sigma_{\ln K}\approx0.3536\) follows specifically from treating the reported endpoints as a central 95% interval for a lognormal quantity.
If 4 and 16 were only the smallest and largest observed values, they would not establish the 2.5th and 97.5th percentiles. The calculation would then have no stated quantile basis, and the standard deviation could not be recovered from those endpoints alone. Treating the half-range, \((16-4)/2=6\), as a standard deviation would be an unsupported assumption, as well as a calculation on the original scale rather than the log scale.
Sampling Must Respect Physical Bounds
Some inputs have hard bounds. Fractions and probabilities lie between 0 and 1; a clearance or volume cannot be negative. A sampling distribution that assigns positive probability to impossible values is not made appropriate merely because most of its draws are plausible. The distribution and any transformations or rejection rules should be checked against the meaning of the input.
For a bounded fraction, a beta distribution is one possible choice. If \(F\) has a beta distribution with shape parameters \(a\) and \(b\), its mean is \(a/(a+b)\), and its variance is \(ab/[(a+b)^2(a+b+1)]\). These expressions permit calibration to a specified mean and standard deviation when a beta distribution is justified. Other bounded distributions may be suitable; the important point is that the model must match the support and evidence.
Worked Example: Avoiding Impossible Fraction Samples
For a hypothetical fraction \(F\), suppose the intended mean is 0.30 and the standard deviation is 0.12. A proposed method draws \(F\) from a normal distribution with those moments. Assess the issue and calculate beta parameters with the same mean and variance as one bounded alternative.
The normal proposal assigns probability below zero. Its standardized lower bound is \(z=(0-0.30)/0.12=-2.50\), so \(P(F<0)=P(Z<-2.50)\approx0.00621\), or about 0.621%. Values above 1 are also mathematically possible, even though their probability under this particular normal model is very small. Thus the proposal has support inconsistent with a fraction.
For a beta distribution with mean \(m=0.30\) and variance \(v=0.12^2=0.0144\), let \(k=a+b\). The beta mean and variance imply \(v=m(1-m)/(k+1)\). Therefore,
A beta distribution with these parameters is supported on values between 0 and 1 and has the specified mean and variance. This calculation establishes a bounded distribution matching two moments; it does not establish that the beta shape is scientifically correct. The distribution should still be assessed against the evidence and the intended interpretation of the fraction.
Clipping and truncation are not neutral fixes. Clipping replaces every draw below zero with zero (and every draw above one with one), creating point masses at the boundaries and changing the moments. Rejection sampling discards out-of-range draws and samples from a conditional, truncated distribution; that also generally changes the mean and variance. If such a procedure is used, the resulting distribution—not just the original proposal—must be checked and described.
Keep Uncertainty and Variability Separate
Uncertainty about a population parameter and variability among individuals answer different questions. Uncertainty concerns what is not known about a shared model quantity; variability concerns differences that remain among individuals even if that shared quantity were known. Combining them can be appropriate for a particular predictive question, but reporting only the combined spread obscures its sources.
A simple additive model shows the distinction. Let an individual outcome be \(Y=2\theta+\varepsilon\), where \(\theta\) is an uncertain population parameter with mean 10 and variance 4, and \(\varepsilon\) represents individual variability with mean 0 and variance 9, independent of \(\theta\). Conditional on a fixed \(\theta\), individuals vary through \(\varepsilon\). Across uncertainty in \(\theta\), the expected population outcome also changes. The law of total variance separates these components:
The total standard deviation for an individual predictive outcome is \(\sqrt{25}=5\) outcome units. The uncertainty standard deviation for the conditional population mean \(E[Y\mid\theta]=2\theta\) is \(\sqrt{16}=4\) outcome units. These are different summaries for different targets. In a PBPK analysis, the analogous distinction is between uncertainty in shared inputs and variability in physiological or drug-related characteristics across individuals.
Worked Example: Reporting Two Sources of Spread
Using the model above, calculate the mean and distinguish the uncertainty in the population mean from the spread of an individual prediction.
Because \(E[\theta]=10\) and \(E[\varepsilon]=0\), the overall mean outcome is \(E[Y]=2(10)+0=20\) outcome units. For a fixed value of \(\theta\), the conditional mean is \(2\theta\), and the individual-level standard deviation is \(\sqrt{9}=3\) outcome units.
Across uncertainty in \(\theta\), the conditional mean has variance \(2^2(4)=16\), hence standard deviation 4 outcome units. For an individual predictive outcome that includes both sources, the variance is \(16+9=25\), hence standard deviation 5 outcome units. The value 5 is not the uncertainty in the population mean: it also includes individual variability. Conversely, reporting 4 as the full individual predictive spread would omit the variability term.
A Good Fit Does Not Guarantee Plausible Parameters
PBPK outputs depend on combinations of inputs. When different combinations produce nearly the same output, a fit to observed data may not identify each input separately. This is a structural reason to check parameter values against independent constraints rather than judging them solely by predictive fit. An implausible input can be compensated by another input, leaving a familiar output while undermining interpretation and extrapolation.
Worked Example: A Compensating Hepatic Parameter Pair
Consider the well-stirred liver expression \(CL_H=Q_H f_{u,B}CL_{int}/(Q_H+f_{u,B}CL_{int})\). For this hypothetical calculation, take hepatic blood flow \(Q_H=90\) L/h. Compare two parameter pairs that both give the same hepatic clearance.
For the first pair, let the unbound fraction in blood be \(f_{u,B}=0.01\) and intrinsic clearance be \(CL_{int}=9000\) L/h. Then \(f_{u,B}CL_{int}=90\) L/h, giving
For the second pair, let \(f_{u,B}=0.80\) and \(CL_{int}=112.5\) L/h. Their product is also 90 L/h, so the calculation again gives \(CL_H=90(90)/(90+90)=45\) L/h. The equation therefore cannot distinguish these two pairs from hepatic clearance alone.
Suppose independent evidence for the drug under consideration supports an unbound fraction near 0.01, not 0.80. The second pair should not be accepted simply because it reproduces the same clearance. Its fit demonstrates compensation in this equation, not evidential support for its component values. The appropriate conclusion is that this output alone does not identify both inputs; independent evidence or additional informative outcomes are needed to evaluate the pair.
Audit Strategy, Diagnostics, and Alternatives
A useful audit starts by asking what each uncertain quantity represents, what values are possible, and what evidence supports its distribution. Then examine sampled parameter sets jointly, not only one input at a time: dependence and compensation can make individually plausible marginal values implausible in combination. Finally, trace unusual outputs to the parameter values and assumptions that generated them. A prediction is not automatically credible because it lies within a numerical range, and a parameter is not automatically credible because the model fits one dataset.
Determine whether a reported range is an observed spread, a plausible bound, or a stated quantile interval. Do not assign a standard deviation until its interpretation and distributional assumptions are explicit.
Confirm that sampled values obey physical bounds. Check the realized mean, spread, tails, and any boundary mass after transformations, truncation, or rejection.
State whether each random quantity represents parameter uncertainty or individual variability, and report the output summary appropriate to the prediction target.
Compare parameter sets with independent evidence and constraints. Investigate whether a good output fit depends on compensating values or an unsupported correlation.
The main alternatives address different errors. Treating range endpoints as quantiles is defensible only when that is what the evidence says; otherwise, retain the range as a bound or use a distribution supported by the available evidence. For bounded inputs, use an appropriate bounded distribution or a justified transformation rather than an unconstrained normal draw followed by unexamined clipping. For predictive questions, report uncertainty and variability separately when possible, and combine them only with a clear target and model. For compensating inputs, use independent constraints and additional informative outcomes; a more flexible fit alone does not resolve the ambiguity.
Diagnostics include checking the frequency of invalid or boundary values, comparing intended and realized distribution summaries, and reviewing parameter sets associated with extreme or influential predictions. A strong change in results after imposing a scientifically supported constraint can reveal that earlier conclusions depended on implausible regions. But sensitivity to a constraint does not, by itself, prove that the constraint is correct; its evidence and scope must also be assessed.
Common Mistakes
- Using half the reported range as the standard deviation. This silently treats endpoints as a particular set of quantiles. Check how the range was defined before deriving a spread.
- Assuming rare impossible draws do not matter. Even a small invalid fraction signals a support mismatch and can affect estimates, especially after nonlinear model transformations.
- Calling a combined predictive spread “parameter uncertainty.” If it includes individual variability, label and decompose the components for the intended prediction target.
- Accepting a good fit as proof that inputs are credible. Multiple parameter combinations can yield the same output. Check each input and the joint set against independent evidence.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A source reports a minimum and maximum of 2 and 10 from a small sample. Can the standard deviation be calculated from those endpoints alone? Explain.
- A proportion is sampled from a normal distribution with mean 0.40 and standard deviation 0.10. What is the probability that a draw is below zero, approximately, and why is this a support concern?
- In \(Y=\theta+\varepsilon\), suppose \(\operatorname{Var}(\theta)=9\) and \(\operatorname{Var}(\varepsilon)=16\), independently. What is the variance of the individual predictive outcome, and what is the variance attributable to uncertainty in the conditional mean?
- Two parameter pairs give the same model output, but one conflicts with independent evidence. What can be concluded from the matching output, and what cannot?
Solutions. (1) No. A sample minimum and maximum do not identify the standard deviation; the sample size and the interior observations also affect it. A distributional assumption and an interpretation of the endpoints would be needed. (2) Standardize: \(z=(0-0.40)/0.10=-4\). Thus \(P(X<0)=P(Z<-4)\approx0.0000317\), about 0.00317%. The probability is small but nonzero, so the normal distribution assigns some probability to an impossible negative proportion. (3) The total predictive variance is \(9+16=25\), assuming the two sources are independent. The conditional mean is \(\theta\), so its variance from parameter uncertainty is 9. (4) The output alone does not distinguish the pairs. The matching value demonstrates non-identifiability from that output, not equal plausibility or evidential support; independent constraints should be used to assess the conflicting pair.