Why Diagnose Prior-Driven Predictions?
A PBPK prediction can be numerically precise and still depend heavily on assumptions about its inputs. This is especially important when parameters are poorly measured, estimated from sparse evidence, or assigned distributions partly through expert judgment. A prediction is prior-driven when changing a plausible assumed distribution for an uncertain input materially changes the output of interest. The term describes sensitivity to the input assumptions; it does not mean that the prediction is necessarily incorrect.
The practical diagnostic is comparative: hold the model structure, scenario, and other inputs fixed; replace one uncertain input distribution with another scientifically plausible choice; then trace and compare the resulting output distributions. This isolates the effect of that choice. It does not establish which distribution is best, nor does it turn an uncertainty analysis into a Bayesian update. Here, “prior” means the distribution assigned to an uncertain model input before considering any new measurement in the analysis.
A useful comparison must be fair. The distributions should represent defensible alternatives, not arbitrary extremes; units and parameter meanings must match; and all other inputs should be treated consistently. If distributions differ in both central location and spread, the comparison diagnoses their combined effect. Separate comparisons can then distinguish sensitivity to location from sensitivity to spread.
A Basic Comparison: Tissue Concentration at Equilibrium
Consider a deliberately simple equilibrium relationship in which tissue concentration \(C_t\) is the product of plasma concentration \(C_p\) and tissue-to-plasma partition coefficient \(K_p\):
Assume \(C_p\) is fixed at \(1.50\ \mathrm{mg/L}\), \(K_p\) is positive and lognormally distributed, and its median is 2.00. In this example, the lognormal geometric standard deviation (GSD) describes the spread: \(\ln(K_p)\) has standard deviation \(\ln(\mathrm{GSD})\). There is no concentration measurement being used to update \(K_p\); the output distribution is produced by propagating the assumed input distribution through this equation.
Worked Example: Propagating a Broad Partition-Coefficient Distribution
Suppose \(K_p\) has median 2.00 and GSD 1.80. Find its central 95% interval and the corresponding interval for \(C_t\). Treat the interval as an uncertainty interval from the specified distribution, not as a confidence interval for an estimated parameter.
For a lognormal distribution with median \(m\) and GSD \(g\), the central 95% interval is \(m\exp[-1.96\ln(g)]\) to \(m\exp[1.96\ln(g)]\). Here,
Therefore,
Multiplying both endpoints by the fixed \(C_p=1.50\ \mathrm{mg/L}\) gives
The median tissue concentration is \(1.50(2.00)=3.00\ \mathrm{mg/L}\). The wide output interval follows directly from the assumed spread of \(K_p\); the calculation alone does not show that measured tissue concentrations support that spread.
This example also shows why reporting a central prediction alone can conceal prior sensitivity. The median is \(3.00\ \mathrm{mg/L}\), but the modeled central 95% range spans roughly \(0.948\) to \(9.49\ \mathrm{mg/L}\). If the scientific question concerns whether concentration exceeds a threshold, the upper tail may be more informative than the median.
Compare Plausible Choices and Trace Their Effects
A distribution comparison is most informative when it states what changed. One comparison can hold the median fixed and vary the spread; another can hold spread fixed and vary the median. If the input is measurement-informed, the distribution should reflect the measurement and its uncertainty as represented in the analysis. If evidence is sparse, contrasting defensible distributions can reveal how much the prediction depends on choices that are not resolved by measurements.
Worked Example: How Prior Spread Changes a Threshold Probability
Use \(C_t=1.50K_p\ \mathrm{mg/L}\), with \(K_p\) lognormal and median 2.00. Compare GSDs of 1.80 and 1.30, and calculate the probability that \(C_t\) exceeds \(6.00\ \mathrm{mg/L}\). Assume the two GSD choices are plausible alternatives and that \(C_p\) is fixed.
The threshold is exceeded when \(1.50K_p>6.00\), or \(K_p>4.00\). For a lognormal \(K_p\) with median 2.00 and GSD \(g\), standardizing \(\ln(K_p)\) gives
For GSD 1.80, the standardized value is \(\ln(2)/\ln(1.80)\approx1.179\), so the exceedance probability is approximately \(1-\Phi(1.179)=0.119\), or 11.9%. For GSD 1.30, it is \(\ln(2)/\ln(1.30)\approx2.642\), giving \(1-\Phi(2.642)\approx0.0041\), or 0.41%.
Both choices have the same median \(K_p\), but the estimated probability of exceeding the concentration threshold differs by about 11.5 percentage points. Under the stated model and assumptions, that threshold probability is strongly sensitive to the assumed spread. It is not a probability that the threshold is biologically safe or unsafe; it is a model-based probability conditional on the chosen input distribution and fixed plasma concentration.
A striking difference in a threshold probability may matter even when the median prediction barely changes. Conversely, two output distributions may have different tails but similar central intervals. Select summaries that match the question: for example, a median for a typical prediction, an interval for spread, or an exceedance probability for a specified decision threshold.
Trace the Effect Through a Dynamic Tissue Model
The effect of an input distribution depends on how the PBPK model maps that input to the output. In a simple perfusion-limited tissue compartment with constant plasma concentration, tissue volume \(V\), blood flow \(Q\), and initial tissue concentration zero, the mass balance is
Assume \(Q\), \(V\), and \(C_p\) are fixed and positive. The model represents delivery from plasma and return from tissue according to the assumed partition coefficient. Solving this linear equation gives the time-dependent concentration:
The input affects both the equilibrium concentration and the rate at which the tissue approaches it. Consequently, an equilibrium calculation alone may not describe sensitivity at a finite time.
Worked Example: Comparing Prior Effects at a Fixed Time
Let \(Q=1.00\ \mathrm{L/h}\), \(V=1.00\ \mathrm{L}\), \(C_p=1.00\ \mathrm{mg/L}\), and \(t=1.00\ \mathrm{h}\). Compare \(K_p\) distributions with median 2.00 and GSDs 1.80 and 1.30. Assume \(K_p\) is the only uncertain input and that all fixed values are known for this hypothetical calculation.
At these fixed values, \(C_t(1)=K_p(1-\exp[-1/K_p])\ \mathrm{mg/L}\). This function is increasing for positive \(K_p\). To verify that, write \(g(K)=K(1-\exp[-a/K])\), where \(a=Qt/V>0\). Differentiating gives
The inequality follows because \(\exp(x)>1+x\) for \(x>0\), applied at \(x=a/K\). Thus, the 2.5th and 97.5th percentiles of \(K_p\) map to the corresponding output percentiles.
For GSD 1.80, the \(K_p\) interval is approximately \((0.63197,6.3294)\), as calculated above. Substitution gives \(C_t(1)\approx0.502\ \mathrm{mg/L}\) at the lower endpoint and \(C_t(1)\approx0.925\ \mathrm{mg/L}\) at the upper endpoint. For GSD 1.30, the interval is \(2.00\exp(\pm1.96\ln1.30)\approx(1.196,3.345)\); substitution gives an output interval of approximately \((0.678,0.864)\ \mathrm{mg/L}\).
The wider assumed input distribution therefore produces a wider output interval, although the finite-time output is less variable than the equilibrium expression \(K_pC_p\) because the tissue has not reached equilibrium. This is a concrete example of tracing prior sensitivity through the model rather than inferring output uncertainty from an input range alone.
Diagnostics, Alternatives, and Limitations
A practical audit begins by identifying the uncertain inputs that plausibly control the output. Then vary one input distribution at a time, using scientifically defensible alternatives, and compare relevant output summaries. If several inputs are changed simultaneously, the output difference cannot be attributed to any one of them. After one-at-a-time comparisons, combined scenarios may be useful for checking whether simultaneous plausible choices amplify or offset effects, but they do not isolate individual causes.
Name the quantity, time point, population, and summary or threshold being evaluated.
Keep units, parameter meaning, and other inputs consistent; document whether the alternatives differ in location, spread, shape, or dependence.
Use the same model and simulation settings where possible, then compare output distributions and the summaries relevant to the question.
State which input choice changed the output and by how much, without claiming that sensitivity identifies the correct distribution.
In a simulation, also check that the number of draws is adequate for the summaries reported, especially for tail probabilities. Repeat or increase the simulation size if Monte Carlo variation is comparable to the difference between scenarios. Check that sampled values satisfy biological and model constraints, that units and transformations are consistent, and that unusual outputs are not caused by numerical failures. These checks distinguish sensitivity to an assumed distribution from errors in implementation or sampling.
The main alternative to distribution comparison is to use one selected input distribution and report only the propagated uncertainty under that choice. This is simpler but can hide dependence on the choice itself. A local sensitivity calculation can show how an output changes near a nominal input value, but it does not generally describe the impact of changing an entire distribution, particularly when the model response is nonlinear. Comparing full propagated distributions captures more of that effect, at the cost of requiring defensible alternatives and adequate computation.
Prior sensitivity is not the same as measurement error, population variability, or uncertainty in model structure. A prediction may be sensitive because data are weak, because different sources imply different distributions, or because the model strongly amplifies a parameter. Conversely, little output change across tested alternatives does not prove that the parameter is well known: the alternatives may be too similar, or the chosen output may be insensitive to that parameter. The conclusion is always limited to the alternatives, model, and output examined.
A Common Pitfall
Wrong reasoning: “The prediction changes greatly under another plausible distribution, so the alternative distribution must be wrong.” Sensitivity alone cannot choose between distributions. It shows that the conclusion depends on a choice not settled by the comparison.
Correction: Report the alternatives and their output effects, then seek relevant measurement evidence or justify the selected distribution on scientific grounds. If such evidence is unavailable, present the sensitivity as a limitation of the prediction rather than hiding it behind a single preferred input distribution.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- If two \(K_p\) distributions have the same median but produce different 95% output intervals, what does that show, and what does it not show?
- In the equilibrium example, how does the threshold \(C_t>6.00\ \mathrm{mg/L}\) translate into a condition on \(K_p\) when \(C_p=1.50\ \mathrm{mg/L}\)?
- Why should the model and all other inputs be held fixed when first comparing alternative distributions for one parameter?
- At finite time in the perfusion-limited example, why is it valid to transform the input interval endpoints into output interval endpoints?
- Name one reason a large output difference between two simulation runs might not represent genuine prior sensitivity.
Solutions. (1) The output spread is sensitive to the assumed input spread; it does not identify which distribution is correct or show that measurements favor either one. (2) \(K_p>6.00/1.50=4.00\). (3) Holding other conditions fixed makes the output contrast attributable to the input distribution being changed. (4) The output function is increasing for positive \(K_p\), so it preserves the ordering of the input quantiles. (5) The runs may differ because of Monte Carlo variation, a coding or unit error, or simultaneous changes to other inputs rather than the distribution under study.